distributed optimization of sustainable power dispatch and … · 2017-11-07 · abstract...

Distributed Optimization of Sustainable Power Dispatch and FlexibleConsumer Loads for Resilient Power Grid Operations

by

Pirathayini Srikantha

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Electrical & Computer EngineeringUniversity of Toronto

c© Copyright 2017 by Pirathayini Srikantha

Abstract

Distributed Optimization of Sustainable Power Dispatch and Flexible Consumer Loads for Resilient

Power Grid Operations

Pirathayini Srikantha

Doctor of Philosophy

Graduate Department of Electrical & Computer Engineering

University of Toronto

2017

Today’s electric grid is rapidly evolving to provision for heterogeneous system components (e.g. in-

termittent generation, electric vehicles, storage devices, etc.) while catering to diverse consumer power

demand patterns. In order to accommodate this changing landscape, the widespread integration of cyber

communication with physical components can be witnessed in all tenets of the modern power grid. This

ubiquitous connectivity provides an elevated level of awareness and decision-making ability to system

operators. Moreover, devices that were typically passive in the traditional grid are now ‘smarter’ as

these can respond to remote signals, learn about local conditions and even make their own actuation

decisions if necessary. These advantages can be leveraged to reap unprecedented long-term benefits that

include sustainable, efficient and economical power grid operations. Furthermore, challenges introduced

by emerging trends in the grid such as high penetration of distributed energy sources, rising power

demands, deregulations and cyber-security concerns due to vulnerabilities in standard communication

protocols can be overcome by tapping onto the active nature of modern power grid components.

In this thesis, distributed constructs in optimization and game theory are utilized to design the

seamless real-time integration of a large number of heterogeneous power components such as distributed

energy sources with highly fluctuating generation capacities and flexible power consumers with varying

demand patterns to achieve optimal operations across multiple levels of hierarchy in the power grid.

Specifically, advanced data acquisition, cloud analytics (such as prediction), control and storage sys-

tems are leveraged to promote sustainable and economical grid operations while ensuring that physical

network, generation and consumer comfort requirements are met. Moreover, privacy and security consid-

erations are incorporated into the core of the proposed designs and these serve to improve the resiliency

of the future smart grid. It is demonstrated both theoretically and practically that the techniques pro-

posed in this thesis are highly scalable and robust with superior convergence characteristics. These

distributed and decentralized algorithms allow individual actuating nodes to execute self-healing and

adaptive actions when exposed to changes in the grid so that the optimal operating state in the grid is

maintained consistently.

ii

Acknowledgements

I would like to sincerely thank my advisor, Professor Deepa Kundur, for her invaluable mentorship

and guidance. Her incredible patience, motivation and insights have allowed me to realize many of my

academic and career goals. She has continuously exposed me to many interesting research and teaching

opportunities that have been extremely rewarding. Being a doctoral candidate at the University of

Toronto has been such a revitalizing and wonderful experience for me mainly due to her. I am forever

indebted to her for her kindness and support.

I would also like to thank Professor Lacra Pavel, Professor Wei Yu and Professor Josh Taylor for

inspiring my work with their courses and research perspectives. I would also like to thank Professor

Hamed Mohsenian-Rad, Professor Reza Iravani, Professor Raymond Kwong, Professor Jorg Liebeherr

and Professor Ben Liang for participating in my thesis committees and providing me with constructive

feedback. Moreover, I am grateful for being able to engage in deep and illuminating discussions about

research, career and academia in general with members of the Communications and Energy Systems

groups.

Finally, I would like to acknowledge my parents and my brother for being such incredibly important

figures in my life. They have lifted me up from so many challenges. I am grateful for their unwavering

faith in me and for continuously bestowing upon me infinite love, support and strength.

