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Control Strategies for the Next Generation Microgrids
by
Ali Mehrizi-Sani
A dissertation submitted in conformity with the requirementsfor the degree of Doctor of Philosophy
Graduate Department of Electrical and Computer EngineeringUniversity of Toronto
Copyright c© 2011 by Ali Mehrizi-Sani
Abstract
Control Strategies for the Next Generation Microgrids
Ali Mehrizi-Sani
Doctor of Philosophy
Graduate Department of Electrical and Computer Engineering
University of Toronto
2011
In the context of the envisioned electric power delivery system of the future, the smart
grid, this dissertation focuses on control and management strategies for integration
of distributed energy resources in the power system. This work conceptualizes a
hierarchical framework for the control of microgrids—the building blocks of the smart
grid—and develops the notion of potential functions for the secondary control for
devising intermediate set points to ensure feasibility of operation of the system. A
scalar potential function is defined for each controllable unit of the microgrid such
that its minimization corresponds to achieving the control goal. The set points are
dynamically updated using communication within the microgrid. This strategy is
generalized to (i) include both local and system-wide constraints and (ii) allow a
distributed implementation.
This dissertation also proposes and evaluates a simple yet elaborate distributed
strategy to mitigate the transients of controllable devices of the microgrid using local
measurements. This strategy is based on response monitoring and is augmented to
the existing controller of a power system device. This strategy can be implemented
based on either set point automatic adjustment (SPAA) or set point automatic ad-
justment with correction enabled (SPAACE) methods. SPAA takes advantage of an
approximate model of the system to calculate intermediate set points such that the
response to each one is acceptable. SPAACE treats the device as a generic system and
monitors its response and modulates its set point to achieve the desired trajectory.
SPAACE bases its decisions on the trend of variations of the response and accounts
for inaccuracies and unmodeled dynamics.
Case studies using the PSCAD/EMTDC software environment and MATLAB pro-
gramming environment are presented to demonstrate the application and effectiveness
of the proposed strategies in different scenarios.
ii
Acknowledgments
Arthur C. Clarke (1917–2008), British science fiction author and futurist, once fa-
mously said:
New ideas pass through three periods: (i) it can’t be done; (ii) it probably
can be done, but it’s not worth doing; and (iii) I knew it was a good idea
all along!
Many ideas crossed my mind during the course of this dissertation, some technically
sound and some not much so. I usually took care of period (i) myself. Whenever I were
stuck between periods (ii) and (iii), my doctoral advisor, Professor Reza Iravani, used
to come to rescue by giving me the encouragement I much needed from a trustable
authority. Every time I left his office, I was more optimistic and more motivated than
when I was going in. As a person, he is the best example of a mentor and role model
whose every action is fully thought through. As an advisor, his vision has no equal;
he is a testament of hard work and humbleness. As a writer, he has a firm dedication
to clear technical writing and chooses his words with exquisite care. His approach
has influenced me so much that I have developed a zeal for correctness and clarity.
His influence is visible in my attempt to make this dissertation more readable.
I wish to express my sincere appreciation for the time and support of my examining
committee members: Professors Alexandar Prodic, Zeb Tate, and Olivier Trescases
of the University of Toronto and Professor Claudio Canizares of the University of
Waterloo. Professor Prodic was always supporting and down-to-earth; Professor Tate
was always approachable and kind; Professor Trescases was always encouraging and
friendly. Professor Peter Lehn, although not directly involved in my committee, was
always welcoming. They made the Energy Systems Group at Toronto a great place
to work.
I was fortunate to be at Toronto at the same time as Maryam (now, Professor
Saeedifard), Mohamed (now, Dr. Kamh), and Amir (soon-to-be Dr. Etemadi). I
thank them for the discussion, encouragement, and friendship. I also thank Professor
Shaahin Filizadeh, my Master’s advisor, who never ceased his support for me, even
after I was out of his jurisdiction.
iii
I had memorable years at Toronto. Now1 I know when I stir my cup of tea,2 there
exists at least one molecule µ for which zi = zf , where zi and zf represent the position
of that molecule before and after stirring, respectively. Now3 I know what the words
“diurnal” and “pedagogy” mean. Now4 I know a heat sink is most of the time bulkier
than the rest of the circuit. Now5 I know “in theory, there is no difference between
theory and practice. But in practice there is.”6 Well, I also know why (most) manhole
covers are round7 (or in the shape of a Reuleaux polygon).
I received generous financial support from Connaught Scholarship, Rogers Gradu-
ate Scholarship, NSERC,8 NSERC/MITACS9 IPS,10 and OGS.11 Without this fund-
ing, my Ph.D. would not have been as smooth as it was. I acknowledge the support
from these sources.
I owe a lot to my parents; my mother poured on me her never-ending love, and
my father gave me his ever-lasting support. From my mother I learned patience, and
from my father I learned persistence.
I certainly can’t thank Shirin enough. Her patience (and gorgeousness) helped
me keep my sanity during the stressful times of this dissertation. Had it not been for
her, my doctorate program would have been longer, harder, and definitely lonelier.
During my times of self-doubt, Shirin was encouraging, caring, and attentive. Not
only was she willing to make many sacrifices for me, she was my “program manager.”
She even proofread this dissertation. In return for all these, “I love you” was the only
romantic thing I could give her most of the time. I look forward to embarking on the
next stage of our life together in Pullman. To her I dedicate this dissertation.
1That is, after taking Distributed Control of Autonomous Robots by Professor Bruce Francis.2I rarely drink coffee.3That is, after taking Engineering Teaching and Learning by Lisa Romkey and after teaching
the DEEP course Everyday Electrical Engineering myself.4That is, after taking Design of High Frequency SMPS by Professor Alexandar Prodic.5That is, after taking Applications of Static Power Converters by Professor Reza Iravani.6The quote is from Jan L. A. van de Snepscheut (1953–1994).7See, for example, http://blogs.msdn.com/b/bgroth/archive/2004/09/27/235071.aspx.8Natural Sciences and Engineering Research Council of Canada.9Mathematics of Information Technology and Complex Systems.
10Industrial Postgraduate Scholarship.11Ontario Graduate Scholarship.
LATEX macros written for my dissertation and VBA snippets developed for my PowerPoint slidesare available online at http://mehrizisani.com/latex. A copy of this dissertation prepared fordouble-sided printing is also available on the same website.
iv
to
Shirin
Table of Contents
List of Tables xii
List of Figures xvi
List of Study Systems xvii
List of Abbreviations xviii
1 Introduction 1
1.1 Power System and the Smart Grid Vision . . . . . . . . . . . . . . . . 1
1.2 Overview of the Control Hierarchy . . . . . . . . . . . . . . . . . . . 3
1.2.1 Grid-Connected Control . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Islanded Control . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2.1 Primary Control . . . . . . . . . . . . . . . . . . . . 4
1.2.2.2 Secondary Control . . . . . . . . . . . . . . . . . . . 5
1.2.2.3 Tertiary Control . . . . . . . . . . . . . . . . . . . . 5
1.3 Statement of the Problem and the Proposed Solution . . . . . . . . . 5
1.4 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Potential Functions for the Secondary Control 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Definition of a Potential Function . . . . . . . . . . . . . . . . . . . . 12
2.3 Components of a Potential Function . . . . . . . . . . . . . . . . . . . 13
2.3.1 Partial Potential for the DER Unit Measurements . . . . . . . 14
2.3.2 Partial Potential for the Constraints . . . . . . . . . . . . . . 14
2.3.3 Partial Potential for the Control Goal . . . . . . . . . . . . . . 14
2.4 Behavior of Potential Terms . . . . . . . . . . . . . . . . . . . . . . . 14
vi
2.4.1 Attractor Terms . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1.1 Point . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.1.2 Circle . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.1.3 Double Circle . . . . . . . . . . . . . . . . . . . . . . 17
2.4.1.4 Bowl . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.2 Repulsor Terms . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.2.1 Point . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.2.2 Circle . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.5 Parameters of the Potential Function Minimizer (PFM) . . . . . . . . 19
2.5.1 Low-Pass Filter Cutoff Frequency . . . . . . . . . . . . . . . . 19
2.5.2 Sampling Interval and Update Time . . . . . . . . . . . . . . . 20
2.5.3 Change Between Two Consecutive Set Points . . . . . . . . . 20
2.5.4 Weight Factors . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.6.1 Theoretical Background . . . . . . . . . . . . . . . . . . . . . 20
2.6.2 Comparison With Conventional Secondary Control . . . . . . 20
2.6.3 Single Versus Multiple Potential Functions . . . . . . . . . . . 21
2.6.4 Infrastructure Requirements . . . . . . . . . . . . . . . . . . . 22
2.6.5 Developed Software . . . . . . . . . . . . . . . . . . . . . . . . 23
2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Application of PFM for Voltage Control 24
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Application Example I . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.1 Study System I . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.3 Potential Function . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.4 Study Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Application Example II . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.1 Study System II . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.2 Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.3 Potential Function . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.4 Study Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3.4.1 Islanded Mode: Step Change in Voltage . . . . . . . 33
vii
3.3.4.2 Grid-Connected Mode: Step Change in Voltage . . . 33
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Generalized PFM Strategy 39
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1.1 Inclusion of Constraints . . . . . . . . . . . . . . . . . . . . . 40
4.1.1.1 GPFM and OPF . . . . . . . . . . . . . . . . . . . . 40
4.1.2 Distributed Implementation . . . . . . . . . . . . . . . . . . . 41
4.2 Decomposition Algorithms . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.1 Decoupled Constraints . . . . . . . . . . . . . . . . . . . . . . 44
4.2.2 Coupled Constraints . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 Formulation of GPFM . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.1 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.2 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3.2.1 Nonvectorized Derivation . . . . . . . . . . . . . . . 50
4.3.2.2 Vectorized Derivation . . . . . . . . . . . . . . . . . 52
4.4 Primal-Dual Interior Point Solver . . . . . . . . . . . . . . . . . . . . 53
4.5 Application Example I . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.5.1 Study System I . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.5.2 Potential Function as the Objective Function . . . . . . . . . . 58
4.5.3 Study Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.6 Application Example II . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6.1 Study System III . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6.2 Potential Function as the Objective Function . . . . . . . . . . 65
4.6.3 Modeling and Implementation Considerations . . . . . . . . . 65
4.6.4 Case Study A: Load Change . . . . . . . . . . . . . . . . . . . 67
4.6.5 Case Study B: Line Outage . . . . . . . . . . . . . . . . . . . 68
4.6.6 Case Study C: Line Outage and Controller Failure . . . . . . . 69
4.6.7 Case Study D: Line Outage, Controller Failure, and Missed
Updates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5 Online Set Point Adjustment for Trajectory Shaping 72
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.2 Concepts and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 75
viii
5.2.1 Predict-Prevent-Publish Paradigm (P4) . . . . . . . . . . . . . 75
5.2.2 Primary and Secondary Control . . . . . . . . . . . . . . . . . 75
5.2.3 Region of Acceptable Dynamic (ROAD) Operation . . . . . . 76
5.2.4 Communication Requirements . . . . . . . . . . . . . . . . . . 78
5.3 Set Point Automatic Adjustment (SPAA) . . . . . . . . . . . . . . . 78
5.3.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.3.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.4 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4.1 Study System IV . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.4.1.1 Start-Up Process . . . . . . . . . . . . . . . . . . . . 85
5.4.1.2 Step Change in Voltage Set Point . . . . . . . . . . . 85
5.4.2 Study System V . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4.2.1 Current Control During Start-Up . . . . . . . . . . . 87
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6 Online Set Point Adjustment With Correction 90
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.2 Set Point Automatic Adjustment With Correction Enabled (SPAACE) 91
6.2.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.2.2 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.3 Comparison of SPAA and SPAACE . . . . . . . . . . . . . . . . . . . 95
6.4 Alternative Methods to SPAACE . . . . . . . . . . . . . . . . . . . . 98
6.5 Effect of SPAACE on Stability . . . . . . . . . . . . . . . . . . . . . . 99
6.6 Existence of a Smooth Response . . . . . . . . . . . . . . . . . . . . . 101
6.7 Upper Bound of m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.8 Measurement and Prediction Enhancement . . . . . . . . . . . . . . . 108
6.9 Physical Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.10 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.10.1 Study System IV . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.10.1.1 Voltage Set Point Change Without Prediction . . . . 111
6.10.1.2 Voltage Set Point Change With Prediction . . . . . . 111
6.10.1.3 Voltage Control Subsequent to Load Energization . . 114
6.10.2 Study System V . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.10.2.1 Current Set Point Step Change Without Prediction . 115
ix
6.10.2.2 Current Set Point Step Change With Prediction . . . 115
6.10.2.3 Current Control Subsequent to Load Energization . . 115
6.10.2.4 Simultaneous Current Set Point Change . . . . . . . 118
6.10.2.5 Current Control During Start-Up . . . . . . . . . . . 118
6.10.3 Study System VI . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.10.3.1 Voltage Control Subsequent to Load Change . . . . . 121
6.11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7 Conclusions 124
7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.4 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Appendices 130
A Working Definition of the Microgrid 130
B Mathematical Treatment of the PF-Based Control 132
B.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . 132
B.1.1 Example 1: Beamer Pursuit . . . . . . . . . . . . . . . . . . . 134
B.1.2 Example 2: Cyclic Pursuit . . . . . . . . . . . . . . . . . . . . 135
B.2 Central Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
C Developed Software Tools 143
C.1 Design of Potential Functions . . . . . . . . . . . . . . . . . . . . . . 143
C.1.1 User Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
C.1.2 Code Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
C.1.3 Test Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
C.2 Design of SPAACE Parameters . . . . . . . . . . . . . . . . . . . . . 146
C.3 Dynamic Simulation of Power Systems . . . . . . . . . . . . . . . . . 148
C.3.1 Component Models . . . . . . . . . . . . . . . . . . . . . . . . 148
C.3.1.1 Synchronous Generators . . . . . . . . . . . . . . . . 149
Time derivatives . . . . . . . . . . . . . . . . . . . . . . 149
Initial values . . . . . . . . . . . . . . . . . . . . . . . . 150
Algebraic network interface equations . . . . . . . . . . 151
x
C.3.1.2 Excitation System . . . . . . . . . . . . . . . . . . . 152
Time derivatives . . . . . . . . . . . . . . . . . . . . . . 152
Initial values . . . . . . . . . . . . . . . . . . . . . . . . 153
C.3.1.3 Load Models . . . . . . . . . . . . . . . . . . . . . . 154
ZIP/Exponential recovery model . . . . . . . . . . . . . 154
Initial values . . . . . . . . . . . . . . . . . . . . . . . . 154
Algebraic network interface equations . . . . . . . . . . 154
C.3.1.4 AC Network . . . . . . . . . . . . . . . . . . . . . . . 155
C.3.2 Simulation Algorithm . . . . . . . . . . . . . . . . . . . . . . . 155
C.3.2.1 Overall Algorithm . . . . . . . . . . . . . . . . . . . 155
C.3.2.2 Numerical Integration . . . . . . . . . . . . . . . . . 156
C.3.2.3 Network Interface . . . . . . . . . . . . . . . . . . . . 157
C.3.2.4 Implementation in MATLAB . . . . . . . . . . . . . 158
D Derivations for the Theory of SPAACE 160
D.1 Justification of the Choice of tp as T2 . . . . . . . . . . . . . . . . . . 160
D.2 Calculation of the Step Response . . . . . . . . . . . . . . . . . . . . 161
D.3 Calculation of the Peak Response . . . . . . . . . . . . . . . . . . . . 162
References 177
xi
List of Tables
2.1 Attractor terms in a potential function . . . . . . . . . . . . . . . . . 16
3.1 Study system I: Parameters . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Parameters of PFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Study system II: Parameters . . . . . . . . . . . . . . . . . . . . . . . 32
4.1 Study system III: Predisturbance steady-state operating conditions . 64
5.1 Study systems IV, V, and VI: Interface parameters . . . . . . . . . . 83
5.2 Study system IV: Parameters . . . . . . . . . . . . . . . . . . . . . . 84
xii
List of Figures
1.1 Hierarchical control levels: primary control, secondary control, and
tertiary control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Dissertation outline. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1 Plot of the potential term as a function of distance R; parameter r is
defined in Section 2.4 and in (a), r = 0; in (b)–(c), r = 1. (a) point;
(b) circle; (c) double circle; (d) bowl. . . . . . . . . . . . . . . . . . . 15
2.2 Different potential function targets. (a) point; (b) circle; (c) double
circle and bowl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Double circle and bowl target areas. . . . . . . . . . . . . . . . . . . . 17
2.4 Schematic diagram of the PFM-based control. . . . . . . . . . . . . . 19
3.1 Nested loop voltage control of a VSC. . . . . . . . . . . . . . . . . . . 25
3.2 Study system I: Three-DG cascade microgrid. . . . . . . . . . . . . . 26
3.3 System I: Step change in voltage from 1 pu to 0.7 pu. . . . . . . . . . 30
3.4 System I: Locus of the PC2 voltage. . . . . . . . . . . . . . . . . . . . 31
3.5 Study system II: Four-DG radial microgrid. . . . . . . . . . . . . . . 31
3.6 System II in the islanded mode: Step change in voltage from 1 pu to
0.7 pu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.7 System II in the islanded mode: Locus of the PC1 voltage. . . . . . . 35
3.8 System II in the grid-connected mode: Step change in voltage from
1 pu to 0.95 pu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.9 System II in the grid-connected mode: Locus of the PC1 voltage. . . 38
4.1 An optimization problem with coupled constraints. . . . . . . . . . . 46
4.2 Study system I decomposed into two areas. . . . . . . . . . . . . . . . 57
4.3 System I: PC1 voltage in response to a step change from 0.90 pu to
1.05 pu under GPFM control. . . . . . . . . . . . . . . . . . . . . . . 61
xiii
4.4 System I: Locus of voltage of PC1. . . . . . . . . . . . . . . . . . . . 62
4.5 System I: Locus of voltage of PC1, PC2, and PC3. . . . . . . . . . . . 62
4.6 Study system III: Twelve-bus system with four generators. . . . . . . 63
4.7 System III: Voltage at generator buses subsequent to a 10% load re-
duction. Traces of V10 and V12 are similar and overlap. (a) without
GPFM; (b) with GPFM. . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.8 System III: Voltage at generator buses subsequent to the outage of line
4-5. (a) without GPFM; (b) with GPFM. . . . . . . . . . . . . . . . 69
4.9 System III: Voltage at generator buses subsequent to the outage of line
4-5 and a change in voltage reference. Traces of V10 and V12 are similar
and overlap. (a) without GPFM restoring the set point; (b) with GPFM. 70
4.10 System III: Voltage at generator buses subsequent to the outage of line
4-5 and a change in voltage set point while updates for generator 4 are
not implemented. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1 Primary and secondary controllers and SPAA and SPAACE. . . . . . 76
5.2 ITI curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3 Region of acceptable dynamic (ROAD) operation curve. (a) two-
dimensional generic ROAD curve, where x1 and x2 are dynamic vari-
ables; (b) one-dimensional ROAD curve pertaining to the ITI curve. . 77
5.4 Variables of SPAA and a representative case. . . . . . . . . . . . . . . 79
5.5 Flowchart representing the SPAA algorithm. . . . . . . . . . . . . . . 82
5.6 An example set of intermediate set points generated by SPAA. . . . 82
5.7 Study system IV: One feeder of the CIGRE medium voltage benchmark
system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.8 System IV: Performance improvement due to SPAA in a start-up sce-
nario. (a) SPAA is not active; (b) SPAA is active and the actual DER
unit is used; (c) SPAA is active and the DER unit is replaced by its
approximate model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.9 System IV: Step change from 0.90 pu to 1.09 pu. (a) without SPAA;
(b) with SPAA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.10 Study system V: IEEE 34-bus test feeder with three augmented DER
units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
xiv
5.11 System V: Start-up response of DER2. (a) without SPAA; (b) with
SPAA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.1 Demonstration of the performance of SPAACE. . . . . . . . . . . . . 93
6.2 Prediction algorithm of SPAACE. . . . . . . . . . . . . . . . . . . . . 94
6.3 Finite state machine representation of SPAACE. . . . . . . . . . . . . 96
6.4 SPAACE application example without prediction. . . . . . . . . . . . 97
6.5 SPAACE application example with prediction. . . . . . . . . . . . . . 97
6.6 SISO representation of a controllable device. . . . . . . . . . . . . . . 99
6.7 Stability of SPAACE. Intermediate set points for n = 2. . . . . . . . . 99
6.8 Definition of T1, T2, and tp. . . . . . . . . . . . . . . . . . . . . . . . 101
6.9 Dependence of the performance of SPAACE on the value of T2 as it
approaches tp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.10 Demonstration of fitness of the choice of T1 and T2 as outlined in the
proposed algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.11 An upper bound for m as a function of damping factor ζ. . . . . . . . 108
6.12 A physical analogy for SPAACE. . . . . . . . . . . . . . . . . . . . . 110
6.13 System IV: Step change in voltage from 1.10 pu to 0.91 pu. (a) without
SPAACE; (b) with SPAACE without prediction; (c) with SPAACE
with prediction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.14 System IV: Step change in voltage from 0.90 pu to 1.09 pu. (a) without
SPAACE; (b) with SPAACE with prediction. . . . . . . . . . . . . . 113
6.15 System IV: Load change from 1 pu to 2 pu at t = 0 s. (a) without
SPAACE; (b) with SPAACE. . . . . . . . . . . . . . . . . . . . . . . 114
6.16 System V (DER2): Step change in current from 0.92 pu to 1.08 pu.
(a) without SPAACE; (b) with SPAACE without prediction; (c) with
SPAACE with prediction. . . . . . . . . . . . . . . . . . . . . . . . . 116
6.17 System V (DER2): Step change in current from 1.08 pu to 0.92 pu.
(a) without SPAACE; (b) with SPAACE without prediction; (c) with
SPAACE with prediction. . . . . . . . . . . . . . . . . . . . . . . . . 117
6.18 System V (DER2): Load energization. . . . . . . . . . . . . . . . . . 118
6.19 System V: Simultaneous step change in current from 0.92 pu to 1.08 pu
in each DER unit. (a) DER1; (b) DER2; (c) DER3. . . . . . . . . . . 119
xv
6.20 System V: Start-up response of DER2. (a) without SPAACE; (b) with
SPAACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.21 Study system VI: IEEE 13-bus unbalanced test feeder with the aug-
mented DER unit and load. . . . . . . . . . . . . . . . . . . . . . . . 121
6.22 System VI: Voltage transient in response to load change in an unbal-
anced system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
A.1 Schematic diagram of a generic multiple-DER microgrid. . . . . . . . 131
B.1 Representation of the disks in the Gersgorin’s Theorem. . . . . . . . . 134
B.2 Example 1: Beamer pursuit. . . . . . . . . . . . . . . . . . . . . . . . 134
B.3 Example 2: Cyclic pursuit. . . . . . . . . . . . . . . . . . . . . . . . . 136
B.4 Gersgorin disks for the central theorem. . . . . . . . . . . . . . . . . . 138
B.5 A sample four-node, four-edge visibility graph. . . . . . . . . . . . . . 139
C.1 The graphical user interface of the developed software for simulation
of autonomous units. . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
C.2 Test scenarios for the developed PFM software. . . . . . . . . . . . . 147
C.3 Developed software for experimenting with parameters of SPAACE. . 148
C.4 Synchronous generator circuit interface. . . . . . . . . . . . . . . . . . 152
C.5 Excitation system model. . . . . . . . . . . . . . . . . . . . . . . . . . 152
C.6 Flowchart of the algorithm for dynamic simulation of power systems. 156
xvi
List of Study Systems
I Three-DG cascade microgrid. . . . . . . . . . . . . . . . . . . . . . . 26
II Four-DG radial microgrid. . . . . . . . . . . . . . . . . . . . . . . . . 31
III Twelve-bus system with four generators. . . . . . . . . . . . . . . . . 63
IV One feeder of the CIGRE medium voltage benchmark system. . . . . 84
V IEEE 34-bus test feeder with three augmented DER units. . . . . . . 88
VI IEEE 13-bus unbalanced test feeder with the augmented DER unit and
load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
xvii
List of Abbreviations
ACE Area Control Error
ADM Alternating Direction Method
ADN Active Distribution Network
AMI Advanced Metering Infrastructure
APP Auxiliary Problem Principle
AVR Automatic Voltage Regulator
BPL Broadband over Power Line
DAE Differential Algebraic Equation
DER Distributed Energy Resource
DG Distributed Generation
DS Distributed Storage
DSO Distribution System Operator
EDF Electricite de France
EMS Energy Management System
FACTS Flexible AC Transmission System
FSM Finite State Machine
GPFM Generalized PFM
GUI Graphical User Interface
HVDC High Voltage Direct Current
ICT Information and Communication Technology
ITI the Information Technology Industry council
KKT Karush-Kuhn-Tucker
MPC Model Predictive Control
OPF Optimal Power Flow
P4 Predict-Prevent-Publish Paradigm
PC Point of Connection
PCC Point of Common Coupling
PCPM Predictor-Corrector Proximal Multiplier
xviii
PFM Potential Function Minimizer
PID Proportional-Integral-Derivative
PPA Proximal Point Algorithm
ROAD Region Of Acceptable Dynamic performance
SCADA Supervisory Control And Data Acquisition
SCR Short Circuit Ratio
SISO Single-Input Single-Output
SNR Signal-to-Noise Ratio
SPA Sequential set Point Assignment
SPAA Set Point Automatic Adjustment
SPAACE Set Point Automatic Adjustment with Correction Enabled
PLL Phase-Locked Loop
PSS Power System Stabilizer
PWM Pulse Width Modulation
TEF Transient Energy Function
UPFC Unified Power Flow Controller
VCO Voltage-Controlled Oscillator
VPP Virtual Power Plant
VSC Voltage-Sourced Converter
ZIP Z (constant impedance), I (constant current), P (constant power)
xix
Chapter 1
Introduction
1.1 Power System and the Smart Grid Vision
The electric power system is undergoing a major paradigm shift to simultaneously
address the requirements of the environment, market, utilities, and consumers. The
envisioned power system is expected to provide a high service standard based on
an array of advanced technologies and new control and operational concepts, e.g.,
distributed energy resource (DER) units, sensors, automated meters, information and
communication technologies (ICT), and control and power management strategies. A
power system with these functionalities falls under the general umbrella of the smart
grid [1]–[5].
The smart grid initiative is proposed to overcome the issues with the existing
power system. These issues include
• Questionable reliability, which results in costly power outages;
• Aging infrastructure, since replacing power system components necessitates ex-
tensive and expensive investment;
• Limited awareness of the events occurring in the system, because of the limited
use of communication;
• Low efficiency of the transmission and distribution systems, which results in
about 10% power loss; and
• Low utilization of power system assets [6], because the system is designed for
the peak demand conditions.
1
Chapter 1. Introduction 2
Smart grid is the vision of the future electric system. However, as a fairly new concept,
it does not have a universally accepted definition. The core elements of the smart
grid are the use of (i) information technology, (ii) communication, and (iii) power
electronic devices. The key technologies required to achieve the vision of the smart
grid include [7]
1. Advanced components, e.g., power electronics and storage systems [1, Appendix
B3];
2. Advanced control technologies, e.g., distributed intelligent agents, analytical
tools such as software algorithms and high speed computers, and operational
applications such as SCADA, substation automation, and demand response;
3. Integrated communications, e.g., WiMAX and broadband over power line (BPL);
4. Sensing and measurement, e.g., advanced metering infrastructure (AMI); and
5. Improved interfaces and decision support systems, e.g., 3D visualization sys-
tems.
This work falls within the smart grid vision because it is related to technologies 1,
2, and 3 mentioned above: (i) it operates based on advanced components such as
DER units, (ii) it proposes an advanced control strategy to address the inadequacy
of existing control strategies, and (iii) it employs integrated communications.
Microgrid [8]–[11] and its various evolved forms, e.g., active distribution system
(ADS), cognitive microgrid, and virtual power plant (VPP) [12]–[16], can be con-
sidered and exploited as the main building block of the smart grid. A working
definition of microgrid used in this dissertation is provided in Appendix A. A mi-
crogrid is an electrical entity that facilitates high depth of penetration of DER units
and relies on ICTs and advanced control/protection strategies. The investigated and
proposed microgrid operational scenarios are mainly based on tracking set points of
DER units [17]–[19]. The first step to enable the microgrid (i) to surpass its current
operational constraints, e.g., to acquire capabilities as a VPP and (ii) to serve as a
building block for the smart grid, is to devise control, protection, and power manage-
ment strategies. These strategies are based on the use of communication and status
monitoring of the microgrid and external information such as market signals. To the
best of the author’s knowledge, this work is the first attempt in this endeavor; it
Chapter 1. Introduction 3
proposes (i) a microgrid secondary control strategy based on the concept of potential
functions and (ii) an auxiliary control strategy to improve the performance of existing
controllers.
1.2 Overview of the Control Hierarchy
The interconnected power system is spread over a large geographical span. This
intricate system can be controlled through either centralized control or decentralized
control. A fully centralized control relies on the data gathered in a dedicated central
controller and requires extensive communication between the controller and other
units. In a fully decentralized control, each unit is controlled by its local controller
that is not fully aware of the system-wide disturbances and is independent of other
controllers [20].
A compromise between fully centralized and fully decentralized control schemes
is the hierarchial control scheme. In the context of power systems, the hierarchial
control scheme includes three control levels: primary, secondary, and tertiary.1 These
control levels differ in their (i) speed of response and the time frame in which they
operate and (ii) infrastructure requirements, e.g., need for communication.
This control hierarchy can also be implemented for the microgrid control. The
principles of operation and control of a microgrid can be best described in two distinct
grid-connected and islanded modes of operation and are described in the rest of this
section.
1.2.1 Grid-Connected Control
In the grid-connected mode, the voltage of the point of common coupling (PCC) of
the microgrid is dominantly determined by the host grid, and the main role of the
microgrid is to accommodate (i) the real or reactive power generated by the DER
units and (ii) the load demand. Reactive power injection by a DER unit can be used
for (i) power factor correction, (ii) reactive power supply, or (iii) voltage control at
the corresponding point of connection (PC).
1This terminology started in the 1980s in Europe by Electricite de France (EDF), which designedand implemented the regional secondary controllers [21].
Chapter 1. Introduction 4
Tertiary Controller
Secondary Controller
Secondary Controller
Secondary Controller
SG1 DG1
DG2
Primary Controllers
Fig. 1.1. Hierarchical control levels: primary control, secondary control, and tertiarycontrol.
The DER units with limited power generation capacity cannot practically assist
a strong utility network in its voltage and/or frequency regulation. In the grid-
connected mode, the host utility may not permit regulation or control of the PCC
voltage by the DER units to avoid interaction with the same functionality performed
by the grid. Therefore, the DER units in the proximity of the PCC (determined by
the electrical distance and SCMVA of the grid) should not actively implement a voltage
control scheme [11].
1.2.2 Islanded Control
In the islanded mode, the microgrid operates as an independent entity and must
provide voltage and frequency control as well as real and reactive power balance.
For example, if the load demand is less than the total generation, the microgrid
central controller should decrease the net generated power. This is accomplished by
assigning new set points to the DER units. On the other hand, if the power generated
within the microgrid cannot meet the load demand, either noncritical load shedding
or activation of storage units must be considered. The hierarchy of control, Fig. 1.1,
is applied to an islanded microgrid as follows:
1.2.2.1 Primary Control
Primary control is the first control level in the control hierarchy and features the
fastest response. Primary control responds to system dynamics and ensures that the
system variables, e.g., voltage and frequency, track their set points [17]–[19], [22], [23].
