Control Strategies for the Next Generation Microgrids · Control Strategies for the Next Generation ... Control Strategies for the Next Generation Microgrids ... of controllable devices

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

http://blogs.msdn.com/b/bgroth/archive/2004/09/27/235071.aspx

http://blogs.msdn.com/b/bgroth/archive/2004/09/27/235071.aspx

http://mehrizisani.com/latex

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

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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


4-5 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


4-5 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


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].


Tertiary 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].


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],


[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].


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.

http://mehrizisani.com


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 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 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.


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.


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.


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.


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)


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)


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)


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.


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-


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.


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.


[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-


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


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,


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.


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.


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.


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.


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


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.


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


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.


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.


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


[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.


• 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.


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.


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)


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


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.


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].


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


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.


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)


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)


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


)∑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


)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


)Vi


)Vi


)∑nb

n=1 Vn

(gin cos(δi − δn) + bin sin(δi − δm)

)

, (4.25)


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)


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


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)


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.


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)


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.


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)



. (4.53)


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)


∂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.


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.


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.


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.


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.


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)


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


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.


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-


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.


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


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


[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


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].


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.


|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.


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.


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


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)


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


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.


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.


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


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.


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.


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.


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


[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,


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


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


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,


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.


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


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)


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


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


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.


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.


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


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)


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.


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


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).


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.


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.


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)


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.


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.


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


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


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.


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.


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.


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.


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.


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


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.


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


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:



• 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.


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.


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)


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.


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)


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.


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.


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 .


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.


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


http://mehrizisani.com/potential


http://mehrizisani.com/spaa

http://mehrizisani.com/loadmodeling

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


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.


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.

http://mehrizisani.com/spaa


(a) (b)(c) (d)(e) (f)

Fig. C.2. Test scenarios for the developed PFM software.


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).

http://mehrizisani.com/loadmodeling


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)


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)


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)


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)


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)


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


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).


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


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,


(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,


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)


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)


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)


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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[127] A. Borghetti, R. Caldon, A. Mari, and C. A. Nucci, “On dynamic load models

for voltage stability studies,” IEEE Trans. Power Syst., vol. 12, no. 1, pp. 293–

303, Feb. 1997.

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