“Life is momentary! Live it completely!” ∼ OSB

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Contents

1 Introduction 1

1.1 Traditional Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Smart Grid and Open Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Distributed Generation and Storage . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Flexible Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 Energy Deregulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.4 Cyber-enablement and Resilience . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Problem Statement and Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 State-of-the-Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.1 Demand Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4.2 Economic Dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.3 Combined DR and DG dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Power Grid Optimizations 14

2.1 Convex Sets and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.1.1 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.2 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.1.3 Convex versus Non-convex Optimization Problems . . . . . . . . . . . . . . . . . . 16

2.2 Optimization for the Electric Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Power Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.2 Steady-State Power Flow Problem Formulation . . . . . . . . . . . . . . . . . . . . 21

2.2.3 Demand Response Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.4 Economic Dispatch Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.5 Linear Power Flow Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.6 Voltage Regulation Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Demand Response 27

3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.1 DR Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.1.3 System Model Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.2 Demand Response via Water-Filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

iv

3.2.1 Demand Response Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.2 Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.3 Dual of the DR Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.4 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.5 Theoretical Convergence Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.6 Analogy to Water-Filling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Demand Response via Population Games . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.1 Set of Transformations to the DR Problem . . . . . . . . . . . . . . . . . . . . . . 39

3.3.2 Computation of EPU Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.3 A Game Theoretic Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.4 Resilience of EPU Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3.5 Strategy Revisions by DR Agents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44



3.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Distributed Generation Dispatch 53

4.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.1.1 Distributed Dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.1.3 Dispatching DGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1.4 System Model Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2 Distributed Dispatch via Dual Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.1 Economic Dispatch Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.2 Master and Agent Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.3 Termination Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2.4 Summary of Proposed Dispatch Algorithm . . . . . . . . . . . . . . . . . . . . . . 61

4.2.5 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61


4.3 Distributed Dispatch via Population Games . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3.1 Dispatch Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.3.2 Master Tier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.3 Secondary Tier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.3.4 Equivalence between Game Characterization and Optimization . . . . . . . . . . . 71


4.3.6 Resilience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74


4.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 Hierarchical Optimization of Distributed Generation and Demand 82

5.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.1.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.2 Transmission Network Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

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5.2.1 Introduction to ADMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2.3 Transformations to ADMM Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.2.4 Summary of Decentralized Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 89


5.2.6 Comparison with State-of-the-art Decentralized Techniques . . . . . . . . . . . . . 95

5.3 Distribution Network Level Coordination . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.3.1 Resilience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6 Conclusions 97

6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.3 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Appendix: List of Acronyms 101

Bibliography 102

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List of Tables

3.1 Distributed algorithm via water-filling for DR agent. . . . . . . . . . . . . . . . . . . . . . 35

3.2 Distributed algorithm via water-filling for the EPU. . . . . . . . . . . . . . . . . . . . . . 36

3.3 Distributed load curtailment via population game theory. . . . . . . . . . . . . . . . . . . 44

3.4 Comparison of DR methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.1 Distributed dispatch algorithm via dual decomposition. . . . . . . . . . . . . . . . . . . . 61

4.2 Strategy revisions by DG agents with voltage rise considerations. . . . . . . . . . . . . . . 69

4.3 Voltage feasibility check by DG agents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.4 Comparison of dispatch methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.1 Decentralized algorithm for bus agent n in the transmission tier. . . . . . . . . . . . . . . 90

5.2 Comparison of results from central versus decentralized algorithms WECC 9 bus system. . 93

5.3 Comparison of results from central versus decentralized algorithms IEEE 39 bus system. . 94

5.4 Decentralized algorithm for bus agent n at the transmission tier. . . . . . . . . . . . . . . 96

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List of Figures

1.1 Illustration of a smart grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Diversity in generation mix in the smart grid. . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Rendering of a smart city. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 NIST Framework representing active players in the smart grid. . . . . . . . . . . . . . . . 5