Chapter 1. Introduction 5
Primary control mostly employs conventional linear control methods and is performed
locally, based on locally measured signals. Because of their speed implications, is-
landing detection and the subsequent change of controller modes lie in this control
level [22]–[24].
1.2.2.2 Secondary Control
Secondary control is the next level of control and is responsible for ensuring power
quality and mitigating longer term voltage and frequency deviations by determining
the set points for the primary control. While this is a common task between a
secondary controller and an energy management strategy [25], the latter lacks (i) the
use of communication between the microgrid components and (ii) the use of possible
distributed storage (DS) units such as spinning reserves.
Secondary control operates on a slower time frame than that of the primary con-
trol, e.g., it has a settling time in the order of a minute in a conventional grid, so that
the initial transients of the microgrid are mostly handled by the primary controller,
and the primary control loop reaches its steady state before the secondary controller
updates the set point. This assists to (i) decouple secondary control from primary
control [26] and (ii) reduce the communication bandwidth, as the secondary control
uses sampled measurements of the microgrid variables.
1.2.2.3 Tertiary Control
Tertiary control is the highest level of control and sets the long term set points de-
pending on the requirements of an optimal power flow, e.g., based on the information
received about the status of the DER units, market signals, and other system require-
ments.
1.3 Statement of the Problem and the Proposed
Solution
The technical literature of the last decade contains an extensive body of research on
the local control of DER units [18], [19], microgrid islanding process [17], [23], [24],
[27], and stable transition from the grid-connected mode to the islanded mode [22],
Chapter 1. Introduction 6
[23]. These control facilities are categorized under primary control and ensure that
the microgrid DER units track their set points, e.g., voltage and real/reactive power.
Following a disturbance, e.g., a set point change, the controllers in a microgrid
should perform two essential tasks:
• Ensure that the trajectory of the microgrid in its transition between the initial
and final set points does not violate the operational constraints. Therefore, the
change in the set points may not be necessarily applied as a step change.
• Ensure that each DER unit of the microgrid tracks its set point as closely as
possible. That is, the deviation between the set point of a unit and its response,
characterized by overshoot and settling time, should be minimal.
These concerns are more pronounced in a microgrid than in a conventional grid be-
cause a microgrid has limited power capacity; thus, it is more prone to failure.
To address the first challenge, this dissertation proposes a microgrid secondary
control strategy based on the concept of potential functions. The proposed strat-
egy [28]–[32] is based on communicating various pieces of information from each DER
unit, e.g., voltage, current, and real and reactive power, to the microgrid central
controller at prespecified time intervals. The central controller defines a potential
function for each DER unit such that its minimum corresponds to the control objec-
tive of the respective DER unit. Therefore, this work proposes and adopts the term
potential function minimizer (PFM) for this central controller. This strategy is based
on the availability of communication in the microgrid.
To address the second challenge, this dissertation proposes a strategy to mitigate
the transients of a microgrid unit to ensure that each unit independently and closely
tracks its set point. This strategy does not require communication; it is based on
response monitoring and can be implemented based on either set point automatic
adjustment (SPAA)2 [33] or set point automatic adjustment with correction enabled
(SPAACE) [33]–[35].3 SPAA takes advantage of an approximate model of the system
to calculate the intermediate set points. SPAACE monitors the trajectory of the
response and bases its decision on the trend of variation of the response. SPAACE
accounts for inaccuracies and unmodeled dynamics by switching the command input
between the original set point and a temporary set point. This strategy is not limited
2Pronounced [spO:].3Pronounced [speIs].
Chapter 1. Introduction 7
to power system applications; it can also be employed for other systems that need
close tracking of their set points. Since many contemporary systems do not include
communication, this strategy can be employed as the backup for the first strategy
should the communication fail.
1.4 Research Objectives
The research objectives of this dissertation are the following:
1. To develop a secondary voltage and/or power controller for enforcing and main-
taining the operation of a microgrid subsequent to disturbances and topological
changes;
2. To develop a distributed implementation for the proposed secondary control
framework, based on the availability and exchange of information between dif-
ferent units;
3. To scrutinize the possible methods and required data for the microgrid commu-
nication;
4. To develop a distributed strategy for trajectory shaping of microgrid units using
response monitoring; and
5. To develop computer tools and software applications to experiment with the
developed strategies and to use in educational settings. These tools are freely
available at http://mehrizisani.com.
1.5 Dissertation Outline
This dissertation is divided into seven chapters and four appendices. The outline
of this dissertation is illustrated in Fig. 1.2. The control strategies proposed in this
dissertation can be categorized based on the availability of communication: In the
presence of communication, potential function–based methods (Chapters 2, 3, and 4)
are employed. Based on the speed of implementation, these methods fall in primary
or secondary control levels. In the absence of communication, trajectory shaping
methods (Chapters 5 and 6) are employed. These methods are augmented to the
primary controllers.
Chapter 1. Introduction 8
One Introduction
Two Potential Functions
Three PFM for Secondary Control
Four Generalized PFM
Five SPAA
Six SPAACE
Seven Conclusions
A Microgrid Definition
B Mathematics of PFM
C Software Tools
Communication: Exchange of
Measurements
No Communication:
Local Measurements
D Theory of SPAACE
Fig. 1.2. Dissertation outline.
Chapter 1. Introduction 9
Chapter 2 introduces the notation of potential functions and presents the behavior
of different terms of a potential function. The advantages and challenges in
implementation of potential functions for microgrid application are evaluated
in this chapter. The different terms of a potential function and the controller
parameters are also defined. Moreover, this chapter discusses the technical
requirements for implementation of the proposed control method.
Chapter 3 demonstrates the application of the potential function notion for the
secondary voltage control. The chapter provides two application examples for
microgrids with different configurations. The proposed strategy calculates in-
termediate set points to avoid introducing a large disturbance in the system.
The performance of the proposed strategy in response to a step change in the
voltage set point is assessed in this chapter.
Chapter 4 generalizes the strategy presented in Chapter 3 for voltage control to
the control of other parameters of the DER units of a microgrid, e.g., real and
reactive power. The salient features of this generalized strategy are (i) based
on availability of information and data exchange, a distributed implementation
is formulated and proposed and (ii) both system-wide and local constraints are
considered to ensure feasibility of the crafted intermediate set points. Extensive
case studies are presented to evaluate the performance of this strategy.
Chapter 5 proposes a method, called SPAA, to improve the set point tracking of
microgrid units in the absence of communication. This strategy employs a sim-
plified model of the unit and ensures that its transient behavior is acceptable,
e.g., the overshoot of the response does not exceed the permissible limits. Sev-
eral case studies on study systems of different sizes are presented to establish
the applicability of this strategy.
Chapter 6 proposes an alternate method, called SPAACE, to the mitigate transients
of a microgrid unit in response to a disturbance such as a set point change or a
remote fault. This method does not require a model of the system; it achieves
a smooth response by temporarily modifying the set point through monitoring
of the response and comparing it to the intended set point. The theoretical
foundation of the proposed strategy is also presented in this chapter. Several
case studies are presented to demonstrate the technical viability of this method.
Chapter 1. Introduction 10
Chapter 7 summarizes the contributions of this dissertation, discusses its conclu-
sions, and recommends feature work.
Appendices provide background and additional information about the dissertation.
Since there is no widely accepted definition of a microgrid, Appendix A pro-
vides the working definition of a microgrid used in this work. The underlying
idea of the potential function–based control, although drastically adapted to
the power system control problem, is borrowed from the field of autonomous
control. A brief overview of the mathematics of the PFM strategy is pre-
sented in Appendix B. Appendix C discusses the software tools developed as
a part of this dissertation. These software tools are publicly available online
at http://mehrizisani.com. Derivations pertaining to the theoretical foun-
dations of SPAACE are presented in Appendix D.
Chapter 2
Potential Functions for the
Secondary Control
2.1 Introduction
In this chapter, the secondary control problem of a microgrid is posed as an application
of the rendezvous problem [36]–[42] to design the trajectory of transition from a set
of initial set points to a set of final set points. The goal of the rendezvous problem
is to have the participating units converge to a prespecified set point. In a typical
rendezvous problem, the units converge to a common set point (or to set points
specified within a range); however, in this work, it is not required that the same set
point is used for all units, and different units can have different set points. Several
approaches to solve the rendezvous problem have been proposed in the technical
literature: (i) the circumcenter control law [43], [44], (ii) aiming toward the farthest
unit approach, and (iii) the potential function method [37]. This work adapts the
Portions of this chapter are published as
[28] A. Mehrizi-Sani and R. Iravani, “Potential-function based control of a microgrid in islanded andgrid-connected modes,” IEEE Trans. Power Syst., vol. 25, no. 4, pp. 1883–1891, Nov. 2010;
[29] A. Mehrizi-Sani and R. Iravani, “Secondary control of microgrids: Application of potentialfunctions,” in CIGRE Session 2010, Paris, France, Aug. 2010;
[30] A. Mehrizi-Sani and R. Iravani, “Secondary control for microgrids using potential functions:Modeling issues,” in 2009 CIGRE Canada Conf., Toronto, ON, Oct. 2009; and
[31] A. Mehrizi-Sani and R. Iravani, “On the educational aspects of potential functions for thesystem analysis and control,” IEEE Trans. Power Syst., vol. 26, no. 2, pp. 878–885, May 2011.
11
Chapter 2. Potential Functions for the Secondary Control 12
potential function method for the secondary control of a microgrid [28]. This choice
is made because potential functions offer more flexibility than the other two methods.
Moreover, the formulation of potential functions is more straightforward. In the
proposed method, a controller called potential function minimizer (PFM) is tasked
with minimization of the potential function associated with each controllable unit.
PFM performs step-by-step minimization of the potential function to calculate the
intermediate set points for each unit.
The potential function approach provides a concise method to convey status infor-
mation of the microgrid as a single entity. This approach can be exploited for different
control levels (primary, secondary, and tertiary) [45]. Although this dissertation pro-
poses the potential function approach for the secondary control of a microgrid, it can
also be used for the microgrid tertiary control if the appropriate time frame and input
signals are employed. Because of its computational burden and signal transmission
delays, the potential function–based method is not readily applicable for the primary
control.
In power systems, potential functions have also been used for the transient sta-
bility analysis [46]–[48], where they are called transient energy functions (TEF) and
constitute a form of the Lyapunov function. Each TEF encodes the energy content
of a synchronous generator [49]. In addition to power systems, potential functions
can comprehensively characterize a number of fields [31] such as computer graph-
ics [37], [50], electric fields [51], mechanics [52], distributed control [37], pattern
recognition [53], electromagnetic field problems [54], and fault diagnosis [55].
In this chapter, the concept of potential function is explained, and the behavior
of different terms in a potential function is explained. Chapter 3 presents application
examples. Chapter 4 generalizes this approach for a distributed implementation and
to handle explicit constraints.
2.2 Definition of a Potential Function
In its most generic form, a potential function φ(·) is a nonnegative scalar function
that measures deviation from the desired state [31].1 In this context, the potential
1Compare this to the definition of the potential energy in physics, e.g., gravitational potentialenergy; the desired state corresponds to the reference level of the system at which the energy of thesystem is minimum.
Chapter 2. Potential Functions for the Secondary Control 13
function φj defined for the DER unit j (denoted hereinafter by DERj) is expressed
as
φj : Rn → R
zj 7→ φj(zj),(2.1)
where zj is the vector of secondary control set points and measurements comprised of
controllable variables (voltage, current, or real/reactive power) of controllable units in
the microgrid. The collection of all potential functions in a microgrid give a measure
of the deviation of the microgrid from its desired state.
A potential function can be designed as a first-order differentiable function such
that it minimization corresponds to achieving the control goal. The direction of the
minimization process for DERj can be determined by a gradient descent method as
z(k)j = z
(k−1)j −K∆T
∂φj∂zj
, (2.2)
where z(k) is the vector of new secondary control set points, z(k−1) is the vector of
previous secondary control set points, ∆T is the time step (the time between two
successive updates of secondary control set points), and K is a constant to adjust the
update magnitude.
2.3 Components of a Potential Function
A potential function defined for DERj conveys information about its measurements,
constraints, and control goal as
φj(zj) = wunu∑i=1
pui (zj) + wcnc∑i=1
pci(zj) + wgpg(zj), (2.3)
where φj and zj are defined in the previous section. Terms pu, pc, and pg are partial
potential functions for the measurements of the DER unit, its constraints, and its
control goal, respectively, and wu, wc, and wg are their corresponding weight fac-
tors. Equation (2.3) is used in conjunction with (2.2) at each controller time step to
determine the updated set points.
Chapter 2. Potential Functions for the Secondary Control 14
2.3.1 Partial Potential for the DER Unit Measurements
The partial potential function pu encodes the measurement information of DERj and
the related units. Generally, this component contains attractive terms to ensure inter
alia (i) a flat voltage profile across the microgrid and (ii) a controlled power flow in
the lines.
2.3.2 Partial Potential for the Constraints
The partial potential function pc specifies the operating conditions that DERj should
avoid to ensure an acceptable operation. Such conditions include limitations on real
and reactive power exports of DERj and the maximum permissible voltage deviation
from the nominal values. Due to their inherent properties, repulsive terms are used
so that this term decreases as DERj moves away from the constraints.
2.3.3 Partial Potential for the Control Goal
The partial potential function pg is responsible for steering DERj toward its set point.
The set point could be a combination of the unit voltage, current, and real/reactive
power. Since it is desired to minimize the deviation of a unit from its set point, this
term should monotonically increase as a function of this deviation.
2.4 Behavior of Potential Terms
Depending on whether a potential term increases or decreases as a DER unit ap-
proaches a predefined area (target), its minimization can lead to attraction to or
repulsion from that area, respectively.
An attractor term, as its name implies, attracts a unit to a prespecified area and
is used for pu and pg in (2.3). This term monotonically increases with the departure
of the unit from the area. Fig. 2.1 shows four types of attractive terms, and Fig. 2.2
shows the corresponding potential function areas—Fig. 2.2(c) corresponds to both
Fig. 2.1(c) and Fig. 2.1(d), depending on the formulation of the potential function.
A repulsor term has the opposite objective and is used for pc in (2.3). That is, it
repulses the unit from the prespecified constraints. A repulsor term decreases as the
unit departs from the prespecified area.
Chapter 2. Potential Functions for the Secondary Control 15
0 1 2 30
1
2
3
4
(a)
0 1 2 30
1
2
3
4
(b)
0 1 2 30
1
2
3
4
(c)
0 1 2 30
1
2
3
4
(d)
R
p(R)
R
p(R)
R
p(R)
R
p(R)
Fig. 2.1. Plot of the potential term as a function of distance R; parameter r is defined inSection 2.4 and in (a), r = 0; in (b)–(c), r = 1. (a) point; (b) circle; (c) double circle; (d)bowl.
zqzqzq
r r r
(a) (b) (c)
zdzdzd
Fig. 2.2. Different potential function targets. (a) point; (b) circle; (c) double circle andbowl.
Chapter 2. Potential Functions for the Secondary Control 16
Table 2.1Attractor terms in a potential function
Type Expression Voltage Control Example p(R) Area
Point R2 1]0 pu (magnitude and angle) Fig. 2.1(a) Fig. 2.2(a)Circle (R− r)2 1 pu (magnitude) Fig. 2.1(b) Fig. 2.2(b)Double circle (2.8) 0.9–1.1 pu (range of magnitude) Fig. 2.1(c) Fig. 2.2(c)Bowl (2.9) 0.9–1.1 pu (range of magnitude) Fig. 2.1(d) Fig. 2.2(c)
Examples of attractor and repulsor terms are given below. For the sake of simplic-
ity and without loss of generality, the rest of this section assumes that z = (zd, zq) ∈R2 represents the vector measurements of a DER unit. However, the results can be
readily generalized to Rn. With some abuse of notation, (zd, zq) and zd + jzq are
used interchangeably. In the following, the term distance is defined as the Euclidean
distance between the vector of unit measurements z and the desired set points zg;
that is, distance equals ‖z−zg‖. R is short form for the distance of z from the origin:
R = ‖z‖ = ‖zd + jzq‖ = (z2d + z2
q )12 . In the general case that the target is at zgd + jzgq
instead of at the origin, a coordinate shift is made by replacing zd and zq with zd− zgdand zq − zgq , respectively.
2.4.1 Attractor Terms
Table 2.1 summarizes the different attractor terms and their example applications for
voltage control.
2.4.1.1 Point
This term causes attraction to a point attractor. Assume a point at the origin. The
simplest differentiable potential function for a unit at z is
p = R2 = z2d + z2
q . (2.4)
The corresponding gradients are
∂p
∂zd= 2zd
∂p
∂zq= 2zq.
(2.5)
Chapter 2. Potential Functions for the Secondary Control 17
r
Target area
r−ε
r+ε
zd
zq
Fig. 2.3. Double circle and bowl target areas.
2.4.1.2 Circle
In this case, the unit seeks to minimize its distance from a circular target. Assume
the circle is at the origin and has a radius of r. This term is defined as the square of
the unit distance from this circular area as
p = (R− r)2. (2.6)
The gradients are
∂p
∂zd= 2zd
(1− r
R
)∂p
∂zq= 2zq
(1− r
R
).
(2.7)
2.4.1.3 Double Circle
In this case, the target area is an annulus with an inner radius of r − ε and an outer
radius of r + ε, as shown in Fig. 2.3. This target area is the locus of all points in
the zdzq-plane that are at a certain distance r from the origin within an acceptable
tolerance ε. The potential function for this target type is the summation of two
potential functions of the single circle type with radii r + ε and r − ε:
p =(R− (r − ε)
)2+(R− (r + ε)
)2
= 2(R2 + r2 + ε2 − 2Rr
).
(2.8)
Chapter 2. Potential Functions for the Secondary Control 18
2.4.1.4 Bowl
This type of attractor is similar to the double circle attractor, except that its defining
potential function is constant for all unit measurements that are within the prescribed
target area. That is,
p =
(R− (r − ε)
)2
; 0 ≤ R < r − ε
0; r − ε ≤ R < r + ε(R− (r + ε)
)2
; r + ε ≤ R.
(2.9)
Note that this potential function is continuous, and its gradient in zd-direction is
∂p
∂zd=
2(R− (r − ε)
)zdR
; 0 ≤ R < r − ε
0; r − ε ≤ R < r + ε
2(R− (r + ε)
)zdR
; r + ε ≤ R.
(2.10)
The gradient in zq-direction is calculated similarly.
2.4.2 Repulsor Terms
Each repulsor term introduced in this subsection is the inverse of the corresponding
attractor term.
2.4.2.1 Point
The point repulsor term is defined as
p =1
R. (2.11)
Using the chain rule, the gradients are
∂p
∂zd= − 1
R2× zdR
∂p
∂zq= − 1
R2× zqR.
(2.12)
Chapter 2. Potential Functions for the Secondary Control 19
PFM DER1Set PointsMeasurementsLPFLPF ... DER2DERn...LPFFig. 2.4. Schematic diagram of the PFM-based control.
2.4.2.2 Circle
In this case, the potential function is defined as
p =1
(R− r)2. (2.13)
The gradients are
∂p
∂zd= − 1
R4× 2zd
(1− r
R
)∂p
∂zq= − 1
R4× 2zq
(1− r
R
).
(2.14)
2.5 Parameters of the Potential Function Mini-
mizer (PFM)
Fig. 2.4 shows the schematic diagram of the potential function–based control for a
microgrid. The measurements of the DER units are communicated to PFM, which
are used to define a potential function for each DER unit. The PFM parameters are
as follows:
2.5.1 Low-Pass Filter Cutoff Frequency
The PFM strategy is implemented as a discrete-time controller. Hence, it uses sam-
pled values of the measurement vector zj to avoid the need for a high bandwidth
communication link between each DER unit and PFM. Thus, instead of the original
vector zj, its low-pass filtered version is transmitted to PFM.
Chapter 2. Potential Functions for the Secondary Control 20
2.5.2 Sampling Interval and Update Time
PFM uses the sampled values of zj; it also updates the set point of DERj at regular
time instants separated by the update time ∆T .
2.5.3 Change Between Two Consecutive Set Points
If the change between two consecutive set points is excessively large such that the DER
unit is not within a certain tolerance of its steady state–acceptable value before the
subsequent set point change occurs, the operation of primary and secondary control
loops of the DER unit become interrelated. To avoid this scenario, the maximum
allowable change in the set point ∆max is limited.
2.5.4 Weight Factors
The weight factors wu, wg, and wc determine the relative importance of the partial
potential functions pu, pg, and pc for each DER unit. For example, if meeting the
power demand is more important than avoiding constraints, the respective weight
factors are selected commensurately to prioritize this requirement. Weight factors can
be assigned based on the size, characteristics, location, and priority of the resources
present in the microgrid, for example through a predetermined look-up table. For
example, in the case studies in the next chapter, more importance is given to achieving
the control goal pg than to maintaining the proximity of measurements of different
units pu; therefore, wu is smaller than wg, as given in Table 3.2.
2.6 Discussion
2.6.1 Theoretical Background
Appendix B provides the theoretical background and a mathematical treatment of
the potential function–based control strategy.
2.6.2 Comparison With Conventional Secondary Control
The conventional secondary voltage control usually uses a constant gain matrix to
map voltage deviations to appropriate control signals [56]. This gain matrix is deter-
Chapter 2. Potential Functions for the Secondary Control 21
mined by offline calculations to minimize a measure of the system voltage deviation
under a set of predefined contingencies. This formulation is based on the assumption
that voltage is mainly controlled by reactive power, for example through droop char-
acteristics. The conventional secondary voltage control assumes steady-state 60 Hz
quantities and does not provide provisions for frequency control.
The advantages of exploiting potential functions over the conventional secondary
control methods [20] include
1. Ease of implementation which enables inclusion of additional terms representa-
tive of the system performance parameters;
2. Online responsiveness to changes in the system;
3. Ability to encode decision boundaries; and
4. Provisions to control the trajectory of the microgrid from the initial operating
point to the desired one based on the characteristics of the units.
2.6.3 Single Versus Multiple Potential Functions
In this work, a scalar potential function φj(·) is defined for each controllable unit of the
microgrid. Therefore, there are as many potential functions as there are controllable
units. An alternative implementation is to define a single global potential function
φ(·) that embeds the information of all units. In the sequel, the first implementation
is referred to as the multiple implementation, and the second is referred to as the
single implementation.
In the single implementation, a central controller receives all measurements and
issues system-wide control commands. However, in the multiple implementation,
each unit is controlled independently, and weight factors and potential terms are
crafted for a unit without directly affecting the performance of the others. The
multiple implementation is more feasible for realization because for each unit, it
requires only the availability of the measurements that are used in the same unit’s
potential function. This implementation also helps avoid the unnecessary coupling
between otherwise unrelated units. It ensures that each potential function merely
uses measurements of units that directly affect each other. In comparison, the single
implementation requires availability of all measurements to calculate the set points.
Chapter 2. Potential Functions for the Secondary Control 22
Regardless of the implementation approach, embedding microgrid information in
a scalar quantity could be inadequate since the status of an individual unit cannot be
readily retrieved. In general, different partial potential terms may exhibit conflicting
behaviors. This situation is exacerbated as the complexity or number of potential
functions increases. This becomes especially important when a unit is about to vi-
olate its safe operating conditions. This issue can be addressed by either employing
barriers [57] to keep the unit from violating its limits or using dynamically changing
weight factors that increase/decrease to reflect the proximity of units to their lim-
its and to reflect operational priorities. The generalized PFM strategy proposed in
Chapter 4 addresses this challenge by explicitly considering the constraints.
2.6.4 Infrastructure Requirements
Although the technical requirements of the proposed strategy, e.g., two-way commu-
nication, computational ability, and high penetration of distributed generation, may
not be readily available at the time of writing this dissertation,2 they lie within the
requirements of the smart grid vision [59]. To realize the vision of the smart grid, the
existing infrastructure and operational philosophies need to be considerably updated.
Therefore, these facilities are expected to become reasonably available in the near
future.
In this work, the communication link is assumed ideal, e.g., with zero transmission
delay. Therefore, system measurements and set points are communicated instanta-
neously to and from PFM. A practical communication link introduces a delay in the
transmission of the control signals [60]. However, the communication delay does not
hamper the performance of PFM if the delay is small compared with the response
time. Moreover, the amount of data exchange is minimal and limited to the set points
and sampled measurements of the units. Since PFM is a secondary control strategy,
its response time is longer than that of the associated primary control, which is in
the order of 40 ms, for a fast voltage-controlled electronically interfaced DER unit.
The response times of some available communication protocols/technologies for a mi-
crogrid in a small geographical span are as follows: 10 ms for CANbus, 5 ms for
IEC61850, 10 ms for WiMAX, and 5 µs/km3 for SERCOS with fiber optic [7], [61],
2For example, currently only 40% of utilities allow remote control of the distributed units [58].3This speed is a function of the speed of light in glass.
Chapter 2. Potential Functions for the Secondary Control 23
[62]. If communication is completely lost, the units can use the last set point received
from the PFM controller.
2.6.5 Developed Software
A software tool is developed to experiment with different potential function types. It
is also proposed to use the software in an educational setting [31]. This software tool
is discussed in Appendix C.1.
2.7 Conclusions
This chapter introduces the concept of potential functions as a new approach to enable
secondary control (and tertiary control) for a microgrid, based on the availability of
communication within the microgrid. A potential function comprises a number of
terms to represent the status, constraints, and control objectives of the respective
unit. To coordinate the behavior of different DER units, each potential function
is controlled through parameters such as weight factors, sampling interval, and the
permissible change between consecutive set points.
The potential function associated with each DER unit of the microgrid embeds
and regularly updates various pieces of information, e.g., current, voltage, power, and
operational constraints, of the DER units. The microgrid central controller minimizes
each potential function, e.g., based on a gradient decent method, to determine the
set point(s) of the corresponding DER unit associated with the minimum of the
potential function. The minimization process is carried out in a discrete-time manner
at predefined time intervals.
The advantages of using potential functions for secondary control include ease of
implementation, ability to represent complex constraints, and ability to craft the tra-
jectory of the unit. For implementation, potential functions require an infrastructure
capable of the increased communication and calculation load. In this work, such an
infrastructure is assumed to be available, as required by the vision of the smart grid.
Chapter 3
Application of PFM for Voltage
Control
3.1 Introduction
This chapter demonstrates the application of potential functions for the microgrid
secondary voltage control in two example scenarios. The case studies assess the
performance of the proposed secondary control for a step change in the voltage set
point of the microgrid in both islanded and grid-connected modes.
A microgrid is at its desired state when all DER units reach their steady-state
conditions with zero set point tracking error. To achieve this goal, the PFM controller
defines a potential function for each DER unit of the microgrid using the information
communicated from the DER unit such that this state corresponds to the minimum of
the respective potential function. Therefore, the effort to reach the control objective of
a DER unit translates to minimization of its potential function. The central controller
determines the direction of the gradient descent of each potential function, calculates
the new set points according to (2.2), and communicates them back to the DER units.
The majority of DER units are interfaced to the microgrid using voltage-sourced
converter (VSC) units [63].1 Voltage control of a VSC in the dq-frame [64] can
Portions of this chapter are published as mentioned in the footnote of Chapter 2.
1There are several microgrid systems in operation that are solely based on VSC-interfaced units.This is mostly the case for urban feeders—as opposed to rural and remote feeders—that integrateelectronically interfaced photovoltaic and battery storage units but no rotating machine–interfacedwind units. Toronto Hydro possesses such urban feeders.
24
Chapter 3. Application of PFM for Voltage Control 25
+–
idq(ref)K(s)
udq
Voltage feedforward terms
Vt,dqVdq(ref)
+−
C(s) DG
idq
Vs,dq
Measurement+
+
Fig. 3.1. Nested loop voltage control of a VSC.
be achieved by controlling the VSC (i) as a voltage source [17] or (ii) as shown in
Fig. 3.1, in a nested loop, based on an inner current control loop and an outer voltage
control loop [65]. In the latter case, the controller C(s) of the inner loop regulates
the converter current, and the controller K(s) of the outer loop regulates its output
voltage. A voltage decoupling term [65] or a current feedforward term [66] can be
used to improve the dynamic performance of the system. Because of its ability in
limiting current should a fault happen, the nested loop approach is employed for the
control of VSC units in the presented application examples.
3.2 Application Example I
To demonstrate the applicability of the proposed PFM method, it is employed for
the secondary control of a microgrid with three DG units in the islanded mode of
operation. Since the grid is strong (230 kV with a high SCMVA), the DG units do not
assist the grid with its voltage regulation.
The studies are conducted in time domain in the PSCAD/EMTDC software [67]
with a simulation time step of 5 µs. The potential functions are implemented in
the Fortran programming language and interfaced to the host simulation software
through a developed PSCAD component [68]. The study system, potential functions,
and results are explained in this section.
3.2.1 Study System I
Fig. 3.2 shows System I, which is a part of the CIGRE North American medium
voltage distribution network benchmark system [69] augmented with three DG units.
Two DG units are dispatchable (DG2 and DG3) and operate as voltage-controlled
units, and one DG unit (DG1) is nondispatchable and operates as a current-controlled
unit.
Chapter 3. Application of PFM for Voltage Control 26∞Rs, LsRL1, LL1, CL1Rf1, Lf1DG1 Rl1, Ll1RL2, LL2, CL2Rf2, Lf2DG2 Rl2, Ll2RL3, LL3, CL3Rf3, Lf3DG3TgT1T2T3 PC1PC2PC3
PCCFig. 3.2. Study system I: Three-DG cascade microgrid.
Each DG unit is represented by a dc source interfaced to the system through
a VSC. Each VSC is connected to the corresponding PC through a filter and an
interface transformer. All transformers are delta-grounded wye. The local load of
each DG unit is connected to the corresponding PC and is represented by a shunt
RL system. A shunt capacitor is used to correct the power factor of each load to 0.95
lagging.
The overhead lines between PCs are represented by a series RL branch per phase.
The microgrid is connected to the main grid at the PCC. The main grid is represented
by an ideal three-phase voltage source behind a series RL branch in each phase. The
grid is strong and has an SCMVA of 12 000 MVA and an X/R ratio of 10. Table 3.1
shows the parameters of System I.
3.2.2 Control Strategy
The goal of this application example is to regulate the voltage of each DG unit at
the corresponding PC at a prespecified magnitude. The voltages of the dispatchable
units are measured, transformed to the dq-frame, and communicated to PFM. PFM
calculates the dq-frame voltage set points and communicates them to the respective
DG units. Since the control of dispatchable units is performed in the dq-frame, the
angle information is used to transform the abc parameters to the dq parameters. In
the studied islanded mode, the angle is generated by a voltage-controlled oscillator
(VCO). In a practical implementation of the proposed method, a periodic synchro-
Chapter 3. Application of PFM for Voltage Control 27
Table 3.1Study system I: Parameters
Fundamental frequency f = 60 HzSwitching frequency fsw = 1620 Hz 27 pu
Grid voltage vs = 230 kVGrid resistance Rs = 0.439 ΩGrid inductance Ls = 11.635 mH
Transformer G 230/12.47 0.013 + j1.55 Ω 0.001 + j0.120 puTransformer 1 12.47/480 500 kVA 0.005 + j0.080 puTransformer 2 12.47/480 300 kVA 0.005 + j0.080 puTransformer 3 12.47/480 350 kVA 0.005 + j0.080 pu
DC bus voltage Vdc = 1200 VFilter impedance 0.025 + j0.040 pu
Line 1 0.846 + j2.112 Ω (Ll1 = 5.603 mH)Line 2 0.517 + j1.292 Ω (Ll2 = 3.427 mH)
Load 1 (240 kVA) R = 810 Ω, L = 2.86 H, C = 1.38 µFLoad 2 (435 kVA) R = 420 Ω, L = 1.80 H, C = 1.83 µFLoad 3 (270 kVA) R = 640 Ω, L = 3.50 H, C = 0.64 µF
nizing signal will be sent from the reference DG to other units to ensure all units
operate with the same angle information.