1.5 NIST Framework representing communication flows in smart grid. . . . . . . . . . . . . . 6

1.6 Interactions between chapters in the thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Examples of convex and non-convex sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Examples of convex and non-convex functions. . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Simple electric grid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Load profile of a home during winter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 Power flow across a line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1 Smart home energy management system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Sample load profile of a home. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Demonstration of water-filling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Aggregate demand in a neighbourhood containing 100 homes. . . . . . . . . . . . . . . . . 37

3.5 Reduced power operation for a single iteration. . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6 Regular versus reduced aggregate demands over a day. . . . . . . . . . . . . . . . . . . . . 38

3.7 Simplex and system potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.8 Illustration of system dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.9 Pairwise comparison revisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.10 Performance of EGT based DR scheme under system limitations. . . . . . . . . . . . . . . 48

3.11 System subjected to perturbation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.12 Ability of the EGT based DR algorithm to maintain demands around a setpoint. . . . . . 49

3.13 Comparison between PC protocol and SG method. . . . . . . . . . . . . . . . . . . . . . . 50

4.1 System model for economic dispatch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Sample system model for DG dispatch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 Power dispatched by DGs for the three types of α computations. . . . . . . . . . . . . . . 62

4.4 Real-time dispatch over a day. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.5 Power Savings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.6 System model for economic dispatch via master and secondary tiers. . . . . . . . . . . . . 66

4.7 Local feasibility checks with voltage rise considerations. . . . . . . . . . . . . . . . . . . . 68

4.8 Low-voltage distribution network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

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4.9 Topology of cyber-physical DN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.10 Projection revisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.11 Impact of population size. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.12 Recovery of DG Agents from Disturbance. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.13 System behaviour (real power and bus voltages) without DG dispatch in place. . . . . . . 78

4.14 Real-time dispatch (real power and bus voltages) over a day with dispatch solution in place. 79

4.15 Comparison of Dispatch Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.1 System model for hierarchical topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.2 WECC 3 bus line diagram, adapted from reference [123]. . . . . . . . . . . . . . . . . . . . 91

5.3 IEEE 39 bus line diagram, adapted from reference [27]. . . . . . . . . . . . . . . . . . . . . 91

5.4 Residual for WECC 9 bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.5 Residual for IEEE 39 bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.6 Impact of Lagrangian step-size on ADMM convergence. . . . . . . . . . . . . . . . . . . . 93

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Chapter 1

Introduction

The electric power grid is a critical infrastructure that empowers a broad spectrum of crucial operations in

modern societies. Recent concerns regarding climate change and energy monopoly have spurred a rapidly

evolving landscape in the power grid. Thus, it is possible to witness the onset of remarkable changes in

fundamental power system components due to increasing diversity in generation mix, consumer demands,

grid protection mechanisms and regulatory processes [1].

1.1 Traditional Grid

For more than hundred years, grid operators have primarily strived to provision for guaranteed uninter-

rupted power supply to consumers [2]. Due to the widespread adoption of a vertically integrated power

grid model with stringent regulations in place, system operators have had incredible insights into various

nuances of the grid. This has enabled the implementation of well-defined planning and control processes

to manage day-to-day operations in the grid [3]. For instance, system operators can forecast consumer

demands using historical data fairly accurately since electricity consumption patterns are traditionally

strongly correlated with diurnal and seasonal changes. Moreover, as the generation mix in the grid has

been primarily composed of bulk generation entities such as fossil, hydro and nuclear plants; generation

capacities of these can be readily inferred from fuel inventories and physical laws (e.g. those associated

with hydro-electric generation). These measures are then leveraged by system operators to centrally

plan well in advance (e.g. hour or day-ahead) the amount of power that can be dispatched by every

active generator to economically meet consumer demands using simplified system models. This process

is referred to as economic dispatch.

Large synchronous power plants, located at remote locations for reasons that include safety (e.g.

nuclear plants) and ease of fuel access (e.g. gas, coal, hydro), produce power at pre-defined levels in

accordance to the centrally computed economic dispatch reference levels. This power is transported over

long distances via high-voltage transmission networks in order to reduce line losses and delivered to con-

sumers residing within distribution networks (DNs) at a lower voltage more suitable for end appliances.

Instantaneous fluctuations in demands that deviate from forecasts are typically accommodated for by

automatic generator control (AGC) built into large synchronous power sources [4]. AGC is a decentral-

ized control mechanism that infers imbalance in demand and supply through local frequency deviations

and adjusts generator power inputs to restore balance. In order to further ensure reliable power supply,

1

Chapter 1. Introduction 2

redundancies in generation and transmission are engineered into the design of the grid so that the sys-

tem will still be able to operate even when one major power component fails (e.g. transmission line or

generator) and this is referred to as N-1 contingency. Grid protection mechanisms such as relays and

circuit breakers are typically passive and are activated by sudden surges in local currents [5]. These

devices isolate incident locations so that issues resulting in instability do not propagate throughout the

grid.

This demand/supply and protection framework has been highly effective in managing the power grid

for many decades until very recently.

1.2 Smart Grid and Open Challenges

Due to mounting concerns regarding climate change and rising energy costs, achieving reliable power

grid operations is no longer the primary goal for system operators. A noticeable shift is now evident

in energy policies that now vigorously promote sustainable, economical and reliable power grid opera-

tions. This has resulted in efforts that strive to minimize dependence on power plants that are expensive

and associated with high carbon-footprint. Renewable distributed energy sources, storage systems and

inherent demand flexibility are considered to comprise of vastly untapped potential for facilitating sus-

tainable grid operations [1]. Energy deregulation has been embraced recently to encourage competitive

energy markets that can then serve to drive electricity prices down to lower rates and prevent energy

monopoly [6]. Moreover, cyber-enablement in the power grid has empowered many individual system

components with the ability to monitor, communicate and intelligently actuate [7]. These capabilities

allow for resilience to be built into the core of the system.

CONTROL

CONTROL

CONTROL

CONTROL

Figure 1.1: Illustration of a smart grid.

These upcoming and emerging trends summarized in Fig. 1.1 have resulted in a movement that has

transformed the traditional power grid into a smart grid. A formal definition of the smart grid presented

by the National Institute of Standards and Technology (NIST) is as follows [8]:

“Smart Grid develops and implements measurement science underpinning modernization of the Na-

tion’s electric grid in order to improve system efficiency, reliability and sustainability, by incorporating

distributed intelligence, bi-directional communications and power flows, and additional advancements to


create a smart grid.”

These changes bring upon many opportunities as well as many challenges due to significant deviations

in the traditional operational paradigm of the power grid and these are discussed next.

1.2.1 Distributed Generation and Storage

Power generation in the smart grid now consists of a diverse mix of energy sources as illustrated in

Fig. 1.2. Renewable generation sources combined with storage systems are highly preferred alternatives

Figure 1.2: Diversity in generation mix in the smart grid.

to traditional fossil-fuel based synchronous plants as these have zero emissions and are not typically

associated with transmission line losses due to the physical proximity of deployment to consumers [9].

Distributed generators (DGs) such as solar panels and wind turbines are typically deployed by individual

power consumers (e.g. rooftop panels and micro wind turbines) or in large dedicated wind/solar farms.

Storage systems are becoming widely commercialized in consumer markets with the advent of electric

vehicles [10]. With significant penetration of DGs that operate in tandem with storage systems, DNs will

be minimally dependent on unsustainable central power plants that are based on fossil-fuels (e.g. coal

and gas generation) to supplement local consumer demands and even have opportunities to make profits

by capitalizing on recent deregulations in energy markets to inject surplus generation back into the

main grid. However, a major deterrent in capitalizing on these opportunities is the inherent generation

variability of DGs.