It is assumed that each DG unit is able to meet the power demand of its local
load, and there is no appreciable power flow between the PCs. Based on this as-
sumption, voltage regulation can be achieved through any combination of direct- and
quadrature-axis voltages, i.e., vd and vq. Therefore, the voltage phase angle is an
available degree of freedom. Thus, for a given voltage magnitude, infinitely many
possible values exist for vd and vq. The values of vd and vq to which PFM con-
verges depend, among other factors, on the grid connection status, power injection of
nondispatchable units, system loads, and configuration of the microgrid.
3.2.3 Potential Function
The vector xj in this application consists of the d- and q-components of the voltage
at the PC of each dispatchable DG unit, i.e., zj = (vdj, vqj). The potential functions
defined for dispatchable DG unit are similar: each potential function has one attractor
term (circle) for the voltage control goal and one attractor term (point) for the voltage
proximity to the other dispatchable DG unit. With reference to (2.3), the partial
Chapter 3. Application of PFM for Voltage Control 28
Table 3.2Parameters of PFM
Parameter System I System II
fc 2 Hz 2 Hz∆T 100 ms 16 msK 4 30∆max 15 3wg 1 1wu 0.1 0.067
potential functions for units, constraints, and control goal for DG2 are
nu∑i=1
pu(z2) =∥∥(vd2 − vd3, vq2 − vq3)
∥∥nc∑i=1
pc(z2) = 0
pg(z2) =((v2d2 + v2
q2)− r2)2,
(3.1)
where r is the desired voltage magnitude. Since no operational constraints are con-
sidered for the DG units, the term pc is nil.
The parameters of the potential function are summarized in Table 3.2. The weight
factors are found by experimenting with a number of case studies. An LPF cutoff
frequency fc of 2 Hz is used in the presented case studies. Therefore, PFM reacts
only to transients with a frequency lower than 2 Hz. This cutoff frequency is chosen
because (i) a transient with a frequency higher than 60 Hz is a power system harmonic
that can be mitigated by passive filters and (ii) a transient with a frequency from
5 Hz to 60 Hz requires a fast response and is mitigated by the primary controller.
The remaining frequency range is the range of interest for a secondary controller.
3.2.4 Study Results
The performance of PFM is evaluated in the islanded mode by imposing a step change
in the microgrid reference voltage. Initially, the target voltage is 1 pu. DG1 injects
id = 0.24 pu and iq = 0.24 pu (pu is based on the rated power of the transformer, not
the rated power of the load). Although not likely to happen in a practical situation,
Chapter 3. Application of PFM for Voltage Control 29
a relatively large 30% reduction in the target voltage (from 1 pu to 0.7 pu) is applied
to evaluate the viability of the proposed PFM method in an extreme condition.
Fig. 3.3 shows the system response following the step command. PFM changes
the set points of vd and vq and as a result, voltage magnitude vmag, in successive steps,
until the voltage magnitude reaches the target set point in 400 ms. Since PFM is a
discrete-time controller and processes the inputs only at specific time instances, there
could be delay before it responds to the step change. The minimum delay is zero (if
the step is applied immediately before the update time) and the maximum delay is
∆T (if the step is applied immediately after the update time). The voltage at the
PC is sinusoidal both before and after the step change, Fig. 3.3(d). The transients
of DG1 currents id and iq are suppressed in 40 ms, Fig. 3.3(e). Fig. 3.3(f) shows the
voltage angle.
Fig. 3.4 shows the locus of the PC1 voltage. Figs. 3.3(f) and 3.4 show that the
magnitude of the voltage reaches the steady state faster than the angle of voltage.
This is expected, since PFM primarily regulates the voltage magnitude.
3.3 Application Example II
The proposed potential function–based control can also be used to overlap the primary
and secondary control functions. In this case, the secondary control helps the system
settle within a certain tolerance of the set points. The primary control then mitigates
any remaining discrepancy. This case is conducted on System II.
3.3.1 Study System II
Fig. 3.5 shows the schematic diagram of the three-phase, three-wire System II adapted
from [70]. The system includes three dispatchable units (DG1, DG3, and DG4) that
operate as voltage-controlled units and one nondispatchable unit (DG2) that operates
as a current-controlled unit. The grid, lines, filters, and DG units are represented
similarly to those of System I. Table 3.3 provides the system parameters.
3.3.2 Control Strategy
System II uses a control strategy similar to that of System I. In the grid-connected
mode, the angle is generated by a PLL synchronized to PC1 in Fig. 3.5.
Chapter 3. Application of PFM for Voltage Control 30
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.6
0.8
1
(a)
v 2,m
ag (
pu)
refPFMmeas
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80
0.5
(b)
v 2d (
pu)
refmeas
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.6
0.8
1
(c)
v 2q (
pu)
refmeas
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8−1
0
1
(d)
v 2 (pu
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.1
0.2
0.3
(e)
i 1 (pu
)
d−axisq−axis
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.880
85
90
(f) Time (s)
/_ v
2 (de
gree
)
Fig. 3.3. System I: Step change in voltage from 1 pu to 0.7 pu. (a) PC2 voltagemagnitude; (b) d-component of PC2 voltage; (c) q-component of PC2 voltage; (d) phase aof PC2 voltage; (e) current injected by DG1; (f) PC2 voltage phase angle.
Chapter 3. Application of PFM for Voltage Control 31
0 0.2 0.4 0.60.7
0.8
0.9
1
vd (pu)
v q (pu
)
Fig. 3.4. System I: Locus of the PC2 voltage.
SPCCPC1 S1DG1 Rf, LfRl, Ll, Cl Rt, LtPC4 S4Rf, LfRl, Ll, Cl Rt, Lt. . .. . .DG2DG3DG4 . . . ∞Rs, Ls
Fig. 3.5. Study system II: Four-DG radial microgrid.
Chapter 3. Application of PFM for Voltage Control 32
Table 3.3Study system II: Parameters
Fundamental frequency f = 60 Hz 1 puSwitching frequency fsw = 1620 Hz 27 puBase power Sb = 10 kVA 1 puBase voltage Vb = 115 V 1 pu
Grid voltage vs = 115 V 1.000 puGrid resistance Rs = 0.025 Ω 0.019 puGrid inductance Ls = 145 µH 0.041 pu
Line resistance Rl = 0.01 Ω 0.008 puLine inductance Ll = 25 µH 0.007 pu
Load resistance RL = 1.322 Ω 1.000 puLoad inductance LL = 1.949 mH 0.556 puLoad capacitance CL = 3610 µF 0.556 pu
DC bus voltage Vdc = 350 VFilter resistance Rf = 0.015 Ω 0.011 puFilter inductance Lf = 637 µH 0.182 pu
3.3.3 Potential Function
Each potential function has an attractor term (double circle) for the voltage control
goal and two attractor terms (point) for the voltage proximity to other two dispatch-
able DG units. With reference to (2.3), the partial potential functions for units,
constraints, and control goal for DG1 are
nu∑i=1
pu(z1) =∥∥(vd1 − vd3, vq1 − vq3)
∥∥+∥∥(vd1 − vd4, vq1 − vq4)
∥∥nc∑i=1
pc(z1) = 0
pg(z1) = 2(R21 + r2 + ε2 − 2R1r).
(3.2)
Radius r is the desired voltage magnitude, and ε is the permissible voltage devia-
tion (15%) for the secondary control. The parameters of the potential function are
summarized in Table 3.2.
Chapter 3. Application of PFM for Voltage Control 33
3.3.4 Study Results
Since the SCMVA of this study system is lower than that of System I, which is indicative
of a weaker grid, (limited) control of voltage in the grid-connected mode is possible.
In both islanded and grid-connected modes, the performance of PFM is evaluated by
applying a step change in the reference voltage.
3.3.4.1 Islanded Mode: Step Change in Voltage
This case study investigates the transient behavior of the microgrid subsequent to a
step change in the target voltage in the islanded mode. Again, although not likely to
happen in a practical situation, a relatively large 30% reduction in target voltage is
applied to evaluate the viability of the proposed PFM method in an extreme condition.
Initially, the target voltage is 1 pu, and DG2 injects id = 1 pu and iq = 0 pu.
The voltage target is changed in a step from 1 pu to 0.7 pu. Fig. 3.6 illustrates the
response of the PC1. Following this disturbance, PFM changes the set points of vd
and vq. As a result, the voltage magnitude vmag reaches its new set point in 50 ms,
Fig. 3.6(a)–(c). Fig. 3.6(a) shows that there is a delay before PFM responds to the
disturbance. This delay is due to the sampled nature of the measurements used by
PFM. Subsequent to mitigation of the disturbance, the PC1 voltage is sinusoidal,
Fig. 3.6(d). The transients of the current of DG2 are suppressed in less than a cycle,
Fig. 3.6(e).
Fig. 3.7 shows the locus of the voltage set point for DG1 in the dq-plane. Figs. 3.6(a)
and 3.7 show that the initial response of the voltage set point of DG1 to the step
change is relatively fast, and the voltage magnitude regulation is achieved in 50 ms.
However, the phase angle of the voltage, Fig. 3.6(f), does not reach the steady state
as fast as the voltage magnitude. This is the reason the d- and q-components of the
voltage, Fig. 3.6(b)–(c), reach the steady state slower than vmag, Fig. 3.6(a). This
case study demonstrates that PFM enables intermediate voltage set point tracking in
the islanded mode.
3.3.4.2 Grid-Connected Mode: Step Change in Voltage
In this case study, the transient behavior of the grid-connected microgrid following
a step change in the set point voltage is studied. Unlike the previous case in which
a large step in voltage was applied, in the grid-connected mode the ability of the
Chapter 3. Application of PFM for Voltage Control 34
0 0.05 0.1 0.15 0.20.6
0.8
1
(a)
v 1,m
ag (
pu)
refPFMmeas
0 0.05 0.1 0.15 0.2
0.4
0.6
0.8
(b)
v 1d (
pu)
refmeas
0 0.05 0.1 0.15 0.20.4
0.6
0.8
(c)
v 1q (
pu)
refmeas
0 0.05 0.1 0.15 0.2−1
0
1
(d)
v 1 (pu
)
0 0.05 0.1 0.15 0.20
0.5
1
(e)
i 2 (pu
)
d−axisq−axis
0 0.05 0.1 0.15 0.250
60
70
(f) Time (s)
/_ v
1 (de
gree
)
Fig. 3.6. System II in the islanded mode: Step change in voltage from 1 pu to 0.7 pu. (a)PC1 voltage magnitude; (b) d-component of PC1 voltage; (c) q-component of PC1 voltage;(d) phase a of PC1 voltage; (e) current injected by DG2; (f) PC1 voltage phase angle.
Chapter 3. Application of PFM for Voltage Control 35
0 0.2 0.4 0.6
0.6
0.7
0.8
vd (pu)
v q (pu
) 0.9
1.0
Fig. 3.7. System II in the islanded mode: Locus of the PC1 voltage.
microgrid to control PC voltages is limited. Therefore, the DG units can only control
the voltage of their corresponding PCs over a narrow range. Thus, instead of a 30%
step change, a 5% step change is applied in the target voltage.
Initially, the target voltage is 1 pu. DG2 injects id = 1 pu and iq = 0 pu. The
voltage target is changed in a step from 1 pu to 0.95 pu. Fig. 3.8 illustrates the
response of the system to this disturbance. Following the step, PFM changes set
points of vd and vq. As a result, the voltage magnitude reaches the new set point in
25 ms, Fig. 3.8(a)–(c).
The delay between the step change and the response of PFM is due to the sam-
pled nature of the voltage measurements used by PFM. Since the grid exchanges
power with the microgrid, the current of DG2 does not undergo noticeable transients,
Fig. 3.8(e). Subsequent to the disturbance, the PC1 voltage becomes sinusoidal,
Fig. 3.8(d).
Fig. 3.9 shows the locus of the voltage set point for DG1 in the dq-plane. Figs. 3.8(a)
and 3.9 show that the initial response of the set point voltage of the DG to the step
change in relatively fast, and voltage magnitude regulation is achieved in 25 ms. How-
ever, the phase angle of the voltage does not reach the steady state as fast as voltage
magnitude, Fig. 3.8(f). This is the reason the d- and q-components of the voltage,
Fig. 3.8(b)–(c), reach the steady state slower than vmag, Fig. 3.8(a). This case study
confirms PFM allows tracking of the voltage set points in the grid-connected mode.
A comparison of Figs. 3.6 and 3.7 to Figs. 3.8 and 3.9 shows that the voltage
regulations is achieved faster and with less transients in the grid-connected mode
Chapter 3. Application of PFM for Voltage Control 36
than in the islanded mode. Comparing Fig. 3.7 to Fig. 3.9 suggests that the existence
of a grid connection assists voltage regulation.
3.4 Conclusions
This chapter demonstrates the technical feasibility of the potential function–based
secondary control for two microgrids in both islanded and grid-connected modes.
The first microgrid has three DER units in a cascade configuration. In the second
microgrid four DER units are radially connected to the point of common coupling.
The study results confirm the viability of the method.
Chapter 3. Application of PFM for Voltage Control 37
0 0.05 0.1 0.15 0.20.9
1
(a)
v 1,m
ag (
pu)
refPFMmeas
0 0.05 0.1 0.15 0.20
0.05
0.1
(b)
v 1d (
pu)
refmeas
0 0.05 0.1 0.15 0.2
−1
−0.9
(c)
v 1q (
pu)
refmeas
0 0.05 0.1 0.15 0.2−1
0
1
(d)
v 1 (pu
)
0 0.05 0.1 0.15 0.20
0.5
1
(e)
i 2 (pu
)
d−axisq−axis
0 0.05 0.1 0.15 0.2−87
−86
−85
(f) Time (s)
/_ v
1 (de
gree
)
Fig. 3.8. System II in the grid-connected mode: Step change in voltage from 1 pu to0.95 pu. (a) PC1 voltage magnitude; (b) d-component of PC1 voltage; (c) q-component ofPC1 voltage; (d) phase a of PC1 voltage; (e) current injected by DG2; (f) PC1 voltagephase angle.
Chapter 3. Application of PFM for Voltage Control 38
0 0.2 0.4 0.6−1
−0.9
vd (pu)
v q (pu
)
Fig. 3.9. System II in the grid-connected mode: Locus of the PC1 voltage.
Chapter 4
Generalized PFM Strategy
4.1 Introduction
The PFM approach for secondary control is proposed in Chapters 2 and 3, where
it is assumed that the transmission and generation requirements can be met. In
this chapter, the developed potential function–based framework is extended in two
aspects:
• Enabling inclusion of explicit local and system-wide constraints1 in the PFM
framework.
• Proposing a distributed implementation of the PFM framework to optimize the
operation of the microgrid as a whole, while preserving the autonomy of DER
units.
Including constraints enables the PFM strategy to extend from voltage-controlled
units to other units such as power-controlled units. The distributed implementation
improves the computational speed and practicality of PFM.
The developed framework is generalized; that is, by appropriately defining the
potential function and selecting decomposition areas, the developed framework can
Portions of this chapter are published as
[32] A. Mehrizi-Sani and R. Iravani, “Constrained potential function-based control of microgridsfor improved dynamic performance,” IEEE Trans. Smart Grid, Special Issue on Microgrids,Jul. 2011, submitted for review (paper no. TSG-00282-2011).
1As opposed to the soft constraints included in the potential function, Subsection 2.3.2.
39
Chapter 4. Generalized PFM Strategy 40
represent a host of different power system optimization scenarios, ranging from a
standard optimal power flow (OPF) implementation to a multi-area OPF problem
and PFM presented in the previous chapter. Therefore, this extended strategy is
called generalized PFM (GPFM).
4.1.1 Inclusion of Constraints
In GPFM, two sets of constraints are added to the PFM problem: (i) power flow
constraints and (ii) real/reactive power and voltage constraints of each DER unit.
Inclusion of constraints turns the PFM strategy into a framework with a two-fold
objective.
• If final set points are not provided, this framework can provide the set points
analogous to OPF.
• If final set points are provided, as in PFM, this framework designs the trajectory
between the initial and final set points while taking into account technical and
operational constraints.
The distinction between these two modes of operation is made by (i) defining the
objective function and (ii) employing intermediate optimization results.
4.1.1.1 GPFM and OPF
Although both GPFM and OPF are expressed as optimization problems, they have
fundamental differences and objectives:
• In OPF, the objective function often represents either the cost of system opera-
tion or transmission losses to devise system set points. However, in GPFM, the
set points are given; the objective function is a function of controller set points
and represents the deviation of the measured system status from its desired
status.
• OPF is an open-loop scheme that provides the final set points. However, PFM is
a closed-loop approach that provides the trajectory between the initial and final
set points. This is especially important in a microgrid with limited resources,
because if all final set points are applied without regards to the system limits,
unacceptable transients may occur in the system.
Chapter 4. Generalized PFM Strategy 41
• In OPF, only the final optimization results are used, while in GPFM, the in-
termediate optimization values are also employed. OPF does not readily allow
compensation for modeling errors and implementation failures. As such, mod-
eling inaccuracies may affect the result and final set points [71].
4.1.2 Distributed Implementation
In this chapter, the constrained PFM is implemented in a distributed manner. In
this implementation, the microgrid is divided into a number of areas, where each
area can include DER units, loads, and transmission lines. The primary motivation
for a distributed implementation is the unwillingness and/or inability of operators of
different areas to share information and data. In some cases, an area operator may
lack the means to share information with other operators. In other cases, an area
operator may not be willing to share the information, especially if the information is
perceived sensitive to the infrastructure operation. A distributed approach preserves
the autonomy of each operator. It is also possible that through decomposing a large
problem into a series of smaller problems, an otherwise unviable solution—using the
available computational facilities—becomes viable. This is especially imperative for
microgrids that, despite their small geographic size, can include several units with
various control strategies. This complicates the formulation of the problem and in-
creases its size. In this aspect, the difference between a microgrid and a large system
is nonsubstantial.2
A conceivable communication coordination scheme is to use some DER units to
relay information from one DER unit to another. This scheme allows communication
between all units and enables the implementation of a centralized scheme while re-
quiring only a limited number of communication links. These communication links
should be selected such that they make the system visibility graph connected, as ex-
plained in Appendix B. However, this scheme suffers from a number of drawbacks and
is not considered further in this dissertation. These drawbacks include the following:
• Validity of the assumption of infinite communication bandwidth (and zero de-
lay), discussed in Subsection 2.6.4, is pertinent on the limited communication
2There are operational microgrid systems in which the number of DER units is comparable tothe number of generating units in a conventional system. For example, the number of units inBornholm island microgrid system in Denmark is comparable to that of the Hydro One system inOntario, even though Ontario is approximately 2000 times bigger than Bornholm.
Chapter 4. Generalized PFM Strategy 42
requirements of the proposed distributed approach; when exchange of large
amounts of information is considered, some DER units may become commu-
nication bottlenecks, thus invalidating this assumption and requiring a fast
communication link.
• A DER unit utilized to relay information may become offline, which can elim-
inate the links that are required for the connectedness of the system visibil-
ity graph. With a higher number of relay units, this scenario becomes more
likely [72].
• Increased dependence on communication, as suggested in the relay scheme,
invariably increases the number of points of failure of the system and reduces
its reliability.
• This scheme leads to the exchange of larger amounts of information; however,
the ability of DER units in handling, organizing, and storing information is
limited. Moreover, since different DER units are owned by different operators,
the issues of privacy, willingness to share, and security of information become
important factors with this scheme.
Several decomposition schemes have been proposed in the literature, including
the auxiliary problem principle (APP), the predictor-corrector proximal multiplier
method (PCPM), and the alternating direction method (ADM) [73]–[85]. These ap-
proaches are based on decomposing the original problem into a number of subproblems
that are solved iteratively until a convergence condition is met. We use the method
proposed in [77] to decompose the optimization problem into areas. Although there
are different methods to define these areas, e.g., based on sensitivity factors and con-
trollability of different buses, in this work partitioning is based on the availability of
information. Roughly, this is equivalent to physical adjacency.
The decomposition method proposed in [77] has the following advantages:
• The implementation is simple and robust. The subproblems associated with
each area are simply slightly modified versions of the optimization problem of
each area.
• The solution and update procedures are straightforward; the central agent is
not required to manipulate the data. Rather, it merely distributes the data.
Chapter 4. Generalized PFM Strategy 43
Therefore, through a careful implementation, it is even possible to alleviate the
need for the central agent.
• This method does not require solution of subproblems until their optimality in
every iteration. Since the results from a single iteration of each subproblem are
sufficient [78], this method offers considerable computational saving.
• There is no need to estimate Lagrangian multipliers. Information about updat-
ing the multipliers is available through the decomposition method itself.
A centralized implementation of GPFM is simpler and not discussed in this chapter.
4.2 Decomposition Algorithms
This section describes the decomposition algorithm employed in this chapter. A
potential function minimization problem stated as a standard optimization problem
can be expressed as
minimizex
φ(x)
subject to g(x) = 0
h(x) ≤ 0,
(4.1)
where x is the vector of variables, g(x) and h(x) represent equality and inequality
constraints, and φ(x) is the potential function employed as the objective function.
This optimization problem is decomposed, and each subproblem is assigned to an
area. The constraints are distributed among the areas such that, even though each
subproblem does not contain all the constraints, the set of all subproblems has the
same constraints as the original problem.
Assuming x = (x1, x2), where x1 and x2 represent the variables for the subprob-
lems 1 and 2, respectively, (4.1) can be expressed in the following equivalent form:
minimizex1,x2
φ(x1, x2)
subject to g(x1, x2) = 0
h(x1, x2) ≤ 0.
(4.2)
Chapter 4. Generalized PFM Strategy 44
This is the centralized form of the optimization problem. It can be written in the
following distributed form:
minimizex1
φ(x1, x∗2) minimize
x2φ(x∗1, x2)
subject to g(x1, x∗2) = 0 and subject to g(x∗1, x2) = 0
h(x1, x∗2) ≤ 0 h(x∗1, x2) ≤ 0
(4.3)
where asterisk denotes the value of the respective variable from the previous iteration.
With regards to constraints, two cases can be identified:
• Decoupled constraints, i.e., constraints g(x) and h(x) can be decomposed into
g1(x1) and h1(x1) that correspond to subproblem 1 and g2(x2) and h2(x2) that
correspond to subproblem 2. Only variables related to one subproblem appear
in each decoupled constraint.
• Coupled constraints, i.e., constraints cannot be decomposed and expressed as a
function of variables of merely one area. If these constraints are removed, the
optimization problem can be trivially decomposed into one independent sub-
problem for each area. Therefore, the term complicating constraints is some-
times used to refer to these constraints.
4.2.1 Decoupled Constraints
With decoupled constraints, it can be shown that if (4.3) is sufficiently iterated, it
will achieve the performance of the centralized implementation (4.2) [86]. The salient
feature of this method is that even if the system resources do not allow iteration
of (4.3) to convergence, optimization can be stopped at any iteration and suboptimal
values of (x∗1, x∗2) can be used. This is because all solutions are in fact feasible [86],
[87]. This immediately follows; if the initial guess (x(0)1 , x
(0)2 ) is feasible, all subsequent
values of (x(k)1 , x
(k)2 ) are also feasible.
As mentioned, the solution of the set of subproblems converges to the solution of
the original problem if a sufficient number of iterations is performed. That is, the
sequence of potential functions φ(k)i (i = 1, 2) in (4.3) are monotonically nondecreasing
Chapter 4. Generalized PFM Strategy 45
with k. To show this, we rewrite (4.2) in the generic form shown below:
minimizexi;i=1,...,n
φ(x) =∑n
i=1 φi(x1, . . . , xn)
subject to xi ∈ Ωi,(4.4)
and we rewrite (4.3) in the generic form shown below:
minimizexi
φi(xi, x∗−i)
subject to xi ∈ Ωi,(4.5)
where Ωi = xi|gi(xi) = 0, hi(xi) ≤ 0 and x∗−i = (x∗1, . . . , x∗i−1, x
∗i+1, . . . , x
∗n). By the
virtue of optimality, it follows that
φi(x∗(k)i , x
∗(k−1)−i ) ≤ φi(x
∗(k−1)i , x
∗(k−1)−i ). (4.6)
4.2.2 Coupled Constraints
In the presence of coupled constraints, no guarantee can be made that optimality
can be reached using the simple decomposition strategy discussed in the previous
subsection. Consider the following problem, illustrated in Fig. 4.1. In this example,
if the original problem is decomposed such that each subproblem employs only local
objectives, the convergence behavior will depend on the initial value of x; the result
may not be optimal, and the algorithm may end in a Nash equilibrium [57].
minimizex1,x2
(x1 − 1)2 + (x2 − 1)2
subject to 0 ≤ x1
x2 ≤ 1
x1 + x2 ≤ 1
(4.7)
Setting the derivative of the objective function to zero yields the nonfeasible expres-
sion x∗1 + x∗2 = 2. This signals that the solutions is on the boundary of the feasible
set. Since the objective function is symmetric with respect to x1 and x2, it can be
concluded that x∗1 = x∗2. It follows from the second constraint that x∗1 = x∗2 = 12.
Chapter 4. Generalized PFM Strategy 46
x1
x2
Feasible Region
x1*, x2
*
Fig. 4.1. An optimization problem with coupled constraints.
This optimization problem can be written as
minimizex1
(x1 − 1)2 minimizex2
(x2 − 1)2
subject to 0 ≤ x1 ≤ 1 and subject to 0 ≤ x2 ≤ 1
x1 + x∗2 ≤ 1 x∗1 + x2 ≤ 1
(4.8)
If the initial guess (x(0)1 , x
(0)2 ) is (1, 0), then for all iterations k, (x
(k)1 , x
(k)2 ) = (1, 0);
that is, the solver does not converge to the optimal value. Instead, it converges to a
Nash equilibrium. The reason x1 and x2 are not further changed is the presence of a
constraint that couples x1 and x2. Note that an initial guess of (x(0)1 , x
(0)2 ) = (0, 0),
for which x1 + x2 6= 1, however, leads to the optimal solution.
The decomposition methodology discussed in this section is based on dual de-
composition technique, and more specifically, on the approximate Newton directions
method [77]. It is an efficient decomposition strategy that allows each controllable
unit in the microgrid to optimize its set point considering both local and system-wide
constraints. This method offers improved computational efficiency, robustness, and
simplicity [78], [88] since subproblems are not needed to be solved until optimality;
this is in contrast to the common Lagrangian procedures that require the optimal
solution of each subproblem in each iteration [82].
Chapter 4. Generalized PFM Strategy 47
Consider the following optimization problem with two subproblems and local vari-
ables x1 and x2.
minimizex1,x2
φ(x1, x2)
subject to c1(x1, x2) = 0
c2(x1, x2) = 0
d1(x1) = 0
d2(x2) = 0,
(4.9)
where xi is the variable for each area, ci represents the coupled constraints, and di
represents the decoupled constraints. For brevity, assume only equality constraints
are present; extension to the case including inequality constraints is straightforward.
In the decomposition algorithm [77], the objective function of each subproblem
is modified to include the coupled constraints of other areas. To enable a separable
implementation, values from the previous iteration are used for nonlocal variables.
To ensure that the duality gap between the primal and dual problems is minimal,
the least possible number of constraints are included in each objective function, e.g.,
decoupled constraints are not included in the modified objective function [76]. The
two subproblems are shown below:
minimizex1
φ(x1, x∗2) + λ∗T2 c2(x1, x
∗2)
subject to c1(x1, x∗2) = 0
d1(x1) = 0
(4.10)
minimizex2
φ(x∗1, x2) + λ∗T1 c1(x∗1, x2)
subject to c2(x∗1, x2) = 0
d2(x2) = 0,
(4.11)
where x∗1 and x∗2 denote the values of x1 and x2 from the previous iteration, and λi is
the Lagrangian multiplier corresponding to constraint ci. Note that x1 appears only
in (4.10) and x2 appears only in (4.11). An update direction for each subproblem is
calculated, and the Lagrangian multipliers are updated by performing an iteration.
The salient feature of this method is that, instead of iterating to convergence, it
suffices to perform a single iteration to update variables. This significantly reduces
Chapter 4. Generalized PFM Strategy 48
the computation burden [79] and is the main difference between this method and a
standard Lagrangian approach.
For a general n-area optimization problem stated as
minimizexi;i=1,...,n
φ(x) =∑n
i=1 φi(x1, . . . , xn)
subject to ci(x1, . . . , xn) = 0
di(xi) = 0,
(4.12)
the subproblems are
minimizexi
φi(xi, x∗−i) +
∑nk=1,k 6=i λ
∗Tk ck(xk, x
∗−k)
subject to ci(xi, x∗−i) = 0
di(xi) = 0,
(4.13)
where x∗−i = (x∗1, . . . , x∗i−1, x
∗i+1, . . . , x
∗n) as defined before.
Details and proof of the convergence of this method can be found in [77]. The
algorithm is as follows:
Step 0: Initialization. Each agent i = 1, . . . , n initializes its variables and param-
eters xi and λi.
Step 1: Single Iteration. Each agent performs a single iteration for its subproblem
to obtain ∆xi and ∆λi.
Step 2: Updating. Each agent updates its variables and parameters as follows.
The information regarding these updates is distributed to other agents.
xi ← xi + ∆xi
λi ← λi + ∆λi(4.14)
Step 3: Stopping Criterion. The algorithm stops if variables do not change sig-
nificantly or if the time limit is reached.
Chapter 4. Generalized PFM Strategy 49
4.3 Formulation of GPFM
As mentioned earlier, the constraints in the GPFM formulation can include system-
wide constraints, e.g., equality constraints due to power flow equations and inequality
constraints due to line flow limits. Power flow equations represent the real and reactive
power balance at each bus. Line flow limits represent limits on current, real power, or
apparent power of a transmission line. There are four variables associated with each
bus: voltage magnitude, voltage angle, and real and reactive power injection from
that bus into the network. Vector x denotes the aggregate variables as defined below:
x =
∆
V
PG
QG
. (4.15)
With nb denoting the number of system buses and ng denoting the number of system
generators, ∆ and V are vectors of nb voltage phase angles and nb voltage magnitudes,
and PG and QG are vectors of ng real and ng reactive power injected at generator
buses.
The remainder of this section presents the formulation of GPFM for a distributed
and constrained implementation.
4.3.1 Objective Function
The objective function is the sum of potential functions, as defined in Chapter 2, of
all DER units:
φ(x) =
ng∑i=1
φi(x1, . . . , xn). (4.16)
Chapter 4. Generalized PFM Strategy 50
4.3.2 Constraints
The power flow constraints are as follows:
Si = Pi + jQi = (|Vi|]δi) I∗i
= Vi
(nb∑k=1
YikVk
)
Pi =
nb∑i=1
|ViVkYik| cos(θik + δk − δi)
Qi = −nb∑i=1
|ViVkYik| sin(θik + δk − δi),
(4.17)
where i = 1, . . . , nb is the bus number, Pi, Qi, and Si are real, reactive, and apparent
power injection from bus i, respectively, Vi and δi are the magnitude and phase angle
of voltage at bus i, and Ii is the current injection from bus i into the system. Yik
and θik are the magnitude and phase angle of the element at row i and column k
(gik + jbik) of the bus admittance matrix Ybus.
The first- and second-order derivatives are derived below in both nonvectorized
and vectorized forms. The vectorized form is used to as an efficient method for
representing the power flow constraints [89]. The nonvectorized form is employed for
coupled constraints.
4.3.2.1 Nonvectorized Derivation
The first-order derivatives of power flow constraints are as follows:
∂Si∂δk
= −|ViVkYik| sin(θik + δk − δi)− j|ViVkYik| cos(θik − δk − δi)
∂Si∂|Vk|
= |ViYik| cos(θik + δk − δi)− j|ViYik| sin(θik + δk − δi).(4.18)
Inclusion of coupled constraints requires calculation of the second-order derivatives.
The second-order derivatives of real power P are as follows:
∂2Pi∂x2
=
[∂2Pi
∂δ∂δ∂2Pi
∂δ∂V
∂2Pi
∂V ∂δ∂2Pi
∂V ∂V
], (4.19)
Chapter 4. Generalized PFM Strategy 51
where ∂2Pi
∂δi∂δm
∂2Pi
∂δ2m∂2Pi
∂δ2i
=
ViVm
(gim cos(δi − δm) + bim sin(δi − δm)
)ViVm
(− gim cos(δi − δm)− bim sin(δi − δm)
)−∑nb
n=1 ViVn
(gin cos(δi − δn) + bin sin(δi − δn)
) , (4.20)
and ∂2Pi
∂Vi∂δm
∂2Pi
∂Vm∂δi
∂2Pi
∂Vm∂δm
∂2Pi
∂Vi∂δi
=
Vm
(gim sin(δi − δm)− bim cos(δi − δm)
)Vi
(− gim sin(δi − δm) + bim cos(δi − δm)
)Vi
(gim sin(δi − δm)− bim cos(δi − δm)
)∑nb
n=1 Vn
(− gin sin(δi − δn) + bin cos(δi − δm)
)
, (4.21)
and ∂2Pi
∂Vi∂Vm
∂2Pi
∂Vi∂Vi
=
[gim cos(δi − δm) + bim sin(δi − δm)
2gii
]. (4.22)
The second-order derivatives of reactive power Q are as follows:
∂2Qi
∂x2=
[∂2Qi
∂δ∂δ∂2Qi
∂δ∂V
∂2Qi
∂V ∂δ∂2Qi
∂V ∂V
], (4.23)
where ∂2Qi
∂δi∂δm
∂2Qi
∂δ2m∂2Qi
∂δ2i
=
ViVm
(gim sin(δi − δm)− bim cos(δi − δm)
)ViVm
(− gim sin(δi − δm) + bim cos(δi − δm)
)−∑nb
n=1 ViVn
(gin sin(δi − δn)− bin cos(δi − δn)
) , (4.24)
and ∂2Qi
∂Vi∂δm
∂2Qi
∂Vm∂δi
∂2Qi
∂Vm∂δm
∂2Qi
∂Vi∂δi
=
−Vm
(gim cos(δi − δm) + bim sin(δi − δm)
)Vi
(gim cos(δi − δm) + bim sin(δi − δm)
)Vi
(gim cos(δi − δm) + bim sin(δi − δm)
)∑nb
n=1 Vn
(gin cos(δi − δn) + bin sin(δi − δm)
)
, (4.25)
Chapter 4. Generalized PFM Strategy 52
and ∂2Qi
∂Vi∂Vm
∂2Qi
∂Vi∂Vi
=
[gim sin(δi − δm)− bim cos(δi − δm)
−2bii
]. (4.26)
4.3.2.2 Vectorized Derivation
In this section, the shorthand notation [A] is used to represent the diagonal matrix
formed using the elements of vector A. The derivative of [A]B with respect to x is
[A]Bx + [B]Ax.
Using x as defined by (4.15) and representing the nb complex bus voltages with
vector V , we have
V|V | =∂V
∂|V |= [V ][|V |]−1
V∆ =∂V
∂∆= j[V ].
(4.27)
The complex power balance equations are expressed as
G(x) = Sbus + Sd − CgSg, (4.28)
where
Sbus = [V ]I∗bus, (4.29)
and Sd and Sg are vectors of load and generator power injections, and Cg is an nb×ngmatrix: the element cij is unity if generator j is connected to bus i and zero otherwise.
E is defined as [|V |]−1V .
The first-order derivatives are as follows:
G∆ =∂Sbus
∂∆
= [V ]∂I∗bus
∂∆+ [I∗bus]
∂V
∂∆
= [V ](Ybusj[V ]
)∗+ [I∗bus]j[V ]
= −j[V ]([V ∗]Ybus − [I∗bus]
),
(4.30)
Chapter 4. Generalized PFM Strategy 53
G|V | =∂Sbus
∂|V |
= [V ]∂I∗bus
∂|V |+ [I∗bus]
∂V
∂|V |= [V ]Y ∗bus[E
∗] + [I∗bus][E]
= [V ](Y ∗bus[V
∗] + [I∗bus])[|V |]−1,
(4.31)
and
GPG= −Cg
GQG= −Cg.
(4.32)
To avoid three-dimensional second-order derivatives, we write Gxy(λ) = ∂∂y
(GTxλ).
We have,
Gxx(λ) =
[G∆∆(λ) G∆|V |(λ)
G|V |∆(λ) G|V ||V |(λ)
]
=
[E + F
(jG(E − F)
)T
jG(E − F) G(C + CT)G
],
(4.33)
where auxiliary variables are defined below:
A = [λ][V ]
B = Ybus[V ]
C = AB∗
D = Y ∗Tbus[V ]
E = [V ∗](D[λ]− [Dλ]
)F = C − A[I∗bus]
G = [|V |]−1.
(4.34)
4.4 Primal-Dual Interior Point Solver
Implementation of the distributed PFM strategy requires availability of internal opti-
mization variables, e.g., Lagrangian multipliers, at each iteration. Therefore, a version
of the primal-dual interior point method [57] is developed to enable implementation
Chapter 4. Generalized PFM Strategy 54
of this work. Consider the general optimization problem
minimizex
f(x)
subject to g(x) = 0
h(x) ≤ 0,
(4.35)
where both linear and nonlinear constraints are present. There are ne (ne ≥ 0)
equality constraints g(x) and ni (ni ≥ 0) inequality constraints h(x). The common
approach to handle inequality constraints is to convert them into equality constraints
using a barrier function. Different barrier functions can be defined [74], [90]. In this
work, a logarithmic barrier function is used that includes slack variables for inequality
constraints. This increases the size of the problem, but improves its convergence [91].
Problem (4.35) is transformed to the following by adding the barrier function to
the objective function:
minimizex
f(x)− γ∑ni
i=1 log(zi)
subject to g(x) = 0
h(x) + z = 0
z ≥ 0.
(4.36)
As the parameter γ approaches zero, problem (4.36) approaches the original prob-
lem (4.35). The Lagrangian of problem (4.36) for a constant value of γ is
L(x, z, λ, µ) = f(x)− γni∑i=1
log(zi) + λTg(x) + µT(h(x) + z
), (4.37)
where λ and µ are Lagrangian multipliers for the equality and inequality constraints,
respectively.
Denote the aggregate optimization variable as u = (x, z, λ, µ). The first order
optimality conditions, Karush-Kuhn-Tucker (KKT), are
0 = Lx(u) = fx(x) + λTgx(x) + µThx(x)
0 = Lz(u) = µT − γ1T[z]−1
0 = Lλ(u) = gT(x)
0 = Lµ(u) = hT(x) + zT.
(4.38)
Chapter 4. Generalized PFM Strategy 55
The goal of optimization is to find the solution u∗ for which (4.38) holds. Assuming
that the current solution guess is u, the direction ∆u, for which u∗ ≈ u + ∆u, can
be approximated using the Newton method. This method is based on a second-order
approximation of the Lagrangian. Note that for some function r(u), r(u + ∆u) =
r(u) + rTu∆u and ruu = rT
uu. Equation (4.38) is linearized at u:
0 = fx(x) + fxx(x)∆x+ (λ+ ∆λ)Tgx(x+ ∆x) + (µ+ ∆µ)Thx(x+ ∆x)
0 = (µ+ ∆µ)T − γ1T[z + ∆z]−1
0 = gT(x) + gTx (x)∆x
0 = hT(x) + hTx (x)∆x+ (z + ∆z)T.
(4.39)
This can be written in matrix form asLxx 0 gx hx
0 [µ] 0 [z]
gTx 0 0 0
hTx I 0 0
∆x
∆z
∆λ
∆µ
= −
LTx
[µ]z − γ1
g(x)
h(x) + z
, (4.40)
where
Lx = fx + λTgx + µThx
Lxx = fxx + λTgxx + µThxx.(4.41)
This set of equations is solved iteratively to find the update direction ∆u. However,
to reduce the problem size and improve algorithm speed, the problem is reformulated
as follows. From the second and fourth rows of (4.40), we have3
[µ]∆z + [z]∆µ = −[µ]z + γ1
hTx∆x+ ∆z = −h(x)− z.
(4.42)
This is solved for ∆µ and ∆z to obtain
∆µ = −µ+ [z]−1(γ1− [µ]∆z
)∆z = −h(x)− z − hT
x∆x,(4.43)
3For simplicity, the variable of a function is not explicitly mentioned if it is clear from the context.
Chapter 4. Generalized PFM Strategy 56
where it is used that [µ]z = [z]µ. Substituting ∆µ from (4.43) in the first row of the
matrix equation (4.40) results in
−LTx = Lxx∆x+ gx∆λ+ hx∆µ
= Lxx∆x+ gx∆λ+ hx(−µ+ [z]−1(γ1− [µ]∆z)
).
(4.44)
Substituting ∆z from (4.43) in (4.44) and simplifying the resulting expression gives
M∆x+ gTx ∆λ = −N, (4.45)
where
M = Lxx + hx[z]−1[µ]hTx
= fxx + λTgxx + µThxx + hx[z]−1[µ]hTx ,
(4.46)
and
N = LTx + hx[z]−1[µ]
(γ1 + [µ]hT(x)
)= fT
x + λTgx + µThx + hx[z]−1(γ1 + [µ]hT(x)
).
(4.47)
In matrix form [M gx
gTx 0
][∆x
∆λ
]= −
[N
g(x)
]. (4.48)
The update direction is calculated by computing ∆x and ∆λ from (4.48) and then
solving for ∆z and ∆µ from (4.43). Variables (x, z, λ, µ) are updated as follows:x
z
λ
µ
←x
z
λ
µ
+
αp∆x
αp∆z
αd∆λ
αd∆µ
, (4.49)
Chapter 4. Generalized PFM Strategy 57
7
DG3
8 9
11
Fig. 4.2. Study system I decomposed into two areas.
where αp and αd are adjustment factors calculated as [91]
αp = min
(ξ min
∆zi<0
(zi−∆zi
), 1
)αd = min
(ξ min
∆µi<0
(µi−∆µi
), 1
),
(4.50)
where ξ is slightly less than unity (0.99995 in our implementation). Parameter γ is
updated as [91]
γ ← σzTµ
ni, (4.51)
where σ is 0.1.
4.5 Application Example I
4.5.1 Study System I
This study system is the 11-bus system of Section 3.2. It has three DER units modeled
as electronically interfaced voltage-controlled units. The system is decomposed into
two areas as shown in Fig. 4.2. Bus 6 is common between these two areas.
The simulation of the power system is performed in the PSCAD/EMTDC soft-
ware, the GPFM algorithm is implemented in the MATLAB programming language,
and the two are interfaced via the method discussed in [68]. A set point update rate
of 0.2 s is chosen.
Chapter 4. Generalized PFM Strategy 58
4.5.2 Potential Function as the Objective Function
The objective function is defined in (4.52). It is composed of six potential functions.
The potential functions in the form (Vi − ri)2 are of circle type and represent control
goals for each DER unit. The potential functions in the form ‖Vi]δi − Vj]δj‖ are of
point type and represent unit measurements:
φ = (V1 − r1)2 + (V2 − r2)2 + (V3 − r3)2
+ ‖V1]δ1 − V2]δ2‖2
+ ‖V1]δ1 − V3]δ3‖2
+ ‖V2]δ2 − V3]δ3‖2
= (V1 − r1)2 + (V2 − r2)2 + (V3 − r3)2
+ (V1 cos δ1 − V2 cos δ2)2 + (V1 sin δ1 − V2 sin δ2)2
+ (V1 cos δ1 − V3 cos δ3)2 + (V1 sin δ1 − V3 sin δ3)2
+ (V2 cos δ2 − V3 cos δ3)2 + (V2 sin δ2 − V3 sin δ3)2 ,
(4.52)
where Vi and ri show the measured and desired magnitude of the voltage at each
DER bus, respectively, and δi denotes the measured voltage phase angle. The ob-
jective function defined for this problem can be modified according to the control
goals of individual problems. The derivations performed in this section serve as a
representative case.
The first-order derivatives are derived below, where x denotes the variables con-
trolled by the potential function and is defined as x = (δ1, δ2, δ3, V1, V2, V3):
∂φ
∂x=
∂φ∂δ1
∂φ∂δ2
∂φ∂δ3
∂φ∂V1
∂φ∂V2
∂φ∂V3
=
2V1V2 sin(δ1 − δ2) + 2V1V3 sin(δ1 − δ3)
2V2V3 sin(δ2 − δ3) + 2V1V2 sin(δ2 − δ1)
2V1V3 sin(δ3 − δ1) + 2V2V3 sin(δ3 − δ2)
2(V1 − r1) + 4V1 − 2V2 cos(δ1 − δ2)− 2V3 cos(δ1 − δ3)
2(V2 − r2) + 4V2 − 2V1 cos(δ2 − δ1)− 2V3 cos(δ2 − δ3)
2(V3 − r3) + 4V3 − 2V1 cos(δ1 − δ3)− 2V2 cos(δ2 − δ3)
. (4.53)
Chapter 4. Generalized PFM Strategy 59
The second-order derivatives are as follows:
∂2φ
∂x2=
∂2φ∂δ21
∂2φ∂δ1∂δ2
∂2φ∂δ1∂δ3
∂2φ∂δ1∂V1
∂2φ∂δ1∂V2
∂2φ∂δ1∂V3
∂2φ∂δ2∂δ1
∂2φ∂δ22
∂2φ∂δ2∂δ3
∂2φ∂δ2∂V1
∂2φ∂δ2∂V2
∂2φ∂δ2∂V3
∂2φ∂δ3∂δ1
∂2φ∂δ3∂δ2
∂2φ∂δ23
∂2φ∂δ3∂V1
∂2φ∂δ3∂V2
∂2φ∂δ3∂V3
∂2φ∂V1∂δ1
∂2φ∂V1∂δ2
∂2φ∂V1∂δ3
∂2φ∂V 2
1
∂2φ∂V1∂V2
∂2φ∂V1∂V3
∂2φ∂V2∂δ1
∂2φ∂V2∂δ2
∂2φ∂V2∂δ3
∂2φ∂V2∂V1
∂2φ∂V 2
2
∂2φ∂V2∂V3
∂2φ∂V3∂δ1
∂2φ∂V3∂δ2
∂2φ∂V3∂δ3
∂2φ∂V3∂V1
∂2φ∂V3∂V2
∂2φ∂V 2
3
. (4.54)
Note that because of symmetry, ∂2φ∂xi∂xj
= ∂2φ∂xj∂xi
; therefore, only one in each pair is
shown below.
∂2φ∂V 2
1
∂2φ∂V1∂V2
∂2φ∂V1∂V3
∂2φ∂V1∂δ1
∂2φ∂V1∂δ2
∂2φ∂V1∂δ3
=
6
−2 cos(δ1 − δ2)
−2 cos(δ1− δ3)
2V2 sin(δ1 − δ2) + 2V3 sin(δ1 − δ3)
−2V2 sin(δ1 − δ2)
−2V3 sin(δ1 − δ3)
, (4.55)
∂2φ∂V 2
2
∂2φ∂V2∂V3
∂2φ∂V2∂δ1
∂2φ∂V2∂δ2
∂2F∂V2∂δ3
=
6
−2 cos(δ2 − δ3)
−2V1 sin(δ2 − δ1)
2V3 sin(δ2 − δ3) + 2V1 sin(δ2 − δ1)
−2V3 sin(δ2 − δ3)
, (4.56)
∂2φ∂V 2
3
∂2φ∂V3∂δ1
∂2φ∂V3∂δ2
∂2φ∂V3∂δ3
=
6
−2V1 sin(δ3 − δ1)
−2V2 sin(δ3 − δ2)
2V1 sin(δ3 − δ1) + 2V2 sin(δ3 − δ2)
, (4.57)
Chapter 4. Generalized PFM Strategy 60
∂2φ∂δ21∂2φ
∂δ1∂δ2
∂2φ∂δ1∂δ3
=
2V1V2 cos(δ1 − δ2) + 2V1V3 cos(δ1 − δ3)
−2V1V2 cos(δ1 − δ2)
−2V1V3 cos(δ1 − δ3)
, (4.58)
[∂2φ∂δ22∂2φ
∂δ2∂δ3
]=
2V1V2 cos(δ1 − δ2) + 2V2V3 cos(δ2 − δ3)
−2V2V3 cos(δ2 − δ3)
, (4.59)
[∂2φ∂δ23
]=[2V1V3 cos(δ1 − δ3) + 2V2V3 cos(δ2 − δ3)
]. (4.60)
The first- and second-order derivatives calculated above are added to the gradient
and Hessian of the Lagrangian, respectively.
4.5.3 Study Results
The performance of GPFM is evaluated in the islanded mode of operation of System I.
The set points of the DER units are step changed simultaneously from 0.90 pu to
1.05 pu. The goal, implied in the objective function, is to provide the trajectory of
the system in this transition while meeting the constraints on transmission lines by
crafting intermediate set points between the initial and final set points.
As a result of this step change, GPFM changes the set points for vd and vq of the
three DG units in successive steps until they reach their final set points. Fig. 4.3(a)
shows the intermediate set points provided by GPFM and the measured magnitude of
DG1 voltage. The d- and q-components of DG1 voltage are shown in Fig. 4.3(b)–(c).
The voltage of PC1 is sinusoidal both before and after the step change, as shown in
Fig. 4.3(d).
The trajectory of PC1 voltage is shown in the dq-plane in Fig. 4.4; it shows that
the voltage magnitude is regulated with a higher priority than the voltage phase
angle. This is expected because the potential function reflects more emphasis on
the regulation of the voltage magnitude than on the regulation of the voltage phase
angle. For comparison, the trajectory of PC1, PC2, and PC3 voltages (corresponding
to DG1, DG2, and DG3, respectively) is shown in Fig. 4.5. It can be seen that the
difference between instantaneous values of voltages is negligible along the trajectory.
Chapter 4. Generalized PFM Strategy 61
0 0.2 0.4 0.6 0.8 1
0.90.95
11.05
(a)
v 1,m
ag (
pu)
Set PointResponse
0 0.2 0.4 0.6 0.8 1
0.90.95
11.05
(b)
v 1d (
pu)
Set PointResponse
0 0.2 0.4 0.6 0.8 1
−0.02
0
0.02
(c)
v 1q (
pu)
Set PointResponse
0 0.2 0.4 0.6 0.8 1
−1
0
1
(d)
v 1 (pu
)
0 0.2 0.4 0.6 0.8 1
−2
0
2
(e)
/_ v
1 (de
gree
)
Time (ms)
Fig. 4.3. System I: PC1 voltage in response to a step change from 0.90 pu to 1.05 puunder GPFM control. (a) voltage magnitude; (b) d-component of voltage; (c)q-component of voltage; (d) phase a voltage; (e) voltage phase angle.
Chapter 4. Generalized PFM Strategy 62
0.6 0.7 0.8 0.9 1 1.1
−0.05
0
0.05
vd (pu)
v q (pu
)
Fig. 4.4. System I: Locus of voltage of PC1.
0.9 1 1.05
0
vd (pu)
v q (pu
)
−1 deg
−0.5 deg
0 deg
0.5 deg
1 deg
DG1DG2DG3
Fig. 4.5. System I: Locus of voltage of PC1, PC2, and PC3.
Chapter 4. Generalized PFM Strategy 63
G1
G2
G4
G3
1
6
10
9
2
12
5 4
7 8
3
11
Fig. 4.6. Study system III: Twelve-bus system with four generators.
4.6 Application Example II
4.6.1 Study System III
The adopted study system is the CIGRE 12-bus test system as illustrated in Fig. 4.6.
The system parameters are provided in [92]. This study system demonstrates that
the proposed control strategy is also applicable to conventional power systems with
integrated synchronous generators instead of integrated electronically interfaced DER
units.
GPFM utilizes a static (steady-state) model of the system using power flow equa-
tions; however, a detailed dynamic model of the power system is implemented to
evaluate the performance of GPFM in more details. This model and the developed
software tool are discussed in Appendix C.3. The study system has four synchronous
generators of which three are salient-pole generators and one is a round-pole gen-
erator. The dynamics of synchronous generators with their exciters, power system
stabilizers, and damper windings are considered. Each generator is represented by
a sixth-order model (one field winding, one damper winding along the d-axis, two
damper windings along the q-axis, rotor angle, and rotor speed). Each AVR is rep-
resented by a fifth-order model (a second-order PSS, a first-order terminal voltage
transducer, and a second-order thyristor-fed exciter). Each load is represented by an
aggregate model consisting of a static component (ZIP) and a dynamic component
(exponential recovery). The real component of each load is modeled as constant-
current, and the reactive component of each load is modeled as constant-impedance.
Chapter 4. Generalized PFM Strategy 64
Table 4.1Study system III: Predisturbance steady-state operating conditions
Bus Type |V | (pu) ]V Pg (MW) Qg (MVAr) Pd (MW) Qd (MVAr)
1 PQ 1.0412 −2.69 0.00 0.00 0.00 0.002 PQ 1.0041 −0.95 0.00 0.00 280.00 200.003 PQ 0.9927 −37.88 0.00 0.00 320.00 240.004 PQ 0.9805 −42.99 0.00 0.00 320.00 240.005 PQ 0.9980 −30.79 0.00 0.00 100.00 60.006 PQ 0.9951 −34.31 0.00 0.00 440.00 300.007 PQ 1.0493 −4.42 0.00 0.00 0.00 0.008 PQ 0.9995 −36.09 0.00 0.00 0.00 0.009 Sl 1.0400 0.00 509.12 0.00 0.00 0.0010 PV 1.0200 1.85 500.00 173.97 0.00 0.0011 PV 1.0100 −36.74 200.00 176.80 0.00 0.0012 PV 1.0200 −30.92 300.00 135.63 0.00 0.00
Conventional quasi–steady state assumptions are made, i.e., the network is repre-
sented by phasor equations forming a complex-valued admittance matrix Ybus. The
power system has 68 state variables. It is assumed that measurements are available
to the controller through either a SCADA system or a state estimator. Therefore, all
components are assumed to be in a single area.
It should be emphasized that the dynamics of the system are not considered in
GPFM, nor is GPFM aware of the load modeling approach. GPFM employs the
steady-state model of the power system. The primary objective of GPFM in this
application example is to craft the trajectory of the power system in response to
a disturbance while minimizing the deviation from the original state. The potential
function is defined to reflect this objective. Table 4.1 gives the predisturbance steady-
state bus voltage magnitudes, voltage phase angles, and real and reactive components
of load and generator powers. The predisturbance state of the system is assumed to
be available; its calculation does not fall into the scope of this work. The test case [92]
includes this information, which is obtained through optimization of some measure
of the system performance.
In this application example, both the power system model and GPFM are imple-
mented in the MATLAB programming language. The differential equations of the
power system are solved using Gill’s version of the fourth-order Runge-Kutta method
(RKG4) [93], which has minimized round-off errors and requires less storage than the
original RK4 method. GPFM provides set point updates every 1 s.
Chapter 4. Generalized PFM Strategy 65
4.6.2 Potential Function as the Objective Function
The objective of the developed potential function is to minimize the deviation of
the generator set points from the predisturbance set points while maintaining bus
voltages within the permissible limits subject to power flow constraints. To maintain
the predisturbance voltage profile, a point-type potential function is used for each
generator. Since the internal dynamics of the generators are not available to GPFM, it
uses the terminal voltages, and not the generator voltage references, in its formulation.
The mapping of terminal voltages to generator voltage references is explained in the
next section. Each potential function is defined as
φj = wj(Vj − Vj0)2, (4.61)
for j = 1, . . . , ng. Weight factor wj represents the relative cost of adjustment of
the reference of each generator. In the presented case study, wj = 1. Vj0 is the
predisturbance bus voltage.
The first- and second-order derivatives are required for the calculation of the
gradient and Hessian of the Lagrangian. The first-order derivatives are
∂φj∂Vi
=
2wj(Vj − Vj0), i = j
0, i 6= j,(4.62)
and the second-order derivatives are
∂2φj∂Vi∂Vm
=
2wj, i = j = m
0, otherwise.(4.63)
4.6.3 Modeling and Implementation Considerations
In GPFM, the transmission lines are represented by power flow equations. In simulat-
ing the dynamics of other power system components, the following model is employed
z = f(z, V )
YbusV = Iinj(z, V ),(4.64)
Chapter 4. Generalized PFM Strategy 66
where z is the vector of states (whose time evolution is given by a set of differential
equations), and V is the vector of bus voltage magnitudes.
GPFM can be expressed as an optimization problem, i.e.,
minimizex,u
∑nj=1 wj(uj − uj0)2
subject to g(x, u) = 0
xmin ≤ x ≤ xmax
umin ≤ u ≤ umax
|u− u0| ≤ ∆umax.
(4.65)
In this formulation, the first set of constraints represents the power flow equations of
the postdisturbance system. Other constraints are the limits on vectors x and u. x is
the vector of power flow variables. Associated with each bus are four variables—phase
angle, voltage magnitude, and real and reactive power generation. u is the vector of
n controlled variables, and wj represents the relative weight of making adjustments
to variable uj. u0 represents the predisturbance value of u. Of the nb bus voltage
magnitudes V , ng correspond to generator buses, are controllable, and are included
in vector u. The remaining elements of V correspond to the load buses, are not
controllable, and are included in vector x. The maximum change in control variables
u is limited to avoid a large disturbance in the system.
In the present application example, only generator voltage references are assumed
to be controllable. However, it is possible to consider other control means, e.g., load
shedding, shunt susceptance, and transformer tap changer settings. Representing
these apparatus in (4.65) is straightforward. The formulation in terms of summation
distributes the control action among different units, which increases the reliability
should a unit fail to implement its command.
GPFM provides intermediate and final set points in response to a disturbance.
The interface between GPFM and the power system is through generator voltage
references. However, GPFM employs a steady-state model of the system that uses
terminal bus voltages. Ideally, terminal voltages and AVR (generator) reference volt-
ages are equal; however, in practice, there is a steady-state error between the AVR
reference voltage and the generator terminal voltage [94]. Therefore, the terminal
voltages obtained from GPFM cannot be directly used as generator voltage refer-
ences. To deal with this discrepancy without having to resort to inclusion of the
Chapter 4. Generalized PFM Strategy 67
system dynamics in GPFM, which will prohibitively increase the problem size, it is
assumed that a change in the voltage reference is proportional to a change in the
terminal voltage. Therefore, the voltage values provided by GPFM are used to cor-
rect the reference values, i.e., they are not directly used as reference values. This is
implemented in the GPFM strategy by updating the voltage references based on
V updateref = V measured
ref +(V GPFM
terminal − V measuredterminal
) V predisturbanceref
V predisturbanceterminal
. (4.66)
Not all voltage phase angles can be given as optimization variables, and one (in
this case, bus 9) has to assume a designated value. In the steady state, it is customary
to assign zero degrees to this phase angle. However, as the rotor angle of a generator
changes in transients, this choice is not valid during dynamics. Prior to each GPFM
run, the voltage phase angle of bus 9 is measured and used throughout the GPFM
update.
Bus voltages are limited between 0.90 and 1.10 pu. However, if a voltage is
too low, a large increase in its set point could result in unstable behavior of the
power system. Therefore, the maximum voltage adjustment in each step is limited to
0.05 pu. A similar statement holds for a bus voltage that is too high. These limits
are incorporated in the GPFM formulation by adjusting Vmin and Vmax for each bus
voltage by
Vmin = min0.90, V + 0.05
Vmax = max1.10, V − 0.05.(4.67)
This ensures that large set point changes that can lead to system instability are
avoided.
4.6.4 Case Study A: Load Change
This case study evaluates the performance of GPFM in response to a load change.
The voltage at generator buses 9, 10, 11, and 12 is controlled to maintain the pre-
disturbance voltage profile. The predisturbance load and voltage values are given in
Table 4.1.
At t = 11 s, all loads are reduced by 10%. Fig. 4.7 shows the evolution of voltage
in the system. For clarity, only the generator bus voltages are shown.
Chapter 4. Generalized PFM Strategy 68
11 20 40 60 80 100 120 1401.00
1.02
1.04
1.06
1.08
(a)
V (
pu)
11 20 40 60 80 100 120 1401.00
1.02
1.04
1.06
1.08
(b)Time (s)
V (
pu)
V
9
V10
V11
V12
Fig. 4.7. System III: Voltage at generator buses subsequent to a 10% load reduction.Traces of V10 and V12 are similar and overlap. (a) without GPFM; (b) with GPFM.
In response to this disturbance, GPFM updates the generator voltage set points.
As a result, the voltage profile reaches the steady state in 50 s, Fig. 4.7(b). Without
GPFM, the voltage profile settles much slower and even after 140 s, the original
voltage magnitudes are not recovered, Fig. 4.7(a). Therefore, GPFM significantly
reduces the settling time—by 64%.
4.6.5 Case Study B: Line Outage
This case study evaluates the performance of GPFM in response to a line outage.
The controller is triggered when a topological change is introduced. Similar to the
previous case, the objective of GPFM is to maintain the predisturbance voltage of
the generator buses.
At t = 11 s, the line between buses 4 and 5 (Fig. 4.6) is disconnected. The line is
reconnected after 1.5 s at t = 12.5 s. Fig. 4.8 shows the evolution of voltage in the
system. For clarity, only the voltage of bus 9 is shown.
In response to this disturbance, GPFM updates the generator voltage set points.
In both cases (without and with GPFM) the voltage reaches the steady state. How-
ever, when GPFM is not active, Fig. 4.8(a), a small error exists between the pre-
Chapter 4. Generalized PFM Strategy 69
11 20 40 60 80
1.038
1.040
1.042
(a)
V (
pu)
11 20 40 60 80
1.038
1.040
1.042
(b)Time (s)
V (
pu)
V
9
Fig. 4.8. System III: Voltage at generator buses subsequent to the outage of line 4-5. (a)without GPFM; (b) with GPFM.
and postdisturbance voltage values. When GPFM is active, Fig. 4.8(b), the voltage
values are corrected much faster, i.e., in its first set point update.