Renewable DG generation is dependent on uncontrollable external environmental factors such as solar

irradiance (affected by cloud cover) and wind speeds (dependent on atmospheric pressure). Renewable

generation forecast models are associated with large error margins that tend to increase over longer

prediction horizons. For instance, day-ahead prediction models for wind generation are associated with

error margins that range between 20% to 40% whereas predictions over one minute intervals are very

accurate [11, 12]. Hence, day-ahead planning involving a large number of DGs will be an extremely

difficult task for system operators due to these perilously high error margins. For this reason, most DGs

are not dispatched by the system operators but are set to operate at maximum power point (MPP)

tracking levels so that all instantaneously generated power is injected back into the grid [13]. However,

with concentrated DG penetration especially in DNs, this maximum power injection approach is not

physically viable as the existing low-voltage infrastructure does not support bi-directional power flows.

Excessive reverse power flow can result in bus over-voltage conditions that can then lead to adverse

equipment damages [14]. Moreover, when power supply from DGs is not accurately accounted for in

day-ahead planning, reliably meeting consumer demands will be very expensive as significant number


of costly ancillary generation entities must be commissioned. Furthermore, balancing demand with

supply can even be infeasible when available generation capacities of standby operational reserves are

unable to offset imbalance stemming from unexpected drops in renewable generation. In these cases,

AGC mechanisms will not be sufficient in maintaining system balance within the prescribed stable

region. Furthermore, contingency planning will be very difficult and passive protection mechanisms such

as circuit breakers and relays will be tripped unnecessarily due to significant volatility in generation.

Recovery from these service interruptions can be very expensive and time-consuming.

1.2.2 Flexible Consumers

With the advent of the internet of things (IoT), it is possible to observe significant changes in consumer

demand patterns as illustrated in Fig. 1.3. With the recent commercialization of electric vehicles, storage

Figure 1.3: Rendering of a smart city.

systems and DGs for homes, consumer demand patterns can no longer be accurately predicted using

only diurnal and seasonal changes or historical data [15]. Unsustainable generation sources such as gas

plants are commissioned typically during peak consumer demand periods as these are able to ramp up

and down generation rapidly as needed to serve fluctuating demand peaks. These generation sources are

typically expensive and inefficient. During non-peak hours, base load demands are primarily served by

nuclear and hydro plants that operate at a constant level and produce no carbon emissions. However,

as these plants are inflexible and require significant time for ramping up/down generation levels, these

are not utilized to supplement highly fluctuating peak demands [16].

As discussed in the previous section, DGs can be utilized to reduce dependence on peak generation

plants. Another alternative will be to curtail consumer demands during these peak periods [17]. Cur-

tailing or shifting consumer demands via direct control or monetary incentives is referred to as demand

response (DR). Typically, direct load control (DLC) techniques are implemented in industrial and com-

mercial sectors. These sectors consist of large load entities that commit to various power reduction levels

well in advance. These are then activated by electric power utilities (EPUs) during times of need in

exchange for monetary incentives. DLC is not widely implemented in the residential sector. Instead, the


time-of-use (TOU) method is used in many regions such as Ontario where electricity prices are set well

in advance for various time periods over a day. Since TOU prices are higher during peak periods, con-

sumers are incentivized to use appliances during non-peak periods. It is however possible for consumers

to become desensitized to these price differentials. Moreover, TOU prices are not reflective of the actual

costs incurred in the power grid (e.g. real cost of generation, congestion) [18].

Most consumers have inherent flexibility in their demand patterns and the degree of this flexibility

is captured by the consumer comfort level measure in this thesis [19]. For instance, consumers may

be tolerant to their thermostat setpoints operating around particular thresholds. Appliances such as

washing machines, dishwashers or dryers can operate at a later time in a day. Moreover, if consumers own

storage systems, these can be used as buffers for storing electricity during inexpensive periods. This also

prevents the wastage of surplus power generated by DGs. Home energy management systems (HEMSs)

can be employed by the EPU to coordinate appliances in a home, based on individual consumer flexibility

and the state of the power grid [9]. However, this is not a straightforward task as there are thousands of

residential consumers with individual tolerance limits and preferences on how various appliances operate.