4.6.6 Case Study C: Line Outage and Controller Failure
This case study evaluates the performance of GPFM in response to both a line outage
and a failure in voltage reference communication. The objective and implementation
of GPFM are similar to the previous two cases.
At t = 11 s, the line between buses 4 and 5 (Fig. 4.6) is disconnected. The line
is reconnected after 1.5 s at t = 12.5 s. Also, at t = 11 s, the voltage reference is
changed to 0.3 pu, which can be due to AVR malfunctioning, a wrongly calculated
set point, or corruption in the communication link. In this case, simply restoring the
set point to the correct value is not a viable solution, as a large change in a set point
can destabilize the system. Fig. 4.9 shows the evolution of the system voltage profile.
In Fig. 4.9(a), the set point is restored after 20 s without GPFM, which results in
oscillatory behavior of the voltages; in Fig. 4.9(b), GPFM corrects the set point.
Subsequent to the line removal, GPFM updates the generator voltage references.
Chapter 4. Generalized PFM Strategy 70
11 20 40 60 80 100 120 140
0.40
0.60
0.80
1.00
1.20
(a)
V (
pu)
11 20 40 60 80 100 120 140
0.40
0.60
0.80
1.00
1.20
(b)Time (s)
V (
pu)
V
9
V10
V11
V12
Fig. 4.9. System III: Voltage at generator buses subsequent to the outage of line 4-5 anda change in voltage reference. Traces of V10 and V12 are similar and overlap. (a) withoutGPFM restoring the set point; (b) with GPFM.
GPFM applies a series of changes in voltage references and restores a stable voltage
profile, Fig. 4.9(b). This results in a settling time of about 90 s.
Note that after the fault, the voltages decrease significantly. However, GPFM
does not apply a voltage correction larger than 0.05 pu in the system. This is a
compromise between stability of the system response and its settling time.
4.6.7 Case Study D: Line Outage, Controller Failure, and
Missed Updates
This case study is similar to case study C but a permanent malfunctioning of the con-
troller of generator 4 at bus 12 is also considered. As a result of this malfunctioning,
the set points that GPFM calculates for generator 4 are not implemented. However,
GPFM is not aware of this failure. This case study is conducted to demonstrate the
robustness of GPFM; in a practical case, generator 4 trips following this event.
Fig. 4.10(a) shows the response of the system without GPFM in which the voltages
collapse at about t = 40 s. Fig. 4.10(b) shows the response of the system with GPFM.
Although the voltage of generator 4 is low and exhibits an oscillatory behavior, the
Chapter 4. Generalized PFM Strategy 71
11 20 40 60 80 100 120 140
0.40
0.60
0.80
1.00
1.20
(a)
V (
pu)
11 20 40 60 80 100 120 140
0.40
0.60
0.80
1.00
1.20
(b)Time (s)
V (
pu)
V
9
V10
V11
V12
Fig. 4.10. System III: Voltage at generator buses subsequent to the outage of line 4-5and a change in voltage set point while updates for generator 4 are not implemented.
voltages of the other three generators are successfully controlled (with some ripple
within the permissible limits). This case study confirms the robustness of GPFM
with respect to failures in implementation of the set points. It should be noted that
this scenario is among the most severe scenarios that may occur.
4.7 Conclusions
In this chapter, the PFM framework introduced in Chapters 2 and 3 is extended to
account for both system-wide and local constraints. This generalized PFM (GPFM)
can be implemented in a distributed manner. Formulation of GPFM is presented, and
case studies are provided to demonstrate implementation of voltage control problem
in the GPFM formulation. These case studies confirm the effectiveness of GPFM in
maintaining and enforcing the satisfactory and feasible operation of the power system.
The developed GPFM framework is general. That is, even though it is developed
for the proposed potential function–based control strategy, it can also be employed
for other power system studies, e.g., standard OPF and distributed optimization of
the power system.
Chapter 5
Online Set Point Adjustment for
Trajectory Shaping
5.1 Introduction
Power system apparatus are increasingly operated closer to their operational limits
to achieve a higher degree of utilization. Power systems, too, are subjected to more
stringent operational constraints and control strategies to maximize technical and
economical utilization due to incentives such as environmental awareness. As a result,
transients may drive the power system close to its operational limits and may cause
violation of operational limits and instability.
This problem is exacerbated in a small-scale power system such as an active dis-
tribution system (ADS) or a microgrid [8], [95], in which resources such as power
generation capacity are relatively limited. In these systems, due to relatively fre-
quent changes in the in-service status of units, a controller designed and tuned for one
system configuration may not perform well for another configuration. Moreover, the
well-utilized characteristics of large power systems, e.g., abundance of inertia provided
by synchronous generators that contributes to system damping, do not necessarily ap-
ply. Thus, the problem of effective and fast mitigation of transients, especially when
Portions of this chapter are published as
[33] A. Mehrizi-Sani and R. Iravani, “Online set point adjustment for trajectory shaping in micro-grid applications,” IEEE Trans. Power Syst., Oct. 2010, accepted for publication (paper no.TPWRS-00823-2010).
72
Chapter 5. Online Set Point Adjustment for Trajectory Shaping 73
the operating point is close to the limits, is of more significance in small-scale power
systems.
In the literature, several methods have been proposed for controller design:
• Systematic design [96], [97]: In this approach, a comprehensive small-signal
model of the microgrid is developed. This model includes the transmission
lines, controllers, filters, generators, and loads. The controllers are designed
using conventional linear and nonlinear control methods, e.g., root locus and
sequential loop closing [98].
• Optimization-based design [99]–[101]: In this approach, the controller parame-
ters are designed using optimization of the system performance through a series
of runs in a simulation software, e.g., PSCAD/EMTDC [67], [102]. The accuracy
of this approach depends on the model and its parameters and the convergence
behavior of the optimization algorithm.
The drawbacks of these approaches are that (i) they rely on the availability of system
models and parameters and (ii) once designed, the controller parameters are appro-
priate only for a specific operating region. When the parameters of the host system
change beyond a certain tolerance, e.g., as a result of a change in load or short circuit
ratio (SCR) of the system, the devised controller parameters become irrelevant. This
hampers the performance of the controller and poses a problem for effective and fast
mitigation of transients, especially when the operating point of a unit is already close
to its limits.
For a modular design that promotes plug-and-play operation in a microgrid, the
aforementioned conventional design methods may be neither the best nor even ade-
quate. Therefore, more applicable are design methods that do not require a detailed
model of the system and are relatively insensitive to changes in the system parame-
ters. Online adjustment of PI-controller gains [103] and application of game theory
for controller design [104] have been reported in the literature. However, the perfor-
mance of the former is limited by the initial choice of controller parameters, and the
operation of the latter is limited to a dc power system with controllable loads.
This chapter recognizes the drawbacks of these approaches and proposes an alter-
native strategy to enhance the set point tracking of controllable power system devices.
The proposed strategy uses local measurements to improve set point tracking through
set point modulation. The term set point in this context refers to the reference value
Chapter 5. Online Set Point Adjustment for Trajectory Shaping 74
for a power system variable, e.g., voltage, current, and power, that a controller acts
upon. The term modulation in this context refers to an adaptive strategy that mon-
itors the response of the system and, based on its trend and sampled values, adjusts
the set point. The proposed strategy is augmented to an existing controller to im-
prove its performance. Instead of applying the set point calculated by the higher level
of control, e.g., secondary control, to the controlled device, the set point is input to
the proposed strategy, which adjusts the set point before passing it to the device. In
the steady state, the applied and original set points are equal.
The proposed strategy enables closer adherence to the dynamic rating of the power
system devices. This strategy can be implemented based on either set point auto-
matic adjustment (SPAA)1 or set point automatic adjustment with correction enabled
(SPAACE).2 This strategy is most useful for the cases that a controller is designed
(and performs satisfactorily) but its performance is deteriorated because of a topolog-
ical change in the system. This is a likely scenario in small-scale power systems, e.g.,
microgrids, that have limited resources. Both SPAA and SPAACE are autonomous,
i.e., they do not need a centralized unit nor do they need a communication channel.
A comparison of SPAA and SPAACE is provided in Section 6.3.
To the best of the author’s knowledge, this is the first time set point modulation
for improving set point tracking is proposed and reported in the power system or
control literature. The salient features of the proposed strategy are as follows:
• It is independent of the control strategy employed for the existing controllers;
• It is robust to changes in system parameters and configuration; and
• It does not need a communication link.
SPAA, which is explained in this chapter, takes advantage of a (reduced-order)
model of the system to design intermediate set points such that the response to each
set point change is satisfactory. SPAA is best suited for mitigating large disturbances
such as the start-up operation.
SPAACE, which is explained in detail in the next chapter, provides virtual damp-
ing for mitigating the transients of a system. To achieve the desired response tra-
jectory for a device, SPAACE observes its trajectory and temporarily manipulates
1Pronounced [spO:].2Pronounced [speIs].
Chapter 5. Online Set Point Adjustment for Trajectory Shaping 75
the set point based on the pattern of behavior, instantaneous value, and operational
limits identified for the response. SPAACE is also capable of mitigating transients
caused by load switching and remote faults. Therefore, SPAACE attempts to bridge
the gap between control and protection by explicitly accounting for the operational
constraints. A software tool is developed to study the behavior of SPAACE; this
software is discussed in Appendix C.2.
5.2 Concepts and Definitions
This section introduces the concepts employed and/or introduced in this chapter.
5.2.1 Predict-Prevent-Publish Paradigm (P4)
This work is inspired by a philosophy referred to here as the predict-prevent-publish
paradigm (P4). In this approach, the response of a system to a change in its set points
is predicted, intermediate set points are crafted to prevent undesired transients, and
the resulting set points are then published to the controllers.
There are several ways to implement the proposed strategy. SPAACE can be
used independently as a predictive-corrective strategy that continuously corrects for
the undesired trajectory excursions. SPAACE can also be used in conjunction with
SPAA in a scheme similar to the model predictive control (MPC) [86], [105] that
consists of a prediction stage and a correction stage: SPAA devises set points by
predicting the system response, and SPAACE corrects for the discrepancy between
the model employed by SPAA and the actual system. Potential function–based control
(PFM) [28] can also be used with SPAACE.
5.2.2 Primary and Secondary Control
The concepts of primary and secondary (and tertiary) control are explained in Chap-
ter 1. As illustrated in Fig. 5.1, SPAA and SPAACE serve as an intermediate step
between the primary and secondary controllers. The set points provided by the sec-
ondary controller are fed into the SPAA/SPAACE block that adjusts the set points
as required. The resulting set points are then input to the primary controller of the
respective unit.
Chapter 5. Online Set Point Adjustment for Trajectory Shaping 76Secondary Controller SPAA orSPAACE Primary Controller Unitxsetpoint x(t)x(t)Regional DataFig. 5.1. Primary and secondary controllers and SPAA and SPAACE.
Δt
|V| (pu)
10 s0.5 s20 ms
1.0
0.8
1.2
1.4
1 ms3 ms
No Interruption
Prohibited Region
No Damage
Fig. 5.2. ITI curve (time axis is not to scale).
5.2.3 Region of Acceptable Dynamic (ROAD) Operation
A characteristic of an electric device is its steady-state rating; nevertheless, it can
withstand short-term violations of this rating. This characteristic is employed, for
example, in relay coordination where a relay reacts to an overcurrent only if its du-
ration exceeds a certain period. Other examples are HVDC operational curves [106],
ratings of power electronic switches, standards and operational requirements, and the
ITI (Information Technology Industry) curve [107] (Fig. 5.2).
We represent this characteristic as a region of acceptable dynamic (ROAD) op-
eration that shrinks or expands based on the time frame of interest, Fig. 5.3(a).
Invariably, this region for the steady state is more restrictive than the region for
transients; therefore, the ROAD for the steady state is a subset of the ROAD for
transients. For example, although a 20% overvoltage is permissible for a duration of
20 ms, only a 10% overvoltage is permissible for the steady state. Fig. 5.3(b) shows
the one-dimensional ROAD curve for overvoltage/undervoltage that is produced from
the ITI curve. It is also possible to develop curves of higher dimensions.
Chapter 5. Online Set Point Adjustment for Trajectory Shaping 77
|V| (pu)0 1 1.10.90.8 1.2 ∆t = ∞ ∆t = 0.5 s 0.7≈ (b)x1
x2 ∆t =∞∆t =T1(a)
Fig. 5.3. Region of acceptable dynamic (ROAD) operation curve. (a) two-dimensionalgeneric ROAD curve, where x1 and x2 are dynamic variables; (b) one-dimensional ROADcurve pertaining to the ITI curve.
Chapter 5. Online Set Point Adjustment for Trajectory Shaping 78
5.2.4 Communication Requirements
In the current implementation, SPAA and SPAACE are associated with one unit and
can be placed physically within or adjacent to that unit, thereby eliminating the need
for a communication link. It is, however, possible to coordinate SPAA/SPAACE for
multiple units, in which case, they require a communication link to the units they
control. Considering the ongoing developments in the communication technology [7]
and the smart grid vision [1], this requirement is not expected to be a major hurdle
in the implementation of the proposed strategy. In the case of communication failure,
the units can automatically revert to the latest update received from the secondary
control to maintain their operation.
5.3 Set Point Automatic Adjustment (SPAA)
5.3.1 Objective
When the set point of a device changes, its response x(t) may momentarily exceed
the minimum/maximum limits. The objective of SPAA is to devise intermediate
set points such that the response x(t) to each fully lies within the ROAD curve
of the device. Therefore, instead of a single set point change, e.g., from x1 to x2,
a sequence of monotonically increasing (or decreasing) intermediate set points, as
shown in Fig. 5.4, is crafted. SPAA requires the availability of a (reduced-order)
model of the device, from the control command to the measured output, to calculate
the overshoot of the response.
5.3.2 Description
SPAA calculates the maximum permissible step change for which the overshoot or
undershoot of the unit is acceptable. According to the ROAD curve of a unit, the
following two scenarios are both acceptable:
• The trajectory of response x(t) remains within the permissible minimum/maximum
limits at all times; and
• The duration of a deviation of x(t) from the permissible limits does not exceed
the prescribed maximum duration.
Chapter 5. Online Set Point Adjustment for Trajectory Shaping 79
time
amplitude
x1
x2x'2
xupper-limit
Δ'x Δx
xpeak
x'peak
Fig. 5.4. Variables of SPAA and a representative case. xupper-limit is the maximumpermissible value for x(t); xpeak and x′peak are the peaks of uncontrolled and controlledresponses, respectively.
To determine the response behavior, SPAA employs the unit information: a model
of the unit that captures the collective dynamics of the primary controllers, filters,
and phase-locked loop (PLL) in a transfer function from the controller input to the
device output x(t). Since SPAA is not intended for high-order transients of a unit,
an approximate second-order transfer function is used.
A second-order transfer function is characterized by its damping factor, natural
frequency, and dc gain. The damping factor determines the percent overshoot, and the
natural frequency determines the settling time. These parameters can be calculated
using curve fitting [108] by observing the step response of the unit. The standard
form of a second-order transfer function with unity dc gain is
G(s) =ω2n
s2 + 2ζωns+ ω2n
, (5.1)
and its response to a unit step is
s(t) = 1− e−ζωnt√1− ζ2
sin(ωdt+ ψ), (5.2)
where ζ is the damping factor, ωn is the natural frequency, ωd is the damped frequency,
and ψ is the phase shift. For the damped sinusoidal waveform (5.2), ωd and ψ are
Chapter 5. Online Set Point Adjustment for Trajectory Shaping 80
defined as
ωd = ωn√
1− ζ2
ψ = tan−1
√1− ζ2
ζ.
(5.3)
The response exhibits an overshoot if ζ < 1. In this case, the peak response is
speak = 1 + exp(−ζπ√1− ζ2
), (5.4)
and the percent overshoot is
Mp = speak − 1 = exp(−ζπ√1− ζ2
). (5.5)
The overshoot and frequency of the damped sinusoidal waveform can be measured
and used to calculate the parameters of the approximate second-order model, i.e., its
damping factor and natural frequency. The relationship between natural frequency
and damped frequency is given in (5.3), and the relationship between overshoot and
damping factor is given in (5.5); see also (5.11).
Assuming the transfer function (5.1) is valid at both initial and final set points,
the response to a step change from x1 to x2 is (see Fig. 5.4)
x(t) = x1 + (x2 − x1)s(t). (5.6)
Without loss of generality, we assume x2 > x1. If the overshoot corresponding to the
step change from x1 to x2 is less than xupper-limit (the maximum permissible value of
x(t)), x2 is passed to the unit without any change. Otherwise, SPAA calculates a
series of intermediate set points x(i)2 , i = 1, . . . , n, where
x1 < x(1)2 < · · · < x
(n)2 < x2, (5.7)
and n depends on the number of required intermediate steps. Each x(i)2 is calculated
such the corresponding peak of the response x(i)peak is less than or equal to xupper-limit.
Note that
x(i)peak = x1 + (x
(i)2 − x1)speak. (5.8)
Chapter 5. Online Set Point Adjustment for Trajectory Shaping 81
Equating the right-hand side of (5.8) to xupper-limit and combining the resulting equa-
tion with (5.5), gives the following expression for the value of the ith intermediate
set point:
x(i)2 =
xupper-limit +Mpx(i−1)2
1 +Mp
, (5.9)
where x(0)2 = x1, Mp is the percent overshoot, and i is the number of intermediate set
point. In the general case, the intermediate set points are calculated from
x(i)2 =
min
xupper-limit +Mpx
(i)2
1 +Mp
, x2
, x2 > x1
max
xlower-limit +Mpx
(i)2
1 +Mp
, x2
, x2 < x1.
(5.10)
Depending on the magnitude of the original step change, initial and final set points,
and their proximity to the limits, it may be necessary that multiple intermediate set
points are calculated. Fig. 5.5 shows the flowchart of the algorithm.
Fig. 5.6 shows the result of applying SPAA to a unit represented by a second-
order transfer function with ζ = 0.2155 and ωn = 82 rad/s. It is desired to change
the set point from 0.90 pu to 1.09 pu. The minimum and maximum steady-state
limits are 0.9 pu and 1.1 pu, respectively. In order to avoid an overshoot, the SPAA
algorithm calculates the following sequence of set points: 0.900, 1.033, 1.078, 1.090.As a result, while the original response shows an overshoot of 1.185 pu, the corrected
trajectory of the unit fully lies within the prescribed ROAD.
5.4 Performance Evaluation
The reported studies in this section are conducted in the PSCAD/EMTDC simulation
software. The algorithm of the proposed strategy is implemented in the Fortran
language and is interfaced to the PSCAD/EMTDC program [68].
While the proposed strategy is equally applicable to traditional power systems,
applications considering microgrids are presented to better discuss the merits of the
proposed strategy. This is because the components in a microgrid are more susceptible
Chapter 5. Online Set Point Adjustment for Trajectory Shaping 82
Start
Yes
Nox2
(i) > x2 ?
x2(i) = (xupper-limit + Mp x2
(i−1)) / (1 + Mp)
Output x2(i) as the
intermediate set point
Output x2 as the final set point
i = 0x2(i) = x1
i = i + 1
End
Fig. 5.5. Flowchart representing the SPAA algorithm.
Time
x(t
)
x(t)
x(t) with SPAA
x1
x2
x2(1)
x2(2)
Fig. 5.6. An example set of intermediate set points generated by SPAA.
Chapter 5. Online Set Point Adjustment for Trajectory Shaping 83
Table 5.1Study systems IV, V, and VI: Interface parameters
System Diagram Transformer Filter
IV: Part of CIGRE system Fig. 5.7 500 kVA, 480 V/12.47 kV 0.03 + j0.12 puV: IEEE 34-bus system Fig. 5.10 400 kVA, 480 V/24.9 kV 0.03 + j0.12 pu
VI: IEEE 13-bus system Fig. 6.21 3 MVA, 480 V/4.16 kV 0.03 + j0.12 pu
and more prone to the violation of their limits.3 The study systems are representative
of practical cases.
Each DER unit in the study systems is modeled as a dc voltage source that is
interfaced to the host system through a two-level VSC. Each DER unit is connected to
the PC through a series RL filter and a step-up transformer. The interface parameters
are given in Table 5.1.
A dq-current control scheme is implemented for DER units [64]. For voltage-
controlled units, an outer loop controls the voltage: the vd-loop provides the set point
for the id-loop, and the vq-loop provides the set point for the iq-loop. To control
the magnitude of voltage, vq is set to zero, and vd represents the desired magnitude.
For voltage control, the ROAD curve corresponding to the ITI curve is chosen as the
reference for constraints.
SPAA is based on the availability of a second-order approximate model of the
system, as presented in (5.1). The parameters of this model are calculated by applying
a step change in the set point of the DER unit and measuring the overshoot Mp and
period ∆t of oscillations of the response. Reordering (5.3) and (5.5) gives the following
expressions for the damping factor ζ and natural frequency ωn:
ζ =− logMp√π2 + log2Mp
ωn = 2π1
∆t
1√1− ζ2
.
(5.11)
5.4.1 Study System IV
The first group of case studies are performed on an islanded portion of the CIGRE
North American medium voltage distribution network benchmark system for DER
3The number of buses in each study system is further limited by the available educational licenseof the simulation software.
Chapter 5. Online Set Point Adjustment for Trajectory Shaping 84∞ DG1Tg T1PC1PCC LLCL RL RfLfRsLs R'LFig. 5.7. Study system IV: One feeder of the CIGRE medium voltage benchmark system.
Table 5.2Study system IV: Parameters
Fundamental frequency f = 60 HzSwitching frequency fsw = 1620 Hz 27 pu
Grid voltage vs = 230 kVGrid resistance Rs = 0.439 ΩGrid inductance Ls = 11.635 mH
Transformer G 230 kV/12.47 kV 0.013 + j1.55 Ω 0.001 + j0.120 puTransformer 1 12.47 kV/480 V 500 kVA 0.005 + j0.080 pu
DC bus voltage Vdc = 1200 VFilter impedance 0.025 + j0.040 pu
Load 1 (240 kVA) R = 810 Ω, L = 2.86 H, C = 1.38 µF
units [69], shown in Fig. 5.7. The scenarios studied in this system serve as the proof
of concept and are performed to demonstrate the capabilities of the proposed strategy.
The parameters of the study system are given in Table 5.2. In this system, the load
is a shunt RLC branch connected to PC1. The load power factor is adjusted to
0.95 lagging. The SCMVA of the grid is 12 000 MVA, and its X/R ratio is 10. The grid
is represented by a three-phase balanced voltage source. The grid imposes the voltage
at PC1 in the grid-connected mode and does not permit effective voltage control by
the DER unit.
The parameters of the approximate second-order transfer function (5.1) are cal-
culated from (5.11): ζ = 0.52 and ωn = 82 rad/s. The performance of SPAA is
evaluated in the islanded mode of this study system. Two case studies that involve a
set point change are considered. These case studies demonstrate the ability of SPAA
to respond to transients caused by a set point change.
Chapter 5. Online Set Point Adjustment for Trajectory Shaping 85
5.4.1.1 Start-Up Process
This case study investigates the black start-up of a DER unit and demonstrates the
effectiveness of SPAA in improving the response. Initially, the output voltage is 0 pu.
At t = 0 s, the voltage set point is step changed to 1.09 pu. Fig. 5.8 shows the
response to this step change.
As shown in Fig. 5.8(a), when SPAA is not active, the voltage has an overshoot
in excess of 1.2 pu for 23 ms. Since this magnitude of overshoot is permissible only
for a duration shorter than 3 ms, the black start cannot be performed by a step
change in the voltage set point. To address this problem, SPAA provides a gradual
change in the set point. When SPAA is active, as shown in Fig. 5.8(b), the set
point change is applied gradually in the following order: 0, 0.9565, 1.0813, 1.0900.The set points are calculated such that the resulting overshoots do not exceed the
upper voltage limit. The settling time in both cases is the same and is about 0.3 s.
It is, however, possible to reduce the settling time by lowering the delay between
subsequent updates of set points. For comparison, Fig. 5.8(c) shows the response
obtained by replacing the electrical system by the same second-order model that is
used in SPAA calculations. Fig. 5.8(b)–(c) shows a close match, indicating that the
employed second-order model can sufficiently represent the system.
5.4.1.2 Step Change in Voltage Set Point
In this case study, the voltage set point is step changed from 0.90 to 1.09. Fig. 5.9(a)–
(b) shows the response of the DG unit without and with SPAA. Although the response
without SPAA, Fig. 5.9(a), does not violate the ROAD curve of the system, the re-
sponse can be made smoother when the set points are provided by SPAA, Fig. 5.9(b).
Moreover, the response obtained with SPAA stays completely inside the steady-state
permissible zone.
5.4.2 Study System V
This study system is the IEEE 34-bus test feeder augmented with three DER units
as shown in Fig. 5.10. DER units are operated in current control mode. The feeder
is converted to a balanced feeder by averaging the line parameters, replacing single-
phase loads with equivalent three-phase loads, and lumping distributed loads at the
Chapter 5. Online Set Point Adjustment for Trajectory Shaping 86
0.00 0.25 0.50 0.750.0
0.4
0.8
1.2
(a)
V (
pu)
0.00 0.25 0.50 0.750.0
0.4
0.8
1.2
(b)
V (
pu)
0.00 0.25 0.50 0.750.0
0.4
0.8
1.2
(c)Time (s)
V (
pu)
Set PointResponse
Fig. 5.8. System IV: Performance improvement due to SPAA in a start-up scenario. (a)SPAA is not active; (b) SPAA is active and the actual DER unit is used; (c) SPAA isactive and the DER unit is replaced by its approximate model.
Chapter 5. Online Set Point Adjustment for Trajectory Shaping 87
0.00 0.25 0.50 0.75
0.9
1.0
1.1
(a)
V (
pu)
0.00 0.25 0.50 0.75
0.9
1.0
1.1
(b)Time (s)
V (
pu)
Set PointResponse
Fig. 5.9. System IV: Step change from 0.90 pu to 1.09 pu. (a) without SPAA; (b) withSPAA.
sending end [109]. The grid is modeled as a three-phase balanced voltage source
connected to bus 800 through a step-down transformer.
In this section, the performance of SPAA for current control in the grid-connected
mode is evaluated.
5.4.2.1 Current Control During Start-Up
This case study investigates the ability of SPAA to improve the start-up process
of DER2. The approximate parameters for the transfer function of DER2 are ob-
tained similarly to the method described previously. For DER2, ζ = 0.361 and
ωn = 8450 rad/s. DER1 and DER3 each inject id = 1 pu and iq = 0 pu. Initially,
DER2 is turned off (id = 0, iq = 0), and the set point of id2 is step changed from 0 to
1.08 pu.
Fig. 5.11 shows the start-up response of DER2. When SPAA is not active, the
response has a 30% overshoot. With SPAA, the overshoot is zero, but the settling
time is similar to the previous case, because SPAA updates the intermediate set points
only when DER2 reaches the steady state.
Chapter 5. Online Set Point Adjustment for Trajectory Shaping 88
800 816≈ 824 828 830 854 852832858 888 890834 860 836 840842844846848DG3DG2DG1
Fig. 5.10. Study system V: IEEE 34-bus test feeder with three augmented DER units.Original loads and shunt capacitors are not shown.
This case study confirms the ability of SPAA is mitigating the overshoot resulting
from the start-up process.
5.5 Conclusions
Power system components are susceptible to the violations of their dynamic limits.
For example, power electronic components can only withstand a very narrow vio-
lations of their currents. With the acceptance of the concepts of microgrid and the
constituting distributed generation units, power systems can be subjected to frequent
disturbances caused by turning devices on and off, connecting and disconnecting
loads, and set point changes. Therefore, it is imperative that controllers ensure that
no excessive excursion from the desired trajectory exists. In this chapter, a strategy
for trajectory shaping of the output response of DER units is proposed and its appli-
cability is demonstrated. This strategy is employed to improve the performance of a
DER unit and its controller.
This strategy can be implemented in two variations: set point automatic adjust-
ment (SPAA) and set point automatic adjustment with correction enabled (SPAACE).
These methods temporarily modify the input to the primary controller of a unit such
that the trajectory of the resulting response exhibits less excursion from the desired
response.
Chapter 5. Online Set Point Adjustment for Trajectory Shaping 89
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.0
0.5
1.1
(a)
I (p
u)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.0
0.5
1.1
(b)Time (s)
I (p
u)
Set PointResponse
Fig. 5.11. System V: Start-up response of DER2. (a) without SPAA; (b) with SPAA.
In this chapter, the SPAA method is discussed, and case studies confirming its
superior performance are presented. SPAA is used when reducing over- or undershoot
is of primary importance, and an approximate model of the system capturing the over-
or undershoot is available. It is shown that SPAA is effective in reducing the excursion
of the response of a DER unit from its permissible limits. SPAA is especially useful
when large set point changes occur in a system, e.g., black start-up.
In the absence of information about the system, the SPAACE variation of the
proposed strategy is employed. This method is discussed in the next chapter.
Chapter 6
Online Set Point Adjustment With
Correction
6.1 Introduction
Chapter 5 introduced a strategy to improve trajectory shaping of power system de-
vices and discussed its first variation (SPAA). This chapter discusses the development
of the second variation of this strategy (SPAACE)1 and assesses its performance.
SPAACE stands for set point automatic adjustment with correction enabled.
SPAACE implements an online monitoring strategy to automatically and tem-
porarily adjust the set point to achieve a response that is generally faster and smoother
than the original response and exhibits a smaller overshoot. In contrast to SPAA,
SPAACE does not require prior knowledge of the system parameters and is adaptive
to the various operational scenarios of a power system. It should be emphasized that
Portions of this chapter are published as
[33] A. Mehrizi-Sani and R. Iravani, “Online set point adjustment for trajectory shaping in micro-grid applications,” IEEE Trans. Power Syst., Oct. 2010, accepted for publication (paper no.TPWRS-00823-2010);
[34] A. Mehrizi-Sani and R. Iravani, “Online set point modulation to enhance microgrid dynamicresponse: Theoretical foundation,” IEEE Trans. Power Syst., Sep. 2011, submitted for review;and
[35] A. Mehrizi-Sani and R. Iravani, “Performance evaluation of a distributed control scheme forovervoltage mitigation,” in CIGRE Int. Symp. Electric Power Syst. Future—Integrating Super-grids and Microgrid, Bologna, Italy, Sep. 2011.
1Pronounced [speIs].
90
Chapter 6. Online Set Point Adjustment With Correction 91
SPAACE, similarly to SPAA, aims to improve the performance of an existing con-
troller and does not replace that controller. This is especially useful in a scenario that
a controller is designed with a series of assumptions about the system operation that
no longer hold, perhaps because of a change in the configuration or the philosophy of
operation of the system.
In a conventional power system, primary controllers are responsible for providing
the damping required for mitigating transients. Several other power system appa-
ratus are also utilized to augment this primary damping. Power system stabilizers
(PSS) [110], unified power flow controllers (UPFC), and shunt and series FACTS de-
vices [111], [112] are among the conventional means employed to increase the primary
damping of a power system; these devices provide extra damping by changing the
power flow or line impedance or by providing a supplementary damping signal. For
example, a PSS increases damping by providing a signal in-phase with the change in a
power system variable, e.g., rotor speed ∆ω, voltage ∆V , or frequency ∆f , to induce
the desired damping. The design of a PSS requires detailed analysis and modeling of
the generator and its excitation system, e.g., automatic voltage regulator (AVR).
The issues with relying on these methods to provide supplementary damping are
that (i) they are relatively costly, hardware-based, and/or intricate, (ii) their design
relies on the availability of parameters and models of devices, and (iii) once designed,
they are effective only for a specific operating region. SPAACE addresses the challenge
of providing auxiliary damping for a power system device without such drawbacks.
In this chapter, SPAACE is introduced and its theoretical foundation is discussed.
The effect of SPAACE on the stability of the system is studied, and the viability
and existence of appropriate instances to update set points are justified. Moreover, a
discussion of the practical considerations for implementation of SPAACE is presented.
Several case studies are presented to establish the technical viability of this method.
6.2 Set Point Automatic Adjustment With Cor-
rection Enabled (SPAACE)
6.2.1 Objective
The objective of SPAACE is online modification of the set point of a unit to achieve a
smooth response x(t), where x(t) represents the voltage of a voltage-controlled unit,
Chapter 6. Online Set Point Adjustment With Correction 92
the current of a current-controlled unit, or the power of a power-controlled unit.
SPAACE monitors the time evolution of x(t) and switches the set point between a
temporary value and the commanded value as required.
Unlike SPAA, SPAACE does not deliberately impose a delay between successive
set point updates; updates can occur before reaching the steady state, i.e., when tran-
sients from the previous set point change are still in effect. In addition to responding
to changes in the set point, SPAACE can respond to other disturbances such as load
switching.
6.2.2 Description
SPAACE applies a temporary change in the set point such that the overall system
response is within the prescribed ROAD. The overall system response is the sum of
the forced response and the natural response. The natural response depends on the
initial condition and cannot be controlled; therefore, SPAACE manipulates the forced
response.
Set point adjustment is performed by applying a scaled version of the original
set point. The change of set points is illustrated in Fig. 6.1. Assume the set point
of x(t) is step changed from x1 to x2, where without loss of generality, it can be
assumed that x2 > x1. In response to this step change, x(t) initially increases. If the
overshoot of x(t) exceeds its maximum permissible value, SPAACE issues a command
to temporarily scale down the set point from x2 to (1−m)x2, where m is a heuristically
determined constant. For the case studies in this chapter, a nominal value of 0.2 is
chosen for m. Since SPAACE is adaptive, its performance is not highly dependent on
m, and changing m will change the timing and duration of the modified set points.
This effectively renders the performance of the strategy insensitive to m.
When the trend of the response—determined either by a predictive scheme de-
scribed next in this section or by the sampled values of the response—indicates that
it will be within the acceptable region, SPAACE releases the set point so that x(t)
settles to x2.
Dealing with a negative step change (x1 > x2) is similar; however, a scaled up set
point (1 +m)x2 is used instead.
SPAACE can be implemented without or with prediction. SPAACE without pre-
diction monitors only the samples of x(t). To improve the accuracy and speed of
Chapter 6. Online Set Point Adjustment With Correction 93
Time
x(t
)
x1
x2
x(t)
x(t) with SPAACE
(1-m)x2
Fig. 6.1. Demonstration of the performance of SPAACE.
SPAACE, a prediction scheme is employed. This scheme turns SPAACE into a pre-
emptive strategy. That is, a second layer of logic issues a change in the set point if
it predicts that x(t) is about to violate a limit. As discussed in Section 6.8, differ-
ent prediction techniques can be employed, e.g., spline curve, exponential fit, Bezier
curve, and moving average methods [113]–[120]. In this work, prediction is performed
by linear extrapolation because of its simplicity, satisfactory performance, and com-
putational efficiency. We have
x(t0 + Tpred) = x(t0) + rTpred, (6.1)
where x(·) is the predicted value of x(·), t0 is the current time, Tpred is the prediction
horizon, and r is the approximate local rate of change of x(t). Values of x(t0) and
x(t0−Tpred) are used to approximate the average rate of change. To limit the required
storage space, only the values of x(t) at regular intervals of Ts (sampling time) are
used. Moreover, only n past values are stored, where n = Tpred/Ts. Since t0 (and as a
result, t0−Tpred) may not be an integer multiple of the sampling time Ts, x(t0−Tpred)
is calculated by interpolating between x(t− nTs) and x (t− (n− 1)Ts).
The procedure to predict the value of x(t0 + Tpred) is illustrated in Fig. 6.2 and is
as follows. An auxiliary variable α is defined as
α =t0 − TkTs
, (6.2)
Chapter 6. Online Set Point Adjustment With Correction 94tTk−n Tk Tk+11TsTk−(n-1) ≈ t0t0−Tpred αTsFig. 6.2. Prediction algorithm of SPAACE.
and x(t0 − Tpred) is calculated from
x(t0 − Tpred) = x(Tk−n) + α(x(Tk−(n−1))− x(Tk−n)
). (6.3)
The average local rate of change of x(t) is then linearly approximated as
r =x(t0)− x(t0 − Tpred)
Tpred
. (6.4)
Finally, the predicted value of x(t0 + Tpred) is calculated from (6.1).
Although SPAACE does not require a model of the system, it needs an estimate of
the value of prediction horizon to be used in the extrapolation algorithm. Prediction
horizon Tpred influences the behavior of SPAACE. Its choice is based on engineering
judgement and the speed of variations of the controlled variable: Tpred is shortest
for current control, longer for voltage control, and longest for power control. Tpred
decreases as the natural frequency of the closed-loop process increases. If a model of
the system is available, it may be used to improve/customize the prediction algorithm
based on the available information.
SPAACE is implemented based on finite state machine (FSM) representation [121],
as illustrated in Fig. 6.3. States are numbered in a binary scheme in which the least
significant bit shows whether x(t) > xmax, the next bit shows whether xpred > xmax,
and the most significant bit shows whether a scaled set point is applied, i.e., (1−m) for
S101, S110, and S100, and (1−m)2 for S111. The states during which a scaled set point
is applied are further distinguished by a dashed circle. As an example, in state S010,
x(t) is within the permissible limits, but its value at the end of the prediction horizon
is going to violate the limit, and the original set point is effective. States shown as wi
are wait states, either for the duration of violation to exceed the permissible limits
(w1 and w2) or for the new measurement sample to become available (w3, w4, and
w5). xmax and Tmax are the maximum value and the respective maximum duration
Chapter 6. Online Set Point Adjustment With Correction 95
permissible for x(t) given by ROAD curve. ∆t is the time since beginning of the
violation.
In the attempt to keep SPAACE as independent as possible from the controlled
unit, it is not provided with the information regarding the details of the unit, e.g.,
its switching frequency. As a result, it is possible that some of the updates issued by
SPAACE cannot be implemented. Such updates include set point changes that are
issued at a rate faster than the switching of an electronically interfaced DER unit.
However, as the case studies in Subsection 6.10.2 confirm, SPAACE compensates
for these unimplemented set point changes by issuing another set point change, if
required, to maintain close tracking of the original set point.
The performance of SPAACE, without and with prediction, applied to a unit
whose dynamics are approximated with a second-order transfer function with a damp-
ing factor ζ = 0.5169 and natural frequency ωn = 82 rad/s, is shown in Figs. 6.4
and 6.5. A step change from 0.90 pu to 1.09 pu is applied at time zero. The mini-
mum and maximum permissible values are 0.90 pu and 1.10 pu, respectively.
Fig. 6.4 shows the performance of SPAACE without prediction. At t = 0.033 s,
the algorithm detects an overshoot. At t = 0.034 s, the applied set point is scaled
down by 20% from 1.090 to 0.872 (that is, (1 −m)x2 with m = 0.2). The set point
is reverted to its original value when x(t) is within the permissible limits. As shown,
the SPAACE-controlled trajectory is superior to the uncontrolled trajectory in terms
of both the settling time and the overshoot.
Fig. 6.5 shows the performance of SPAACE with prediction. Because the algo-
rithm additionally monitors the trend of the signal, the set point shift is performed
earlier (at t = 0.030 s) than the previous case. The set point is reverted to its original
value at 0.032 s. The trajectory is fully within the prescribed limits. Moreover, the
response reaches the steady state considerably faster than the case without prediction.
6.3 Comparison of SPAA and SPAACE
While SPAA and SPAACE have similar objectives, their different approaches make
each suited for a specific class of systems and scenarios. The differences between
SPAA and SPAACE include the following:
• SPAA requires availability of a simplified second-order model of the system that
captures its dynamics from the reference set point to the response. In contrast,
Chapter 6. Online Set Point Adjustment With Correction 96
x(t) > xmax
S000
S001
S010
S101
S111
w1
xpred > xmax
Δt < Tmax
Δt > Tmax
Δt > Tmax
w2
Δt < Tmax
w4
SVWait
S100
x(t) < xmax
w3
SV
Wait
S110
xpred > xmax
x(t) > xmax
w5
SV
Waitxpred < xmax
xpred < xmax
x(t) < xmax
Wait
Fig. 6.3. Finite state machine representation of SPAACE for a positive step. A reversestep is dealt with similarly. States are defined in the text. ∆t is the time passed sinceviolation of limits; Tmax is the maximum permissible duration of the violation asprescribed by the ROAD curve. x(t), xpred, and xmax denote the sampled value of themonitored signal, its predicted value, and its maximum permissible value, respectively. SVis short for “still violation” and signifies a persisting violation.
Chapter 6. Online Set Point Adjustment With Correction 97
0.00 0.05 0.10 0.15 0.20
0.87200.9000
1.09001.1185
Time (s)
|V| (
pu)
Set PointNo SPAACESPAACE
Fig. 6.4. SPAACE application example without prediction.
0.00 0.05 0.10 0.15 0.20
0.87200.9000
1.09001.1185
Time (s)
|V| (
pu)
Set PointNo SPAACESPAACE
Fig. 6.5. SPAACE application example with prediction.
SPAACE does not require a model of the system, but needs an estimate of the
prediction horizon for the extrapolation algorithm; SPAACE assumes that the
system is—and remains—stable during its operation. Section 6.5 discusses the
validity of this assumption.
• SPAA updates the set point only after the system response reaches the steady
state. This is to avoid interaction between the apparatus set point and its re-
sponse. However, this also invariably decreases the speed of response of SPAA.
SPAACE, however, does not have this limitation and can update the set point
repeatedly and at any time instant and even faster than the speed of the con-
trolled unit. Therefore, SPAACE is generally faster than SPAA.
• SPAA assumes a priori availability of an approximate system model. Therefore,
it is essentially an open-loop control method. In contrast, SPAACE continuously
corrects for control actions and inaccuracies in its prediction, which renders it
a closed-loop approach.
• SPAA is more effective for a large step change, which is likely to cause a large
Chapter 6. Online Set Point Adjustment With Correction 98
overshoot; SPAA calculates the temporary set points to specifically avoid such
large overshoots.
• SPAA primarily monitors the control command for a step change and acts
accordingly. However, SPAACE monitors both the control command and the
system output. Therefore, SPAACE can respond to an array of disturbances
in addition to a set point change, e.g., load change and line faults. In general,
SPAACE responds to any disturbance that causes a deviation of the response
from the set point.
6.4 Alternative Methods to SPAACE
An alternate method to SPAACE is gradual ramping of the set point. However, this
method is not considered in this work because of the following drawbacks:
• Information about the necessity of adjustment of set point is not always available
a priori. SPAACE activates only when it senses that an improvement in the
system response is needed. In ramping up the set point, the set point is modified
regardless of the performance of the existing controller. Such modification is
not always required, nor is it desirable.
• Selection of the ramp slope requires knowledge of the system characteristics,
e.g., settling time, which are not necessarily readily available.
• Considering the wide acceptance of PI-based controllers in the power system
that are designed to track dc commands, it is institutive to apply a a step rather
than a ramp.
Another method to improve set point tracking is inclusion of a derivative term
in the controller, i.e., a PID-based controller. Although the derivative term can ef-
fectively be considered as a predictive scheme, implementing a PID-based controller
necessitates changing the existing controller. This is in contrast to the underlying
objective of the proposed strategy that is designed to be an add-on controller that
improves the performance of an existing controller. Moreover, it is not feasible to con-
tinually run studies to adjust the controller gains in response to changes in the system
parameters. Unavailability of the system data and communication requirement for
updating the controller parameters are other hurdles.
Chapter 6. Online Set Point Adjustment With Correction 99H(s)r(t) x(t)Fig. 6.6. SISO representation of a controllable device.
Time
1
t0 t1 t2 t3 t4
1-mr(t)
Fig. 6.7. Stability of SPAACE. Intermediate set points for n = 2.
6.5 Effect of SPAACE on Stability
This section demonstrates that SPAACE does not alter the stability behavior of the
system. Assume that the system is single-input single-output (SISO), as shown in
Fig. 6.6, and the transfer function from input to output is stable and has a unity dc
gain. Assume that a step change is applied to the unit. With appropriate scaling and
time shift, and without loss of generality, the set point change can be represented with
a step change of unity magnitude applied at t = 0 s. SPAACE provides a sequence of
set points r(t), Fig. 6.7, which can be represented as the summation of time-shifted
step functions. Since SPAACE ensures that the last set point is the same as the
original set point, there is an odd number of terms (2n+ 1) in the series:
r(t) = u(t− t0) +2n∑i=1
(−1)imu(t− ti), (6.5)
where n is the number of applied step changes, t0 = 0, and ti’s form a monotonically
increasing series (ti > tj⇐⇒i > j). For example, if there is one pair of intermediate
set point changes, r(t) is expressed as
r(t) = u(t)−mu(t− t1) +mu(t− t2). (6.6)
Chapter 6. Online Set Point Adjustment With Correction 100
In Laplace domain, (6.5) can be written as
R(s) =1
s+m
2n∑i=1
(−1)ie−tis
s. (6.7)
Therefore, the Laplace transform of output x(t) is
X(s) = H(s)R(s)
= H(s)
(1
s+m
2n∑i=1
(−1)ie−tis
s
),
(6.8)
where H(s) represents the system transfer function. From the final value theorem,
limt→∞
x(t) = lims→0
sX(s)
= lims→0
sH(s)
(1
s+m
2n∑i=1
(−1)ie−tis
s
)
= lims→0
H(s)× lims→0
(1 +m
2n∑i=1
(−1)ie−tis
)
= 1×
(1 +m
2n∑i=1
(−1)i
)= 1,
(6.9)
where the last equality holds because the number of terms in the summation is even
(2n). Therefore,
limt→∞
x(t) = 1. (6.10)
Note that from the above, it follows that in general the sufficient condition for
stability of SPAACE is that the overall dynamics of the device from R(s) to X(s)
(including control loops) can be expressed as a stable transfer function with a nonzero
bounded dc gain. That is, SPAACE does not alter the stability behavior of a system;
if the system is stable, it remains stable under augmented SPAACE.
The underlying assumption for the operation of SPAACE is that the overall closed-
loop system, consisting of the DER unit and its primary controller, is stable and so
remains during the operation of SPAACE. The discussion above justifies the valid-
ity of this assumption for linear systems. However, strictly speaking, power systems
Chapter 6. Online Set Point Adjustment With Correction 101
Time
x(t)
1−m
0 T1 T2tp
1xp
Fig. 6.8. Definition of T1, T2, and tp.
are nonlinear due to factors such as magnetic saturation, nonlinear loads, and pulse
width–modulated converters. Nevertheless, power systems are not highly nonlinear;
for example, (i) the mathematical nonlinearity of synchronous generators due to mul-
tiplicative frequency terms is not significant because the frequency does not change
rapidly and (ii) the structural nonlinearity of pulse width modulators is widely ac-
cepted to be represented with linear operation. Thus, nonlinearity of power systems
for the range of dynamics considered for SPAACE is minor; power system is closer to
a linear system than to a nonlinear system. This assumption is also implicit in the
design of primary controllers, which is based on system linearization. If power system
was highly nonlinear, the primary controllers would not be able to follow their set
points when a significant deviation from the original operating point occurs. In this
case, SPAACE cannot improve the tracking behavior either.
6.6 Existence of a Smooth Response
This section discusses the timing of a temporary change in the set point that results in
a response with shorter settling time and smaller overshoot than the unmanipulated
response. Assume the set point is changed from 1 to (1 −m) at T1 and returned to
1 at T2, Fig. 6.8.
Determining T1 and T2 can be posed either as an exhaustive search or as an op-
timal control problem to minimize∫∞
0(x(t)− r(t))2 dt. However, deriving analytical
expressions for T1 and T2 is of limited practical use, because it requires the knowl-
edge of system parameters a priori, e.g., damping factor ζ and natural frequency
ωn for a second-order system. Therefore, the focus of this section is to demonstrate
the existence of T1 and T2 for a second-order system for any value of ω > 0 and
Chapter 6. Online Set Point Adjustment With Correction 102
0 < ζ < 1. The software described in Appendix C.2 is utilized to study the behavior
of a second-order system equipped with SPAACE in response to changes in the set
point.
An acceptable response can be achieved if the set point is switched to (1 − m)
before the first peak of x(t). Assume x(t) reaches its first peak xp at tp. Then, since
x(t) is a causal signal, the choice of T2 has no effect on tp as long as T2 > tp. Moreover,
having T2 < tp is not desired because it increases the peak (overshoot) of x(t). At a
time instant infinitesimally close and after tp, x(t) is negative. Since the set point is
increased from (1−m) to 1 at T2, the choice T2 = tp reduces the oscillations of x(t).
The negative value of x(t) can (at least partially) cancel the effect of increasing the
set point and can lead to a smoother response. A discussion and comparison of x(t)
with and without switching of set point at T2 is presented in Appendix D.1. This is
further confirmed by experimenting on many systems with different values for ζ and
ωn, using the developed software introduced in Appendix C.2. This approach also
requires an appropriate value of T1 such that the peak of x(t) is unity. This strategy
is summarized below:
1. Choose T1 such that xp = 1.
2. Calculate A1 and A2 coefficients, defined in Appendix D.2, corresponding to T1
to obtain a closed-form expression for x(t) for T1 < t < T2.
3. Choose T2 to be tp for x(t) obtained in the previous step.
This set of choices ensures that x(T2) = 1 and x(T2) = 0. Therefore, from (D.1) in
Appendix D.1, it follows that x(T2) = 0. Thus x(t) = 1 for t ≥ T2 is the solution to
the differential equation, as can be deduced from Appendix D.2, (D.9) and (D.13).
The settling time is, therefore, tp.
Fig. 6.9 shows the dependence of the performance of SPAACE on the choice of
T2, when T1 is the value prescribed above. Fig. 6.9(a) shows the unmanipulated
response of the system to a step change. Fig. 6.9(b)–(e) shows that the settling time
and overshoot of the response improve as T2 approaches the prescribed value of tp.
Finally, in Fig. 6.9(f), T2 equals tp, and the response exhibits no overshoot.
The rest of this section shows that a value for T1 does in fact exist. Without loss
of generality, it can be assumed that x(t) is initially zero, and a unit step is applied
to the system at t = 0 s, followed by a step of magnitude (1−m) at t = T1, Fig. 6.8.
Chapter 6. Online Set Point Adjustment With Correction 103
T1 T2tp(a)
(c)
(e)
(b)
(d)
(f)
T1 T2 T1 T2
T1 T2 T1 T2
t
x(t)
t
x(t)
t
x(t)
t
x(t)
t
x(t)
t
x(t)
Fig. 6.9. Dependence of the performance of SPAACE on the value of T2 as it approachestp. In (a), SPAACE is not active. In (b)–(e), T1 is the prescribed value, while T2 is not. In(f), T2 is also the prescribed value, tp.
Chapter 6. Online Set Point Adjustment With Correction 104
For a second-order system, e.g.,
H(s) =ω2n
s2 + 2ζωns+ ω2n
(6.11)
the response x(t) can be expressed as
x(t) = α + Ai1S(t) + Ai2C(t), (6.12)
where S(t) and C(t) are defined in (D.8) of Appendix D.2, and
α =
1, t < T1
1−m, T1 ≤ t < T2
1, T2 ≤ t,
(6.13)
and (A11, A12) and (A21, A22) are constants that are used in (6.12) to calculate the
value of x(t) prior and subsequent to T1, respectively. Appendix D.2 provides a
method to calculate (A11, A12) and (A21, A22) from x(t) and its derivative x(t) at a
given time instant.
The peak value of x(t) occurs at some tp > T1. Time tp can be calculated by
setting the derivative of x(t) to zero, Appendix D.3:
tp =tan−1(d) + ψ
ωd, (6.14)
where wd = wn√
1− ζ2, ψ = cos−1(ζ), (D1, D2) = (A21, A22), and
d =−D2
D1
=xωn sin(ωdT1) + x sin(ωdT1)
xωn cos(ωdT1) + x cos(ωdT1),
(6.15)
where, to simplify the notation, x and x are short forms for x(t)|T1 and x(t)|T1 ,respectively. x is defined as x = x − α, and its value is calculated by substituting
tp from (6.14) in x(t). Detailed derivation is presented in Appendix D.3, and the
Chapter 6. Online Set Point Adjustment With Correction 105
simplified form of xp is obtained as
xp(T1) =sin(ψ)
ωde−ψ cot(ψ)eζωnT1e−γ cot(ψ)
√(xωn)2 + x2 + 2xxωnζ, (6.16)
where γ = tan−1(d).
We use the intermediate value theorem to show the existence of some T1 for which
xp = 1. Define:
f(T1) = x(T1)− 1
= x+ α− 1
= x−m.
(6.17)
Since x(·), x(·), and γ(·) are continuous functions, f(·) is also continuous, and the
conditions of the intermediate value theorem hold. If there exist t1 and t2 such that
f(t1)f(t2) < 0, the intermediate value theorem states that f(t) has a root between t1
and t2. We choose t1 = 0; then
x|0 = x|0 − α = −α
x|0 = 0
γ|0 = tan−1(d|0) = −ψ.
(6.18)
Substituting for x from (6.18) in (6.17) and using (D.9) and (D.13) from Appendix D.2,
we have
f(t1) =√α2ω2
ne−γ cot(ψ) sin(ψ)
ωde−ψ cot(ψ) −m
= αωnsin(ψ)
ωde− cot(ψ)(γ+ψ) −m
= αωnsin(ψ)
ωn√
1− ζ2−m
= α−m
= 1− 2m
> 0.
(6.19)
Chapter 6. Online Set Point Adjustment With Correction 106
The last inequality holds because 0 < m < 1 and m is chosen to be less than 0.5 (usu-
ally between 0.1 to 0.3). A discussion on the choice of m is presented in Section 6.7.
We choose t2 = 5tsettling. Since the system is stable,
x|t2 = 0
x|t2 = 0.(6.20)
Substituting for x from (6.20) in (6.17) and using (D.9) and (D.13) from Appendix D.2,
we have
f(t2) = 0 + α− 1
= −m
< 0.
(6.21)
Since f(t1)f(t2) < 0, the requirements of the intermediate value theorem are met
and f(t) = 0 for some t1 < t < t2. We choose this root as T1. Note that finding an
analytical expression for T1 may be difficult and of no practical use, since the exact
values of ωn and ζ are not necessarily known a priori.
Finally, T2 is chosen to be tp, where tp is calculated from (6.14).
This algorithm shows the viability of SPAACE, but cannot be used in a practical
implementation of SPAACE, because
• Deriving closed-form expressions for T1 and T2 is not practical, even for a second-
order system; and
• This algorithm requires knowledge of the system transfer function, which is not
necessarily available.
For comparison, Fig. 6.10 shows the trajectory of the response with and without
SPAACE. A step change from 0.90 to 1.09 is applied to a second-order system with ζ =
0.2 and ωn = 100 rad/s at t = 0.01 s. We have T1 = 0.025 s, T2 = 0.033 s, and m =
0.2. SPAACE is successful in eliminating the oscillations of x(t) and substantially
decreasing the settling time from 0.20 s to 0.02 s—a ten-fold reduction.
Chapter 6. Online Set Point Adjustment With Correction 107
0 0.05 0.1 0.15 0.20.8
0.9
1
1.1
Time
x(t)
x(t)
x(t) with SPAACE
T1 T2
Fig. 6.10. Demonstration of fitness of the choice of T1 and T2 as outlined in the proposedalgorithm.
6.7 Upper Bound of m
Equation (6.19) can be used to calculate an upper bound for m. Since
tan(γ) = tan(−ψ), (6.22)
there are infinitely many solutions for γ: γ = kπ − ψ, where k is an integer. In
(6.18), the solution corresponding to k = 0 was selected because the first root was
desired. Since in practice T1 cannot be zero, a more reasonable value for γ is π − ψ.
Substituting this value for γ in (6.19) and noting γ + ψ = π gives
f(t′1) = αe−π cot(ψ) + α− 1
= (1−m)e−π cot(ψ) −m(6.23)
For the inequality f(t′1) > 0 to hold, the following condition should be met:
m <e−π cot(ψ)
1 + e−π cot(ψ). (6.24)
Since ζ can change between 0 and 1 (and correspondingly, ψ can change between π
and 0), m changes between 0 and 0.5. This gives an upper bound for m as a function
of ζ as shown in Fig. 6.11. With a critically damped response (ζ = 1), there is no
Chapter 6. Online Set Point Adjustment With Correction 108
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
mup
per−
boun
d
ζ
Fig. 6.11. An upper bound for m as a function of damping factor ζ.
need to modulate the set point, and the upper bound for m is zero. As ζ decreases,
a higher value of m is permissible to compensate for the reduced damping.
6.8 Measurement and Prediction Enhancement
The measurements of x(t) are, in practice, noisy and contain switching ripple, es-
pecially if electronically interfaced DER units are employed. Therefore, it is desir-
able to eliminate this noise and increase the signal-to-noise ratio (SNR). Further,
this will allow us to use a more accurate extrapolation method for the prediction-
enabled SPAACE. As a solution to this curve fitting problem, general-purpose cubic
splines [113]–[116], French curve, exponential fit [117], [118], and system identification
methods, e.g., [119], [120], can be employed. Spline implementation has the desired
characteristic that it does not assume a specific time dependence for x(t) and is suit-
able for different response behaviors. In spline implementation, x(t) can be expressed
as the weighted sum of base splines, e.g., [122]
x(t) =K∑k=1
wkSk(t), (6.25)
where Sk’s represent base splines that are identical except for a time shift. Since only
sampled values of x(t) are available, (6.25) can be written in matrix form as
x = Aw, (6.26)
where x is the vector of n measurements of x(t) with xi = x(ti), w is the vector of K
weight factors, and A is the n×K matrix of values of splines defined as Aij = Sj(ti).
Chapter 6. Online Set Point Adjustment With Correction 109
To obtain the value of w, an optimization problem is solved to minimize the fitting
error as defined in
Φ =1
2
n∑i=1
(x− x)2, (6.27)
where x is the sampled values of x(t) and x is the fitted value of x and is equal to
Aw as defined in (6.26). The solution of the minimum square error problem is
w = (ATA)−1AT x, (6.28)
which gives the required weight factors for (6.25). This fit can be used to improve the
proposed prediction algorithm (linear extrapolation) and enhance the performance of
SPAACE.
6.9 Physical Analogy
To help better understand the algorithm of SPAACE, this section draws an analogy
between SPAACE and a physical system. Fig. 6.12 shows a representative mechanical
system. Although depicted is a second-order system, this discussion can be readily
extended to higher-order systems. The system consists of a damper B, a mass m,
and a spring k. This system is equivalent to a series RLC circuit that consists of a
resistor R, an inductor L, and a capacitor C, respectively [31]. The mass and the
spring store energy; the damper dissipates energy.
Assume a force u(t) = F+f(t) is applied to the mass at t = 0 s. This force consists
of two terms: a constant term F and a time-varying term f(t), where f(0) = 0 and
f(∞) = 0. In its displacement from the initial position x1 to the final position x2, the
damper-spring-mass system experiences transient oscillations. Distance x is measured
from the hinge to the center of the mass.
Due to F at t = 0 s, the mass moves toward x2, but since the system has inertia,2
it does not stop at x2. Consequently, the mass moves past x2 and experiences an
overshoot. The operation of the SPAACE strategy is equivalent to changing the time-
varying component of force f(t) at appropriate time instants to control the position
x(t). Therefore, to counteract the overshoot of the displacement, f(t) is applied in
the direction opposite of x(t). This additional force should be applied before the mass
2The system possesses inertia because it includes components that store energy.
Chapter 6. Online Set Point Adjustment With Correction 110
B
k
u(t) = F + f (t)
x(t)x1
x2
B
k
F
m
m
f (t)
+
Fig. 6.12. A physical analogy for SPAACE.
reaches x2, because due to inertia, the mass continues its displacement in the same
direction for some time even after applying the force. This is the reason that the
direction of the response trajectories of the presented case studies does not change
immediately following the set point update. Therefore, SPAACE needs to employ a
prediction facility to apply the update before the response experiences an overshoot.
Nevertheless, the speed and acceleration of x(t) decrease following the application of
f(t). If required, f(t) is subsequently updated to achieve the desired trajectory until
the mass settles to x2.
Changing the input command f(t) does not modify the internal dynamics of the
system. There is still energy exchange between the mass and the spring, and there is
still energy dissipation by the damper; however, the amount of these energies reduce
as a result of SPAACE, as confirmed by the reduced oscillations in the controlled
traces. Although it is possible to manipulate the system by increasing or decreasing
mass m, this changes the internal dynamics and damping properties of the system
and unnecessarily increases the complexity of the approach.
Chapter 6. Online Set Point Adjustment With Correction 111
6.10 Performance Evaluation
6.10.1 Study System IV
This study system is one lateral of the CIGRE North American benchmark system
as described in Section 5.4.1, Fig. 5.7, and Table 5.2, with electronically interfaced
DER units. The performance of SPAACE is evaluated in the islanded mode of this
study system. The following case studies demonstrate the ability of SPAACE to
mitigate the transients caused by set point changes and power system disturbances.
The permissible range of voltage variations is from 0.90 pu to 1.10 pu.
6.10.1.1 Voltage Set Point Change Without Prediction
This case study investigates the transient behavior of the DER unit subsequent to a
step change in the voltage set point. To demonstrate the ability of SPAACE in the
worst-case scenario, the maximum possible step size, 0.20 pu, is applied.
Initially, the DER voltage is 1.10 pu. Its set point is step changed to 0.91 pu.
Fig. 6.13 illustrates the response of the system. In Fig. 6.13(a), SPAACE is not active,
and the trajectory of voltage experiences an undershoot lower than the minimum
permissible value. SPAACE is active in Fig. 6.13(b) and, by temporarily adjusting the
set point, keeps the voltage trajectory within the steady-state ROAD. When SPAACE
detects that the voltage exceeds the 0.90 pu limit, it temporarily scales up the set
point by (1 + m), where m = 0.20 and is chosen heuristically. To avoid the settling
of voltage to this temporary set point, SPAACE restores the original set point once
it detects that the voltage is within the permissible limits. The excursion of voltage
below the minimum permissible value is mitigated, and the voltage settles in 100 ms.
Comparison of Fig. 6.13(a) with Fig. 6.13(b) confirms the satisfactory operation of
SPAACE in preventing the voltage trajectory from exceeding the steady-state ROAD.
6.10.1.2 Voltage Set Point Change With Prediction
Fig. 6.13(c) shows the performance of SPAACE in response to the same voltage step
change as that of Fig. 6.13(a)–(b) when the prediction facility of SPAACE is enabled.
Comparing Figs. 6.13(b) and 6.13(c) shows that prediction enables SPAACE to
(i) detect an anomaly earlier and respond to it in a timely manner, while without
prediction, SPAACE updates the set point only after it violates a limit; and (ii)
Chapter 6. Online Set Point Adjustment With Correction 112
0 0.1 0.2 0.3 0.4
0.9
1
1.1
(a)
V (
pu)
0 0.1 0.2 0.3 0.4
0.9
1
1.1
(b)
V (
pu)
0 0.1 0.2 0.3 0.4
0.9
1
1.1
(c)
V (
pu)
Time (s)
Set PointResponse
Fig. 6.13. System IV: Step change in voltage from 1.10 pu to 0.91 pu. (a) withoutSPAACE; (b) with SPAACE without prediction; (c) with SPAACE with prediction.
Chapter 6. Online Set Point Adjustment With Correction 113
0 0.1 0.2 0.3 0.4
0.9
1
1.1
(a)
V (
pu)
0 0.1 0.2 0.3 0.4
0.9
1
1.1
(b)Time (s)
V (
pu)
Set PointResponse
Fig. 6.14. System IV: Step change in voltage from 0.90 pu to 1.09 pu. (a) withoutSPAACE; (b) with SPAACE with prediction.
restore the original set point faster. As expected, prediction decreases the settling
time and leads to a smoother response.
A reverse step change is also applied to the DER unit, and the response of the
system is illustrated in Fig. 6.14. Initially, the DER voltage is 0.90 pu when its
set point is step changed to 1.09 pu. In Fig. 6.14(a), SPAACE is not active, and
the step change causes the trajectory of voltage to exceed the steady-state ROAD.
SPAACE is active in Fig. 6.14(b) and, through temporary adjustments of set point,
keeps the voltage trajectory within the steady-state ROAD. When SPAACE detects
that voltage is about to exceed the 1.10 pu limit, it temporarily scales down the set
point by (1 − m), where m = 0.20 and is chosen heuristically. To avoid settling of
voltage at this temporary set point, SPAACE restores the original set point once it
detects the voltage is not further increasing.
It should be mentioned that because of the nonlinear nature of SPAACE (and
system), it responds differently to the step changes depicted in Figs. 6.13 and 6.14,
although they have the same magnitude.
Chapter 6. Online Set Point Adjustment With Correction 114
0 0.1 0.2 0.3 0.40.5
0.7
0.9
1.1
(a)
V (
pu)
0 0.1 0.2 0.3 0.40.5
0.7
0.9
1.1
(b)Time (s)
V (
pu)
Set PointResponse
Fig. 6.15. System IV: Load change from 1 pu to 2 pu at t = 0 s. (a) without SPAACE;(b) with SPAACE.
6.10.1.3 Voltage Control Subsequent to Load Energization
In this case study, the performance of SPAACE in response to a load change is
evaluated. The voltage is initially 0.91 pu and the load is 1 pu. With no change in
the voltage set point, a second load is switched on at t = 0 s to increase the total
load to 2 pu. Fig. 6.15 shows the voltage response, where the load increase causes
a voltage sag. In Fig. 6.15(a), SPAACE is not active, and the voltage returns to its
set point value in 127 ms. In Fig. 6.15(b), SPAACE is active and temporarily scales
up the set point to restrict the voltage reduction. Due to the system time constant,
SPAACE cannot completely prevent excursion of the voltage outside steady-state
ROAD; however, it significantly reduces the duration of this excursion—by 70%,
from 127 ms to 38 ms.
6.10.2 Study System V
This study system is the IEEE 34-bus test feeder augmented with three electronically
interfaced DER units as described in Section 5.4.2 and illustrated in Fig. 5.10. In this
section, the performance of SPAACE for current control in the grid-connected mode
Chapter 6. Online Set Point Adjustment With Correction 115
of this study system is evaluated. It is assumed that an excursion of less than 10%
from 1 pu is desired.
6.10.2.1 Current Set Point Step Change Without Prediction
This case study evaluates the performance of SPAACE in response to a step change
in the current of DER2 while DER1 and DER3 each inject 1 pu of current. All DER
units operate at unity power factor, with vq = 0 and iq = 0.
Fig. 6.16 shows the response of DER2 to a step change in its current set point from
0.92 pu to 1.08 pu. Fig. 6.16(a) shows that when SPAACE is not active, the current
experiences an overshoot of 37%, which may exceed the rating of DER2. However, as
shown in Fig. 6.16(b), when SPAACE is active (without prediction), the maximum
current is limited to 1.10 pu as desired, and its trajectory is confined within the
permissible range. There is no significant change in the speed of response.
The reverse step change, from 1.08 pu to 0.92 pu, is also applied to DER2. Without
SPAACE, Fig. 6.17(a) shows that the current has an undershoot of 37%. However,
when SPAACE is active (without prediction), the undershoot is significantly reduced,
as shown in Fig. 6.17(b). There is no appreciable change in the settling time.
6.10.2.2 Current Set Point Step Change With Prediction
The tracking capability of SPAACE can be improved by enabling the prediction
mechanism. The same step change scenarios as the previous case are applied to
System V, and the transient behavior of DER2 is studied; see Figs. 6.16 and 6.17.
Since the prediction facility of SPAACE is active in this case study, SPAACE detects
an increase in current prior to exceeding the limit and issues a temporary set point
earlier than that of the previous case. Consequently, the settling time is improved
compared with both previous cases. Comparison of Fig. 6.16(c) to Fig. 6.16(a)–(b)
and comparison of Fig. 6.17(c) to Fig. 6.17(a)–(b) show the effectiveness of prediction-
enabled SPAACE in mitigating transients and improving the response time. For the
remaining case studies, SPAACE with prediction is employed.
6.10.2.3 Current Control Subsequent to Load Energization
The performance of SPAACE in response to a disturbance caused by load energization
is investigated in this case study. A resistive 0.5 pu load is connected to bus 844 of
Chapter 6. Online Set Point Adjustment With Correction 116
0 1 2 3 4 5
0.9
1.0
1.1
(a)
I (p
u)
0 1 2 3 4 5
0.9
1.0
1.1
(b)
I (p
u)
0 1 2 3 4 5
0.9
1.0
1.1
(c)
I (p
u)
Time (ms)
Set PointResponse
Fig. 6.16. System V (DER2): Step change in current from 0.92 pu to 1.08 pu. (a)without SPAACE; (b) with SPAACE without prediction; (c) with SPAACE withprediction.
Chapter 6. Online Set Point Adjustment With Correction 117
0 1 2 3 4 5
0.9
1.0
1.1
(a)
I (p
u)
0 1 2 3 4 5
0.9
1.0
1.1
(b)
I (p
u)
0 1 2 3 4 5
0.9
1.0
1.1
(c)
I (p
u)
Time (ms)
Set PointResponse
Fig. 6.17. System V (DER2): Step change in current from 1.08 pu to 0.92 pu. (a)without SPAACE; (b) with SPAACE without prediction; (c) with SPAACE withprediction.
Chapter 6. Online Set Point Adjustment With Correction 118
0 1 2 3 4 5 6 7 8 9 10
0.9
1.0
1.1
I (p
u)
Time (ms)
No SPAACESPAACE
Fig. 6.18. System V (DER2): Load energization.
System II in Fig. 5.10. The current of DER1 and DER3 is 1 pu and the current of
DER2 is 1.08 pu. Initially, the load is offline; it is energized at t = 0 s.
Fig. 6.18 depicts the effect of load change on the current tracking of DER2. With-
out SPAACE, the current waveform experiences a peak of 1.15 pu and settles in 4 ms.
However, with SPAACE (with prediction), the current waveform does not experience
a significant change, and in particular, the peak does not violate the 1.10 pu limit.
This case study demonstrates the ability of SPAACE in recovering the current and
rejecting the load energization disturbance.
6.10.2.4 Simultaneous Current Set Point Change
This case study investigates the ability of SPAACE to improve the current tracking
of all three DER units in response to a simultaneous step change in their current
set points. This scenario can represent a sudden variation in the power exchange
of the DER units, e.g., due to a change in the wind speed. Fig. 6.19 shows the
trace of currents subsequent to a step change from 0.92 pu to 1.08 pu. Note that
the waveforms and intermediate set points are different in the three cases. This case
study confirms the effectiveness of SPAACE for a multi-DER microgrid, which is
particularly important because it is desired that the SPAACE controllers operate
independently.
6.10.2.5 Current Control During Start-Up
This case study investigates the ability of SPAACE to improve the start-up process
of DER2, from 0 pu to 1.09 pu. Fig. 6.20 shows the start-up response of DER2. This
scenario is similar to the start-up scenario using SPAA discussed in Subsection 5.4.2.1
and Fig. 5.11.
Chapter 6. Online Set Point Adjustment With Correction 119
0 1 2 3 4 5
0.9
1.0
1.1
(a)
I (p
u)
0 1 2 3 4 5
0.9
1.0
1.1
(b)
I (p
u)
0 1 2 3 4 5
0.9
1.0
1.1
(c)
I (p
u)
Time (ms)
Set PointResponse
Fig. 6.19. System V: Simultaneous step change in current from 0.92 pu to 1.08 pu ineach DER unit. (a) DER1; (b) DER2; (c) DER3.
Chapter 6. Online Set Point Adjustment With Correction 120
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.0
0.5
1.1
(a)
I (p
u)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.0
0.5
1.1
(b)Time (ms)
I (p
u)
Set PointResponse
Fig. 6.20. System V: Start-up response of DER2. (a) without SPAACE; (b) withSPAACE.
When SPAACE is not active, the response has an overshoot of 30%. When
SPAACE is active, it initially increases the set point to speed up the start-up pro-
cess. Because this causes an overshoot, SPAACE adjusts the temporary set point to
achieve an acceptable response trajectory. This reduces the overshoot to 17% and
the settling time to 55% of the unmanipulated case.
This case study confirms the applicability of SPAACE to reduce both the over-
shoot and the settling time in this application. Based on the performance of SPAA
in the same scenario, Subsection 5.4.2.1, either SPAA or SPAACE can be utilized
depending on which of the settling time or the overshoot is the priority.
6.10.3 Study System VI
This study system is the IEEE 13-bus test feeder, Fig. 6.21, which includes one-, two-,
and three-phase unbalanced lines with unbalanced spot and distributed loads; loads
include constant-impedance, constant-current, and constant-power components [123].
The feeder is disconnected from the grid, and an electronically interfaced DER unit
is connected to bus 675 through a three-phase transformer and a series filter. The
Chapter 6. Online Set Point Adjustment With Correction 121646 645 634633632611 684 675692671680652650
DG1Add’l LoadFig. 6.21. Study system VI: IEEE 13-bus unbalanced test feeder with the augmentedDER unit and load. Loads and shunt capacitors of the original test feeder are not shown.
interface parameters are given in Table 5.1. The objective of using this system is to
demonstrate the robustness of the proposed strategy to a high degree of imbalance.
6.10.3.1 Voltage Control Subsequent to Load Change
Although the SPAACE method is developed with the assumption of balanced opera-
tion, it can be applied to an unbalanced system as well. The DER unit supports the
real and reactive power demand of the microgrid.
The voltage is initially 1.09 pu, and the microgrid is operating with its unbalanced
loads and laterals and an additional three-phase balanced load of 0.9 pu at bus 675.
The additional load is switched off at t = 0 s; Fig. 6.22 shows the resulting voltage
waveform at the respective bus. When SPAACE is not active, the disturbance results
in a fairly sustained oscillatory operation of the microgrid. However, when SPAACE
is active, it manages to settle the voltage in 1 s by applying temporary voltage set
points. This case study confirms the ability of SPAACE in stabilizing the microgrid
subjected to a load change.
It is possible to improve the results by extending the proposed strategy to explicitly
handle an unbalanced system. In this case, the unbalanced system is represented in
the sequence network, and appropriate sequence network controllers and a generalized
PLL [124] are added to the system. SPAACE provides separate set points for each of
the three sequence networks.
Chapter 6. Online Set Point Adjustment With Correction 122
0.0 0.2 0.4 0.6 0.8 1.0
0.5
1.0
1.5
V (
pu)
Time (s)
No SPAACESPAACE
Fig. 6.22. System VI: Voltage transient in response to load change in an unbalancedsystem.
6.11 Conclusions
The requirement for higher degree of utilization of the utility grid infrastructure ne-
cessitates the power system to operate close to its limits. During transients, a power
system, particulary a small-scale power system such as a microgrid, may be exposed to
excessive power transfer and/or voltage values and angle/voltage/frequency instabil-
ity issues. Therefore, it is important that stability, control, and operational limits of
a power system are adequately addressed, monitored, and enforced. In conventional
power systems, this problem is addressed by designing controllers that ensure the
transients are within the acceptable limits. However, in a microgrid, where relatively
large changes in the system load, generation, and even topology frequency occur,
the controllers have limited robustness to system topology, operating point, and sys-
tem parameters. Thus, such changes can render the performance of the controllers
unsatisfactory.
SPAACE monitors the trend and instantaneous values of the response of a unit
and modulates its set point to achieve a desirable response, which is characterized
by having a small settling time and small excursion from the set point, e.g., small
overshoot. The salient features of SPAACE are that (i) it is robust with respect to
system parameters, (ii) it does not need a model of the system, and (iii) it is based
on local signals, i.e., it does not need a communication link.
This chapter discusses theoretical considerations regarding the SPAACE method.
The existence of appropriate time instances to switch the set point is proven, and it
is demonstrated that SPAACE does not alter the stability behavior of the system.
An upper bound for the choice of m is also derived.
A number of case studies are presented to confirm the effectiveness of the proposed
Chapter 6. Online Set Point Adjustment With Correction 123
strategy in improving the dynamic behavior of the system during transients. The
simulation results show that SPAACE is able to reduce the excursion of the response
from the allowable region under different control schemes (voltage and current) and
under both balanced and unbalanced test conditions.
Chapter 7
Conclusions
7.1 Summary
This dissertation proposes control strategies for the next generation microgrids (and
power systems) to improve their dynamic behavior during transients. These strategies
pertain to the next generation power system because they can take advantage of the
availability of communication and/or computation facilities.
This dissertation proposes two sets of strategies:
• A hierarchical control framework, in which the notion of potential functions is
developed for secondary control using communication (PFM and GPFM); and
• An algorithm for response shaping, in which the notion of set point modulation
is developed for achieving a desired response trajectory without communication
(SPAA and SPAACE).
These strategies craft the trajectory of the system from an initial set point to a final
set point. Although the first strategy can also be utilized to find appropriate final set
points, this dissertation does not focus on this mode of operation. SPAA/SPAACE
can be used as the backup of PFM/GPFM in case communication fails. To the best
of the author’s knowledge, this dissertation is the first attempt in developing such
strategies.
A diverse set of study systems, ranging from balanced small systems to unbalanced
large systems, are employed to evaluate the performance of the proposed strategies.
The studies are performed in PSCAD/EMTDC and MATLAB environments, and the
124
Chapter 7. Conclusions 125
algorithms are implemented in Fortran and MATLAB languages. Systems are mod-
eled with different levels of detail: generic stable system, steady state, electromagnetic
transients, and electromechanical transients representation.
7.2 Conclusions
A microgrid and its constituent DER units have practical limitations in withstand-
ing excursion of electrical variables, e.g., voltage and current, from the prescribed
regions. Therefore, it is imperative that strategies exist to limit such excursions. The
proposed methods, i.e., PFM, GPFM, SPAA, and SPAACE, ensure this by designing
the trajectory of the microgrid and/or DER units. The main conclusion of this dis-
sertation is that by appropriately designing the trajectory, it is possible for a power
system to operate closer to its limits—a trend that is necessitated by the increase in
power demand.
The general conclusions of this dissertation are as follows.
• A distributed implementation of the PFM strategy is required when data ex-
change between units is limited. In this case, allocation of constraints to sub-
problems needs to be carefully considered; otherwise, convergence of the decom-
position algorithm cannot be guaranteed. The performance of the distributed
implementation is comparable to that of the centralized implementation.
• Effective response shaping is possible through the SPAACE method. SPAACE
reduces the overshoot and/or settling time of the system response to a distur-
bance. The main objective of SPAACE is to reduce the duration in which the
response violates the prescribed limits. SPAACE is augmented to an existing
controller and improves its performance. SPAACE is successfully applied for
both voltage and current control.
• Employing SPAA permits the controlled unit to undergo a large change in its
set point, e.g., black start-up, without its response violating the limits. SPAA
calculates appropriate intermediate set points for the controlled unit. This
method outperforms gradually increasing the set point, i.e., ramping up the
set point, since many power system units employ PI-based controllers. Both
voltage and current controllers are shown to benefit from this strategy.
Chapter 7. Conclusions 126
Some specific conclusions of this dissertation based on the reported case studies
are as follows:
• GPFM is effective in maintaining and enforcing satisfactory operation of the
power system. In a case study involving a load change (Subsection 4.6.4),
GPFM reduces the settling time by 64%. In another case involving line outage
and controller failure (Subsection 4.6.6), GPFM stabilizes an otherwise collaps-
ing system.
• SPAACE is effective in mitigating transients. Some representative results are as
follows: In the scenarios studying a step change in the current set point (Sub-
section 6.10.2.2), SPAACE is able to mitigate an overshoot of 37% without in-
creasing the settling time. In a load energization scenario (Subsection 6.10.2.3),
SPAACE is able to eliminate a current peak of 1.15 pu when a 1 pu load change
occurs. In another load energization scenario (Subsection 6.10.1.3), SPAACE
reduces the duration of the excursion of voltage beyond the permissible limits
by 70%. Moreover, SPAACE is successful in stabilizing an otherwise oscillatory
behavior of voltage in 1 s when a 1 pu load shedding occurs (Subsection 6.10.3.1).
• In black start-up scenarios, both SPAA and SPAACE are effective. In the
current-controlled scenario presented in Subsection 5.4.2.1, SPAA reduces an
overshoot of 30% to zero while maintaining the original settling time of the
system. In the same scenario repeated for SPAACE in Subsection 6.10.2.5, the
overshoot is reduced from 30% to 17% and the settling time is reduced by 45%.
7.3 Contributions
The main contributions of this dissertation are as follows:
• This dissertation introduces the concept of devising intermediate set points for
designing the trajectory of the system from one set point to another. The
notion of potential functions is defined and employed. Each controllable unit is
associated with a potential function that conveys information about the unit’s
measurements, constraints, and control objectives. This dissertation proposes
potential function minimization (PFM) as an approach to steer these units
toward their final set points. This dissertation also proposes a generalized PFM
Chapter 7. Conclusions 127
strategy (GPFM) to (i) explicitly accommodate both system-wide and local
constraints and (ii) enable a distributed implementation of the optimization
algorithm.
• This dissertation proposes and evaluates a strategy to improve set point tracking
of power system apparatus. This strategy calculates and issues temporary set
points to manage the over- and undershoot of the response. This strategy does
not replace the existing controllers; rather, it augments them and improves their
performance. Therefore, this strategy is particularly effective when the perfor-
mance of a controller, designed for a specific system with certain operational
assumptions, deteriorates as a results of changes in the system topology and/or
generation and demand profiles. This strategy is implemented in SPAA and
SPAACE variations depending on the availability of models and depending on
the considered disturbances. Although this strategy is presented in the context
of power system applications, it can also be used for other control systems that
need close tracking of their set points.
This dissertation also introduces/redefines a multitude of concepts, e.g., hierar-
chical control for a microgrid (primary, secondary, and tertiary control), the region
of acceptable dynamic operation (ROAD) that shrinks or expands based on the time
frame of interest, and trajectory shaping. Moreover, a number of software tools are de-
veloped to assist in understanding the underlying concepts introduced in this disserta-
tion. These software tools are accessible publicly online at http://mehrizisani.com.
7.4 Future Work
This dissertation proposes novel strategies and introduces new concepts. In addition
to the microgrid application, these ideas are applicable to other small-scale power
systems, e.g., naval ships, military systems, and aircrafts. In these systems, spinning
reserve may not exist, inertia is not significant, and the notion of slack bus is not nec-
essarily valid. Consequently, an operation philosophy that shares the regulation and
control responsibilities among the constituent components is sought. The proposed
strategies are equally applicable to large interconnected power systems.
It is suggested that the future work focuses on the following areas:
Chapter 7. Conclusions 128
• The proposed distributed implementation of GPFM assumes that the availabil-
ity of communication links between units does not change over time. However,
scenarios are conceivable in which the availability of these links changes over
time, e.g., when a DER unit is disconnected from the host system for main-
tenance. In this scenario, algorithms that can handle time-varying connection
between units are required to enable distributed optimization when subproblems
have different constraint sets. Currently, this an active area of research [84], [85]
and such methods do not yet exist. Once these methods are developed, they
can be employed in the GPFM framework.
• SPAACE relies on local measurements to modulate the set point, and its per-
formance is shown to be satisfactory even when multiple disturbances occur in
the system. It is suggested to enable SPAACE to utilize system-wide informa-
tion and wide-area measurements in a coordinated effort to further increase the
system damping.
• The proposed strategies assume that the power system and its components are
balanced. Since unbalanced systems are abundant at the distribution level, an
extension of this work is to systematically design these strategies for unbalanced
systems.
Appendices
129
Appendix A
Working Definition of the
Microgrid
Fig. A.1 shows the schematic diagram of a generic microgrid, which can have any
arbitrary circuit configuration. A microgrid is a cluster of collocated DER units
(DG and DS units) and loads. Each DER unit is interfaced to the microgrid at
its respective point of connection (PC). A microgrid (i) is served by a distribution-
voltage class network, (ii) is interfaced to the main grid at the point of common
coupling (PCC), (iii) can operate in the grid-connected mode, the islanded mode,
and the transition between these two modes, and (iv) is able to meet the demand of
at least a major portion of its local loads [125].
An active distribution network (ADN) is a microgrid that is equipped with power
management and supervisory control for the loads, DG units, and DS units. The
formal definition of an ADN is [58]
Active networks are distribution networks with the possibility of control-
ling a combination of distributed energy resources (generators, loads, and
storage). A distribution storage operator (DSO) has the possibility to
manage electricity flows using a flexible network topology. DER units
take some degree of responsibility for system support, which will depend
on a suitable regulatory environment and connection agreements.
A cognitive microgrid is an intelligent microgrid that features an adaptive ap-
proach for the control of the microgrid components. In the context of the virtual
power plant (VPP) [14], the cognitive microgrid is presented to the host grid at the
130
Appendix A. Working Definition of the Microgrid 131
Microgrid ∞S
PCC
Main Grid
DERn
PCn
DER3
PC3
DER2
PC2
DER1
PC1
Fig. A.1. Schematic diagram of a generic multiple-DER microgrid.
PCC as a single controllable entity that has a prespecified performance. The internal
mechanics of the VPP is hidden from the host power system.
Appendix B
Mathematical Treatment of the
Potential Function–Based Control
The underlying idea of the potential function–based control is borrowed from the field
of autonomous control; however, it is significantly modified and developed to adapt
to the power system control problem. This appendix presents the mathematical
foundation of the potential function–based control: it states (i) the conditions under
which the control is possible and (ii) the reason minimizing the potential function is
equivalent to satisfying the control goal.
B.1 Definitions and Examples
This section provides the background required for proving the central theorem. Two
examples are presented as specific cases of the central theorem.
Definition B.1. Each node in graph GµG represents a controllable unit of the mi-
crogrid. The term graph is used in the extended sense, i.e., the position of nodes is
important. For example, the position of a node representing a voltage-controlled DG
unit can be represented by the d- and q-components of its measured voltage. In R2,
a node is represented by z = (zd, zq) or equivalently, z = zd + jzq.
Definition B.2. A point kinematic unit represented by a node in GµG is either a unit
with no internal dynamics or a unit with negligible internal dynamics in the time
frame of interest. For example, the dynamics of primary controllers are neglected in
the time frame of interest to the secondary controller; therefore, a secondary controller
132
Appendix B. Mathematical Treatment of the PF-Based Control 133
deals with the combination of a unit and its associated primary controller as a point
kinematic unit.
Definition B.3. A visibility graph is a graph in which an edge between nodes i and
j means that units i and j can “see,” i.e., have access to the measurements of, each
other.
Definition B.4. A digraph (directed graph) is a graph with directed edges: the
ability of unit i to see unit j does not imply that unit j can also see unit i. In this
work, a weight factor is associated with each edge.
Definition B.5. The adjacency matrix A associated with graph GµG is a square
matrix for which element aij of A is unity iff there is an edge (directed edge, if GµG
is a digraph) between nodes i and j; otherwise, aij = 0.
Definition B.6. The outdegree of a node in graph GµG is the number of edges (di-
rected edge, if GµG is a digraph) that leave that node. Matrix D is defined as the
diagonal matrix of outdegrees of nodes in GµG.
Definition B.7. The Laplacian of graph GµG is defined as D − A.
Definition B.8. The graph GµG has a globally reachable node zi if zi is reachable
from all other nodes of the graph, considering directed edges.
Definition B.9. In a digraph GµG a directed path is a finite sequence of directed
edges.
Definition B.10. A digraph GµG is strongly connected if every node is reachable from
every other node. Node j is reachable from node i if there is a directed path from
node i to node j.
Definition B.11. A matrix A is nonnegative if all elements aij ≥ 0. We use the
notation A ≥ 0.
Definition B.12. The spectral radius of matrix A is the maximum magnitude of all
the eigenvalues. That is, ρ(A) = max|λ| : λ ∈ σ(A).
Definition B.13. A matrix A is irreducible if there is a permutation matrix P such
that PAPT is block upper triangular.
Appendix B. Mathematical Treatment of the PF-Based Control 134
Im λ
Re λ
a33
a22a11
Fig. B.1. Representation of the disks in the Gersgorin’s Theorem.
z1
c1
c2
z2
zd
zq
Fig. B.2. Example 1: Beamer pursuit.
Theorem B.1. Let A ≥ 0. Then A is irreducible iff GµG is strongly connected.
Theorem B.2. If the diagraph is strongly connected, zero is a simple eigenvalue of
its associated Laplacian L.
Theorem B.3 (Gersgorin’s Theorem). Assume an n× n matrix A = [aij]. For each
row i of A, construct a disk centered at aii with a radius equal to the sum of absolute
values of the remaining elements in that row, Fig. B.1. The eigenvalues of A are in
the union of such disks. That is,
σA ⊂n⋃i=1
λ : |λ− aii| ≤
n∑j=1j 6=i
|aij|
B.1.1 Example 1: Beamer Pursuit
Consider two voltage-controlled DG units, Fig. B.2, where each unit has exactly two
controllable (and measurable) variables. The goal is to stabilize each DG unit to
its predetermined set point ci = (z∗di, z∗qi). As mentioned, the units are considered
kinematic.
Appendix B. Mathematical Treatment of the PF-Based Control 135
The following potential function is suggested:
φ1(z) =1
2(c1 − z1)2
φ2(z) =1
2(c2 − z2)2,
(B.1)
and the corresponding control law is
z1 = (c1 − z1)
z2 = (c2 − z2),(B.2)
which can be rewritten as
u1 = −u1
u2 = −u2,(B.3)
where ui = (zi − ci). With u defined as the aggregate vector (u1, u2), the solution is
u(t) = Ae−t or
z(t) = Ae−t + c, (B.4)
where A is a matrix calculated from the initial values of the units. The units always
stabilize to c regardless of the value of A. Therefore, this control law is valid.
B.1.2 Example 2: Cyclic Pursuit
Consider four units similar to example 1, Fig. B.3. A set of predetermined set points
exists in this example. Because of limited communication links, each unit i seeks to
maintain its proximity to only unit (i+ 1) mod 4, i.e., to unit i+ 1 when i = 1, 2, 3
and to unit 1 when i = 4. The following control law is suggested.
z1 = (c1 − z1) + w(z2 − z1)
z2 = (c2 − z2) + w(z3 − z2)
z3 = (c3 − z3) + w(z4 − z3)
z4 = (c4 − z4) + w(z1 − z4),
(B.5)
Appendix B. Mathematical Treatment of the PF-Based Control 136
z1
c1c2
z2 c3c4
z3
z4
zd
zq
Fig. B.3. Example 2: Cyclic pursuit.
where w is the weight factor (real and positive).
In matrix form,
z = c+ (wW − I)z, (B.6)
where
c =
c1
c2
c3
c4
, z =
z1
z2
z3
z4
, W =
−1 1 0 0
0 −1 1 0
0 0 −1 1
1 0 0 −1
. (B.7)
The solution of this differential equation is
z = (I − wW )−1c+ e−(I−wW )t. (B.8)
The units converge to the steady-state solution z0 = (I−wW )−1c if the time-varying
term e−(I−wW )t is stable to zero. Since λW ∈ −2,−1 ± j, 0, the real part of the
eigenvalues of W is nonpositive, <λW ≤ 0. Therefore,
∀w ∈ R≥0, <wλW − 1 ≤ 0, (B.9)
meaning that the time-varying term decays to zero for all values of w. By dynamically
adjusting w, z0 can be modified. For example, w can be set to zero for the units to
converge to c.
Appendix B. Mathematical Treatment of the PF-Based Control 137
B.2 Central Theorem
The central theorem is the generalization of the previous examples. In general, each
unit can have communication links to any number of other units, and different weights
can be used for each link. For example, for a three-unit pursuit
z1 = w12(z2 − z1) +w13(z3 − z1)
z2 = w21(z1 − z2) +w23(z3 − z2)
z3 = w31(z1 − z3) +w32(z2 − z3)
, (B.10)
where wij ∈ R≥0. In matrix form, z = Wz, where
W =
−w12 − w13 w12 w13
w21 −w21 − w23 w23
w31 w32 −w31 − w32
. (B.11)
Notice that irrespective of the weight values wij, the rows of W always sum to zero.
It is easy to confirm that this is also valid for the case of n units.
The central theorem is stated as follows:
Theorem B.4 (Central Theorem). Let GµG be a digraph in which each node z repre-
sents a point kinematic unit, and the location of the node represents the measurements
of that unit. The control law
z = (c− z) +Wz, (B.12)
where W is defined similarly to (B.11), with wij ∈ R≥0, leads to stabilization of the
units to the aggregate vector c, if at least one node in GµG is globally reachable.
Proof. For n units, the ith row of the matrix W = W − I is
Wi =[wi1 wi2 · · · win
], (B.13)
Appendix B. Mathematical Treatment of the PF-Based Control 138
Re λ
Im λ
−1
Fig. B.4. Gersgorin disks for the central theorem.
where
wij = wij (i 6= j)
wii = −n∑j=1j 6=i
wi1 − 1. (B.14)
According to Gersgorin’s Theorem, the eigenvalues of W are in the union of n disks,
each centered at wii with a radius of |wii| for 1 ≤ i ≤ n. Such disks are always
entirely in the left-hand plane. Consequently, all eigenvalues of W are stable. Since
λW = λM − 1, the eigenvalues of W are also stable, Fig. B.4. Therefore, this control
law is stable.
Theorem B.5 (Central Theorem for Potential Functions). Let GµG be a digraph
in which each node z represents a point kinematic unit, and the location of each
node represents its measurements. Assume the visibility graph is symmetric, i.e., if
unit i has access to the measurements of unit j, then unit j also has access to the
measurements of unit i. Assume the graph is time invariant and strongly connected
with m links. Let ei (i = 1, . . . ,m) denote the links represented as vectors, i.e.,
ei = zj − zk for some units j and k that have access to the measurements of each
other. Define a potential function as φ = 12‖e‖2. Then, the control law u = −Jφ(z)T
results in the closed-loop system z = −Lz, where L is the 2n × 2n matrix obtained
by multiplying each element of L (the Laplacian of the visibility graph) by the 2 × 2
identity matrix.
Appendix B. Mathematical Treatment of the PF-Based Control 139
1
2
3
4
5
e1
e2
e3e4
Fig. B.5. A sample four-node, four-edge visibility graph.
Proof. We will present the proof for the case that units are in R2. Extension to higher
dimensions is straightforward.
Let z denote the aggregate state (z1, z2, . . . , zn) and e denote the aggregate links
(e1, e2, . . . , em). We have e = Pz for a certain 2m×2n matrix P . Consider an example
with n = 5 units and m = 4 links as shown in Fig. B.5. The matrices, when x- and
y-components of each node are shown separately, are as follows:
e1x
e1y
e2x
e2y
e3x
e3y
e4x
e4y
︸ ︷︷ ︸
e8×1
=
1 0 0 0 −1 0 0 0 0 0
0 1 0 0 0 −1 0 0 0 0
1 0 0 0 0 0 −1 0 0 0
0 1 0 0 0 0 0 −1 0 0
0 0 1 0 0 0 −1 0 0 0
0 0 0 1 0 0 0 −1 0 0
0 0 0 0 0 0 1 0 −1 0
0 0 0 0 0 0 0 1 0 −1
︸ ︷︷ ︸
P8×10
z1x
z1y
z2x
z2y
z3x
z3y
z4x
z4y
z5x
z5y
︸ ︷︷ ︸
z10×1
. (B.15)
Note that there is exactly one 1 and one −1 in each row of P . This is because each link
connects exactly two distinct nodes. And because there is at most one link between
two nodes, the pattern of each row is different from others. Also, the number of
nonzero elements in each column shows the outdegree (or indegree) of the respective
node. Moreover, row i+ 1 (i = 1, 3, 5, 7) is obtained by shifting row i one column to
right.
Appendix B. Mathematical Treatment of the PF-Based Control 140
The Laplacian of the visibility graph can be found from its outdegree and adja-
cency matrices:
L =
2
1
1
3
1
︸ ︷︷ ︸
D5×5
−
1 1
1
1
1 1 1
1
︸ ︷︷ ︸
A5×5
=
2 0 −1 −1 0
0 1 0 −1 0
−1 0 1 0 0
−1 −1 0 3 −1
0 0 0 −1 1
.
(B.16)
Multiply each element of L by I2×2 to obtain the 2n× 2n matrix L as shown below:
L =
2 0 0 0 −1 0 −1 0 0 0
0 2 0 0 0 −1 0 −1 0 0
0 0 1 0 0 0 −1 0 0 0
0 0 0 1 0 0 0 −1 0 0
−1 0 0 0 1 0 0 0 0 0
0 −1 0 0 0 1 0 0 0 0
−1 0 −1 0 0 0 3 0 −1 0
0 −1 0 −1 0 0 0 3 0 −1
0 0 0 0 0 0 −1 0 1 0
0 0 0 0 0 0 0 −1 0 1
10×10
(B.17)
which is equal to PTP .
Appendix B. Mathematical Treatment of the PF-Based Control 141
For a general proof, consider the n× n matrix Psmall:
Psmall =[c1 c2 · · · cn
], (B.18)
where ci’s are the columns of Psmall. Therefore,
PTsmall =
cT
1
cT2
...
cTn
. (B.19)
Note that elements of Psmall are 1, 0, or −1. The element lii of matrix L is the sum of
square values of elements of the vector ci, which is equal to the outdegree of the ith
node. Therefore, the elements on the main diagonal of L are the same as elements
dii of the outdegree matrix D. The element lij = ci · cj is either −1 (when there is a
link between zi and zj, or respective x’s, and y’s) or 0 (when there is no link between
zi and zj). As columns of Psmall are distinct and there is exactly one or zero links
between each two nodes, −1 and 0 are the only possible values for lij. Therefore, the
off-diagonal elements equal negative of those of the adjacency matrix A. Thus,
PTsmallPsmall = D − A = L. (B.20)
Note that multiplying each element of L by I2×2 is the same as adding a shifted
version of each row just below itself and adding a shifted version of each column just
to its right. Therefore, the eigenvalues of the resulting matrix L are the same as those
of the original matrix L but with an algebraic multiplicity of 2. P is obtained from
Psmall by multiplying each of its elements by I2×2.
Define the potential function as
φ =1
2
m∑i=1
‖ei‖2 =1
2‖e‖2. (B.21)
Therefore, φ = 0 iff all units are collocated. This suggests a gradient control law
using the Jacobian of φ.
Appendix B. Mathematical Treatment of the PF-Based Control 142
In terms of z,
φ(z) =1
2‖e‖2
=1
2‖Pz‖2
=1
2(Pz)T(Pz)
=1
2zTPTPz
=1
2zTLz.
(B.22)
Recall that the Jacobian of xTQx is xT(Q + QT), which equals 2xTQ when Q is
symmetric (as is the case for L). Therefore, Jacobian Jφ(z) of φ(z) is,
Jφ(z) =1
2D(zTLz)
= zTL.(B.23)
With the control law u = −Jφ(z)T, we have
z = −Jφ(z)T
= −(zTL)T
= −Lz,
(B.24)
where the last equality follows because LT = L. L is nonnegative and irreducible—its
corresponding graph is strongly connected. Thus, it has a simple eigenvalue of 0 and
the rest of its eigenvalues are in the right-hand plane (by Gersgorin’s Theorem). L
has 0 as a repeated eigenvalue, but for the two decoupled variables x and y. Hence,
x’s rendezvous and so do the y’s, meaning that z’s (or equivalently, units) rendezvous.
Therefore, we derived the control law u = −Lz using a potential function.
Appendix C
Developed Software Tools
In this appendix, three developed stand-alone software tools—developed as tools
for the comprehension of the concepts developed in this dissertation—are presented.
These softwares are developed in the MATLAB programming language, feature an
easy-to-use graphical user interface (GUI), and rely on the matrix analysis capabili-
ties of MATLAB. The p-codes of the developed softwares are freely available online
for download at http://mehrizisani.com. The codes are self-contained and run by
simply entering their names at the MATLAB prompt.
The first software, discussed in Section C.1 and available at http://mehrizisani.
com/potential, serves as a test platform to assess different potential function types
in various scenarios. It can also be used in educational settings. The second software
tool, discussed in Section C.2 and available at http://mehrizisani.com/spaa, al-
lows experimenting with different parameters of SPAACE. The third software tool,
discussed in Section C.3 and available at http://mehrizisani.com/loadmodeling,
allows detailed dynamic simulation of a power system including generators and control
circuitry.
C.1 Design of Potential Functions
In this software, potential functions are employed to control a number of kinematic
points, e.g., DG units and autonomous robots. In this context, each unit has a
monitoring device (e.g., a camera for a robot and a voltmeter for a DG unit) that
enables it to measure its distance from the target, other units, and obstacles. Using
such distance information, each unit defines a potential function whose minimization
143
Appendix C. Developed Software Tools 144
translates to reaching a common target. The software can also be used as an educa-
tional tool for experimenting with the potential functions using autonomous units in
a two-dimensional space [31].
C.1.1 User Interface
The software is written with usability and visual appeal in mind and as such, it
features a GUI as shown in Fig. C.1. The interface allows adjustment of various
parameters of the software as well as interaction with the elements (units, obstacles,
and the target). The interface consists of two main sections. The left section is a
canvas that shows the position of the elements and allows the user to relocate them
using the drag-and-drop functionality. The upper-right corner of the GUI hosts three
tabs for controlling the behavior of the units, obstacles, and target. The midright
section contains the parameters of the potential function. The simulation can be
started, paused, or stopped, and the software window content can be saved/printed
using the controls located in the lower-right area.
A nonexhaustive list of the software parameters that can be adjusted through the
GUI is as follows:
1. Target size;
2. Mobility of the obstacles;
3. Maximum speed of units;
4. Proximity penalty for units;
5. Position of elements on screen;
6. Number of units and obstacles;
7. Size and location of the canvas;
8. Type of the potential functions;
9. Visibility of the trajectory of the units;
10. Weight factors of potential function terms;
11. Type of units and obstacles (point or circular); and
Appendix C. Developed Software Tools 145
Fig. C.1. The graphical user interface of the developed software for simulation ofautonomous units.
12. Behavior of obstacles at the boundary (whether to bounce off the edge or to
continue from the opposite edge).
C.1.2 Code Details
Each unit is represented by its position (a complex number z = x + jy) and its
heading. The heading of each unit is determined from a gradient descent method by
minimizing its respective potential function. Each potential function has one term to
drive the unit toward the target, (nv − 1) terms to keep the unit in close proximity
of the other unit, and no terms to keep the unit away from the obstacles; nv is the
number of units, and no is the number of obstacles.
The new position of a unit is calculated from its current position and heading.
The software calculates the speed of each unit and compares it with the maximum
allowable speed. If the speed exceeds the maximum, the software adjusts the speed
but retains its heading. When a unit reaches a boundary, its heading or position
is adjusted (depending on the requested behavior) to restrain the unit within the
boundaries.
Appendix C. Developed Software Tools 146
C.1.3 Test Scenarios
Two test scenarios are presented in this section. The screen capture movies of both
scenarios can be downloaded from [51].
In the first test scenario, the software simulates seven units and three stationary
obstacles. The target is circular and initially in the middle of the canvas. Fig. C.2(a)
shows a snapshot of the beginning of the simulation. The element positions are
initialized randomly. Fig. C.2(b) shows that the units stabilize to an equilibrium
formation around the target. Note that the units are cluttered around the top of the
target because of the presence of a circular obstacle close to the bottom of the target.
The trajectories of the units are shown in Fig. C.2(c). The target is moved using
the drag-and-drop facility of the GUI to the corner of the canvas. Fig. C.2(d) shows
that units follow the target. Subsequently, the behavior of obstacles is changed from
stationary to moving. Fig. C.2(e) shows that when an obstacle moves toward this
group of units, they rearrange to avoid the obstacle while keeping their proximity.
Another scenario is depicted in Fig. C.2(f). In this scenario, 30 units are present
and follow a point target. No obstacles is present. As shown, the units position
themselves symmetrically in the equilibrium formation.
C.2 Design of SPAACE Parameters
A software tool is developed to evaluate the viability of the concept of SPAACE and
study the effect of selection of time instances T1 and T2 on the system response. This
software tool is available online at http://mehrizisani.com/spaa. Fig. C.3 shows
the GUI of the developed software. The left pane shows the traces for the original
response, the manipulated response, and the set points. The right pane contains
sliders that adjust the initial and final set points, time instances T1 and T2, scaling
factor m, and damping factor ζ and natural frequency ωn for a second-order system.
When these parameters change, the graph in the left pane is updated automatically.
This enables fast and efficient observation of the effect of these parameters on the
performance of SPAACE.
Appendix C. Developed Software Tools 147
(a) (b)(c) (d)(e) (f)
Fig. C.2. Test scenarios for the developed PFM software.
Appendix C. Developed Software Tools 148
Fig. C.3. Developed software for experimenting with parameters of SPAACE.
C.3 Dynamic Simulation of Power Systems
This section provides details about the simulation tool developed for the study sys-
tem III discussed in Subsection 4.6. Dynamics of synchronous generators with their
exciters, power system stabilizers, and damper windings are considered. Loads are
represented with an aggregate model consisting of a static component (ZIP) and a
dynamic component. The developed tool is general and can simulate the transient
behavior of any power system given appropriate data files. Components models and
their differential algebraic equations (DAE), the simulation algorithm, and implemen-
tation in MATLAB are discussed in the remainder of this section. This software tool
is available online at http://mehrizisani.com/loadmodeling.
C.3.1 Component Models
In this section the models used for synchronous generators, exciters, and loads are
discussed. A detailed discussion of the models can be found in [93]. All quantities
are in per unit values (Lad-base reciprocal per unit system).
Appendix C. Developed Software Tools 149
C.3.1.1 Synchronous Generators
In this work, a sixth-order model is used for round-rotor generators (GENROU model
in PSS/E [126]), which includes one field winding ψfd, one damper winding aligned
with the d-axis ψ1d, two damper windings aligned with the q-axis ψ1q and ψ2q, and
two mechanical variables: rotor angle δ and rotor speed ωr. For salient-pole machines,
a fifth-order model (GENSAL model in PSS/E) with one damper winding ψ1q on the
q-axis is employed
Time derivatives The equations of motion are as follows:
∆ω =1
2H(Tm − Te −D∆ωr)
δ = ω0∆ωr,(C.1)
where time t is in seconds, rotor angle δ is in electrical radians, and other quantities
are in per unit. The electrical torque is given by
Te = ψadiq − ψaqid. (C.2)
In the per unit system, power and torque have the same numerical values. The
dynamic equations of the rotor circuit are
ψfd = ω0
(efd +
ψad − ψfdXfd
Rfd
)ψ1d = ω0
(ψad − ψ1d
X1d
R1d
)ψ1q = ω0
(ψaq − ψ1q
X1q
R1q
)ψ2q = ω0
(ψaq − ψ2q
X2q
R2q
),
(C.3)
where ω0 is the steady-state rotor speed. ψad and ψaq are the d- and q-axis mutual
flux linkages, which are calculated as
ψad = X ′′ads
(−id +
ψfdXfd
+ψ1d
X1d
)ψaq = X ′′aqs
(−iq +
ψ1q
X1q
+ψ2q
X2q
),
(C.4)
Appendix C. Developed Software Tools 150
where
X ′′ads =1
1/Xads + 1/Xfd + 1/X1d
= X ′′d −Xl
X ′′aqs =1
1/Xaqs + 1/X1q + 1/X2q
= X ′′q −Xl,(C.5)
and Xads and Xaqs are saturated values of the d- and q-axis mutual reactances given
by
Xads = KsdXadu
Xaqs = KsqXaqu.(C.6)
In this work, the saturation constants Ksd and Ksq are assumed to be unity.
Initial values Initial values of the states can be calculated from the power flow
solution of the system before applying the disturbance. The power flow solution gives
terminal real power Pg (which is equal to the mechanical torque Tm), reactive power
Qg, and voltage Vbus. The terminal current, expressed in the common RI-frame, is
Ibus =Pg − jQg
V ∗bus
. (C.7)
The rotor angle (equivalently, the q-axis angle) with respect to the RI-frame is
δ0 = ](Vbus + (Ra + jXq)Ibus
). (C.8)
From Park’s transformation and with δ and Vbus in the RI-frame, the dq-axis voltages
are calculated as [ed
eq
]=
[sin δ0 − cos δ0
cos δ0 sin δ0
][VR
VI
], (C.9)
and dq-axis currents are calculated similarly from RI-axis currents. Field current is
calculated from
ifd =eq +Raiq +Xdid
Xad
, (C.10)
and field voltage is calculated from
efd = Rfdifd. (C.11)
Appendix C. Developed Software Tools 151
The initial values of states are calculated from the following set of equations:
∆ω = 0
δ = δ0
ψfd = (Xad +Xfd)ifd −Xadifd
ψ1d = Xad(ifd − id)
ψ1q = −Xaqiq
ψ2q = −Xaqiq
(C.12)
For the fifth-order model, ψ2q is ignored.
Algebraic network interface equations Stator transients ψd and ψq are ne-
glected because stator and network transients are fast and reach the steady state
before the simulation time is advanced by one time step. By further neglecting speed
variations and subtransient saliency, each synchronous machine can be modeled as
a variable voltage source behind a constant impedance, as shown in Fig. C.4. The
impedance is given given by
Z ′′gen = Ra + jX ′′, (C.13)
where X ′′ = X ′′d = X ′′q . Transforming the local dq-components of the generator into a
global RI-reference frame, the RI-components of the voltage source can be calculated
from [E ′′R
E ′′I
]=
[sin δ cos δ
− cos δ sin δ
][E ′′d
E ′′q
], (C.14)
where dq-voltages are given by
E ′′d = −X ′′aqs(ψ1q
X1q
+ψ2q
X2q
)E ′′q = X ′′ads
(ψfdXfd
+ψ1q
X1q
).
(C.15)
Finally, the interface equation is
Iinj =E ′′RI − Vbus
Z ′′gen
. (C.16)
Appendix C. Developed Software Tools 152
X"Ra
EqÐδ = E"R +jE"I
EtÐ0
Iinj
Fig. C.4. Synchronous generator circuit interface.
W
W
sT
sT
1STABK
2
1
1
1
sT
sT
RsT1
1
B
A
sT
sT
1
1
E
A
sT
K
1
Δω
Et
vs,max
vs,min
v3v2
v1
+
Vref
+− vx v4
Efd
EF,max
EF,minPhase
CompensationWashoutGain
Transducer ExciterRegulator
vs
Fig. C.5. Excitation system model.
C.3.1.2 Excitation System
A fifth-order model is used to represent the excitation system. This model is a
combination of PSS/E SCRX and IEEE ST1A models and consists of a second-order
PSS, a first-order voltage transducer, and a second-order exciter, as shown in Fig. C.5.
Time derivatives The differential equations governing the PSS are
v1 =1
TR(Et − v1)
v2 = KSTAB∆ωr −1
TWv2
v3 =1
T2
(T1v1 + v2 − v3),
(C.17)
where
vs = v3
vs,min ≤vs ≤ vs,max.(C.18)
Appendix C. Developed Software Tools 153
The remaining states of the excitation system are
v4 =1
TB(vx + TAvx − v4)
Efd =1
TE(KAv4 − Efd),
(C.19)
where
vx = Vref − v1 + vs
vx = −v1 + vs,(C.20)
and
EF,min ≤ Efd ≤ EF,max, (C.21)
and the non-windup limits are
Efd ← 0 if
Efd ≥ EF,max, Efd > 0
Efd ≤ EF,max, Efd < 0.(C.22)
Initial values The initial values of the states of the excitation system are
Efd =Xadu
Rfd
efd
v1 = Et
v2 = 0
v3 = 0
v4 =EfdKA
,
(C.23)
and the AVR reference voltage is given by
Vref =EfdKA
+ Et. (C.24)
Appendix C. Developed Software Tools 154
C.3.1.3 Load Models
The developed program can handle loads that are a combination of constant impedance
(Z), constant current (I), constant power (P), and exponential recovery (ER) compo-
nents [127].
ZIP/Exponential recovery model The real and reactive components of a load
represented by the ZIP model are
Pd = P0
(Kpz
(V
V0
)2
+Kpi
(V
V0
)+Kpp
)+ Pd,ER
Qd = Q0
(Kqz
(V
V0
)2
+Kqi
(V
V0
)+Kqq
)+Qd,ER,
(C.25)
where P0, Q0, and V0 are steady-state values of the load real and reactive power
and voltage, respectively, and are obtained from power flow solution. The sum of
Kp and Kq coefficients is unity. Pd,ER and Qd,ER represent the exponential recovery
component of the model and are defined as
Pd,ER = xp + P0Kpe
(V
V0
)npt
Qd,ER = xq +Q0Kqe
(V
V0
)nqt
,
(C.26)
where xp and xq are load state variables, defined by the following differential equation:
xp =1
Tp
(P0Kpe
(V
V0
)nps
− Pd,ER)
xq =1
Tq
(Q0Kqe
(V
V0
)nqs
−Qd,ER
).
(C.27)
The parameters of the exponential recovery model can be found from measurements.
They can be also approximated by typical values in the absence of more specific data.
Initial values The initial values of xp and xq are zero.
Algebraic network interface equations The linear portion of each load (Kpz
and Kqz) is modeled as a constant impedance connected between the respective bus
Appendix C. Developed Software Tools 155
and ground. This component is added to the Ybus matrix by updating the respective
diagonal element as
Y updatedbus (i, i) = Ybus(i, i) +
KipzP
id0 − jKi
qzQid0
|V i0 |2
, (C.28)
where variables with zero subscript are obtained from power flow.
The nonlinear component of each load (constant current, constant power, or ex-
ponential recovery) is modeled as a (negative) current injection to the respective bus.
The value of this injected current is calculated from
Iinj,load = −(PdIP + PdER)− j(QdIP +QdER)
V ∗bus
, (C.29)
where PdIP is the contribution of the constant power and constant current portions
of the load. PdER is the portion modeled as exponential recovery. Reactive power
terms are defined similarly.
C.3.1.4 AC Network
The AC network is represented by the complex-valued admittance matrix Ybus.
C.3.2 Simulation Algorithm
C.3.2.1 Overall Algorithm
The simulation algorithm is given below. Fig. C.6 shows the corresponding flowchart.
1. Obtain Pg, Qg, and Vbus from power flow.
2. Calculate the vector of initial states x0 at t = t−0 from the power flow solution.
This includes initial rotor fluxes of synchronous generators, initial voltages of
the exciter system, initial states of exponential recovery loads, and initial states
of induction motors. At this stage, all time derivatives are zero.
3. Apply the contingency at t = t0. At t = t+0 , the system states remain unchanged
but network voltages and currents may change instantaneously.
4. Find the new voltages and currents of the network from the algebraic equations
of the network and devices (as discussed in Subsection C.3.2.3).
Appendix C. Developed Software Tools 156
Start
End
Obtain initial values of P, Q, and V from power flow
Yes
No
Calculate initial states x0
Interval simulated?
Apply the contingency
Find the network V and I from interface equation
Calculate derivatives
Find new states x(k+1)
Advance time by dt
Fig. C.6. Flowchart of the algorithm for dynamic simulation of power systems.
5. Find derivatives x(k) based on the states x(k) obtained from the previous step
and voltages obtained from the algebraic interface equations.
6. Advance time by the time step dt and find new states x(k+1) from x(k) and
x(k) by an appropriate integration method. In this work, Gill’s version of the
fourth-order Runge-Kutta method (RKG4), discussed below, is used.
7. Go to step 4 and repeat until the whole time interval is simulated.
C.3.2.2 Numerical Integration
The differential equations in this simulation form a set of first-order nonlinear ordinary
different equations with known initial conditions. In this work, the Gill’s version of
Appendix C. Developed Software Tools 157
the fourth-order Runge-Kutta method (RKG4) is employed. Assume x is the vector
of states; the system of equations can be express as
x = f(x, t). (C.30)
In each time step, the following equation is iterated four times, for j = 1, 2, 3, 4, to
find the state x. The solution at the end of each time step is given by x4.
xj = xj−1 + kj∆t, (C.31)
where
kj = aj(f(xj−1, t)− bjqj−1
)qj = qj−1 + 3kj − cjf(xj−1, t).
(C.32)
Constants aj, bj, and cj are defined as
a1 = 1/2 a2 = 1−√
0.5 a3 = 1 +√
0.5 a4 = 1/6
b1 = 2 b2 = 1 b3 = 1 b4 = 2
c1 = 1/2 c2 = 1−√
0.5 c3 = 1 +√
0.5 c4 = 1/2.
(C.33)
For the first time step, q0 = 0; subsequent to the first time step, q0 is initialized with
q4 of the previous time step.
C.3.2.3 Network Interface
Network interface requires solution of a set of algebraic equations relating voltages and
currents of the network to those of the devices. The injected currents are calculated
from the network equations:
Iinj = YbusVbus. (C.34)
The injected currents obtained from above are set equal to those obtained from
the system generators and nonlinear loads, as given in (C.16) and (C.29), respectively.
Assuming that buses are ordered such that load buses appear before generator buses,
Appendix C. Developed Software Tools 158
(C.16) and (C.29) can be rewritten in matrix form as
Iinj =
Iinj,1
...
Iinj,nl
0
...
0
+
01,1 · · · 01,nl01,1 · · · 01,ng
.... . .
.... . .
0nl,1 0nl,nl0nl,1 0nl,ng
01,1 · · · 01,nlYG1 0 01,ng
.... . . 0
. . . 0
0ng ,1 0ng ,nl0ng ,1 0 YGng
0
...
0
E ′′1...
E ′′ng
−
V1
...
Vnl
Vnl+1
...
Vnl+ng
(C.35)
or
Iinj = Idevice + YG × (E ′′d − Vbus) . (C.36)
Equating (C.34) and (C.35), we get
YbusVbus = Idevice + YG × (E ′′d − Vbus) . (C.37)
Solving for bus voltages,
Vbus = (Ybus + YG)−1 × (Idevice + YGE′′d ) (C.38)
or
Vbus = ZGbus × (Idevice + YGE′′d ) , (C.39)
where
ZGbus = (Ybus + YG)−1 . (C.40)
Equation (C.39) is used for interfacing the power system components with the
network. Because Idevice is a function of the bus voltage, obtaining the correct values
of voltages may require a few iterations using the Gauss-Seidel method.
C.3.2.4 Implementation in MATLAB
The algorithm is implemented in MATLAB because of its vector calculation capa-
bilities. The code is divided into a number of modules that perform a specific job.
The main module solves the set of time domain differential equations describing the
transient behavior of the power system. Dynamics of generators and their exciters,
Appendix C. Developed Software Tools 159
damper windings, and power system stabilizers as well as loads are considered in this
program. Calculation of initial conditions is based on the solution of power flow.
The dynamic data (and power flow data) should be in the PSS/E format. Cur-
rently, GENROU, GENSAL, and SCRX models are supported.
Appendix D
Derivations for the Theory of
SPAACE
This appendix provides derivations related to the theoretical foundations of SPAACE,
which is explained in Chapter 6.
D.1 Justification of the Choice of tp as T2
Section 6.6 provides justification of a suitable choice of T1. This appendix justifies
the choice of T2 = tp; this choice corresponds to restoring the set point immediately
after the peak of x(t). Approximating the time evolution of x(t) with a second-order
response, we have
x(t) + 2ζωnx(t) + ω2nx(t) = ω2
nu(t), (D.1)
where u(t) = α, and α is defined in (6.13). Immediately subsequent to the (first)
peak of x(t), two cases can be considered: (i) set point is returned to unity and (ii)
set point is kept at (1−m). x(t) is denoted by x+r and x+
n , respectively, for these two
cases. Equation (D.1) is written for the mentioned two cases as
x+r + 2ζωnx
+r + ω2
nx+r = ω2
n × 1
x+n + 2ζωnx
+n + ω2
nx+n = ω2
n × (1−m).(D.2)
160
Appendix D. Derivations for the Theory of SPAACE 161
Because x(t) and its first-order derivative x(t) are state variables and hence continu-
ous, we have
x+r = x+
n = xp
x+r = x+
n = xp = 0.(D.3)
From (D.2) and (D.3) it follows that
x+n < x+
r < 0. (D.4)
That is,
|x+n | > |x+
r |. (D.5)
Therefore, by releasing the set point, the local convexity of x(t) decreases resulting
in a smoother curve. This choice results in the solution x(t) = 1 for t > T2.
D.2 Calculation of the Step Response
The solution of the second-order differential equation (D.1) describing a simplified
system is
x(t) = xh(t) + xp(t), (D.6)
where xh(t) is the homogenous (natural) response, and xp(t) is the particular (forced)
response due to u(t). For u(t) = α, where α is a constant,
xh(t) = A1e−ζωnt sin(ωdt) + A2e
−ζωnt cos(ωdt)
xp(t) = α.(D.7)
To simplify the notation, define
S(t) = e−ζωnt sin(ωdt)
C(t) = e−ζωnt cos(ωdt)
x(t) = x(t)− α,
(D.8)
which gives
x(t) = A1S(t) + A2C(t). (D.9)
Appendix D. Derivations for the Theory of SPAACE 162
The derivative of x(t) is
x(t) = A1
(− ζωnS(t) + ωdC(t)
)+ A2
(− ζωnC(t)− ωdS(t)
). (D.10)
Constants A1 and A2 are calculated from the known values of x(t) and x(t) at some
instant, e.g., t0, from the following system of linear equations:[S(t0) C(t0)
−ζωnS(t0) + ωdC(t0) −ζωnC(t0) + ωdS(t0)
][A1
A2
]=
[x(t0)
x(t0)
]. (D.11)
Using Cramer’s rule and the following identities
−ζωnS(t) + ωdC(t) = −ωne−ζωnt sin(ωdt− ψ)
−ζωnC(t) + ωdS(t) = −ωne−ζωnt cos(ωdt− ψ)
S2(t) + C2(t) = e−2ζωnt,
(D.12)
and observing that the determinant of the coefficients matrix is −ωde−2ζωnt, A1 and
A2 are calculated from
A1 =eζωnt0
ωd
(x(t0)ωn cos(ωdt0 − ψ) + x(t0) cos(ωdt0)
)A2 =
−eζωnt0
ωd
(x(t0)ωn sin(ωdt0 − ψ) + x(t0) sin(ωdt0)
).
(D.13)
D.3 Calculation of the Peak Response
To find an expression for the peak xp, we set the derivative of x(t) to zero. From
(D.10) and (D.12), we have
0 = x(tp)
=(− ζωnS(tp) + ωdC(tp)
)D1 +
(− ζωnC(tp) + ωdS(tp)
)D2
= −ωne−ζωntp sin(ωdtp − ψ)D1 − ωne−ζωntp cos(ωdtp − ψ)D2,
(D.14)
where D1 and D2 are functions of T1. Rearranging (D.14) gives
D2
D1
= − tan(ωdtp − ψ). (D.15)
Appendix D. Derivations for the Theory of SPAACE 163
The peak time is calculated from
tp =tan−1(d) + ψ
ωd, (D.16)
where
d =−D2
D1
. (D.17)
Substituting tp from (6.14) in (D.9) gives
xp = D1S(tp) +D2C(tp) (D.18)
and
S(tp) = e−ζωntp sin(ωdtp)
= e−ζωntp sin(ψ + γ),(D.19)
where γ = tan−1(d). Similarly,
C(tp) = e−ζωntp cos(ψ + γ). (D.20)
Substituting (D.17), (D.19), and (D.20) in (D.18) yields
x(tp) = D1e−ζωntp
(sin(ψ + γ)− d cos(ψ + γ)
). (D.21)
Divide and multiply the right-hand side of (D.21) by√
1 + d2 to get
xp = D1e−ζωntp
√1 + d2
(1√
1 + d2sin(ψ + γ)− d√
1 + d2cos(ψ + γ)
)= D1
√1 + d2e−ζωntp sin(ψ).
(D.22)
The last equality follows because 1/√
1 + d2 = cos(γ) and d/√
1 + d2 = sin(γ).
Equation (D.22) can be further simplified by substituting the values of (D1, D2) =
(A1, A2) from (D.13) for t0 = T1. Define
M2 =e2ζωnT1
w2d
. (D.23)
Appendix D. Derivations for the Theory of SPAACE 164
We have
D21(1 + d2)
M2=D2
1 +D22
M2
=(xωn cos(ωdT1 − ψ) + x cos(ωdT1)
)2
+(xωn sin(ωdT1 − ψ) + x sin(ωdT1)
)2
= (xωn)2 + x2
+ 2xxωn(
cos(ωdT1 − ψ) cos(ωdT1)
+ sin(ωdT1 − ψ) sin(ωdT1))
= (xωn)2 + x2 + 2xxωnζ,
(D.24)
where x and x are short forms for x(t)|T1 and x(t)|T1 , respectively, and the last equality
holds because cos(ψ) = ζ.
Further, using the value of tp from (D.16), −ζωntp can be expressed as
−ζωntp = −ζωn(γ + ψ
ωd)
= −cos(ψ)
sin(ψ)(γ + ψ)
= − cot(ψ)(γ + ψ).
(D.25)
Substituting (D.23), (D.24), and (D.25) in (D.22) simplifies the expression for xp to
xp =sin(ψ)
ωde−ψ cot(ψ)eζωnT1e−γ cot(ψ)
√(xωn)2 + x2 + 2xxωnζ. (D.26)
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