Coordinating all of these entities centrally is not a tractable task for the EPU. Hence, tapping onto the

significant potential of demand flexibility by system operators can be a daunting task.

1.2.3 Energy Deregulation

Energy deregulation has allowed entities such as private generation sources and investors who attempt

to take advantages of energy arbitrage to directly compete in electricity markets [20]. Modern grid

Figure 1.4: NIST Framework representing active players in the smart grid.

interactions of various traditional components with the energy markets, as conceptualized by NIST,

are illustrated in Fig. 1.4. The competitive structure of energy markets prevents energy monopoly

and serves to reduce electricity prices. However, many challenges regarding how ownership can be

assigned to non-revenue generating grid maintenance processes (e.g. transmission line expansion) and

how a large number of individual private entities can be coordinated at a system-wide level to ascertain

reliability are major concerns. Traditionally, demand and supply planning has been straightforward as

individual generation characteristics of synchronous plants were readily available to a central managing

authority. Moreover, infrastructure expansions required to support growing consumer demands are

typically managed by these central public entities as well. With deregulation, transmission lines can

operate close to physical limits and congestion can result in cascading system-wide failures. As it is


not possible to segregate power flowing through the transmission lines, assigning ownership for causing

congestion to various private generation entities is a difficult task. Moreover, committing public funds for

expensive infrastructure enhancement projects based on private generation entities that may not exist

in the future are not sound investment plans. Hence, the coordination of diverse individual participants

that takes into account physical infrastructure limits in addition to economical benefits is imperative for

the uninterrupted operation of the grid.

1.2.4 Cyber-enablement and Resilience

The widespread integration of communication technologies with the physical grid has resulted in an

information-rich cyber-physical system that consists of interconnected components equipped with the

ability to make intelligent decisions [21]. Various communication networks in the smart grid are illus-

trated in Fig. 1.5 which is a diagram rendered by NIST. For instance, phasor measurement units (PMUs)

Figure 1.5: NIST Framework representing communication flows in smart grid.

collect local grid measurements at a high frequency (around 50 Hz) and relay these to data concentrators

(DCs) that then apply analytics on this data for grid monitoring purposes. Homes are equipped with

smart meters that record energy consumption measurements at high granularity and communicate these

to the corresponding EPU. This data is then used by the EPU for billing purposes and/or to provide

consumer feedback for incentivizing sustainable energy consumption patterns [22]. Smart appliances

in homes such as thermostat controls, refrigerators, lights, washers, and dryers are equipped with the

ability to intelligently adopt local operations based on user preferences and behaviours [23]. Moreover,

critical power system components such as generation systems (both traditional synchronous and dis-

tributed generation), storage and grid protection components are also equipped with communication

and intelligent decision-making capabilities.

This elevated level of monitoring and actuation capabilities facilitated by the cyber-enabled power

grid can be leveraged to coordinate heterogeneous system components via advanced data analytics

and control techniques. Furthermore, adaptive decision-making by individual entities allows for the

automated recovery from any stress imposed due to unexpected circumstances and thereby enhancing


the resilience of the system. According to reference [24], a system is resilient if it is able “to prepare for

and adapt to changing conditions and withstand and recover rapidly from disruptions”. In this thesis,

resilience to perturbations arising from noise in communication signals (injection of false information) and

unresponsiveness of actuating nodes is considered. To alleviate adverse effects from these perturbations,

the system must be robust to perturbations whereby unaffected active nodes are able to actively adapt

and respond to disturbances in the system. This allows for automated recovery and in-built resilience

in the system.

However, it is important to note that the standard communication protocols utilized by cyber-enabled

devices also inherit well-known vulnerabilities [25]. These can be exploited by adversaries to gain access

to actuating components and stealthily manipulate these to cause major system disruptions [26, 27].

Thus, it is imperative to ensure that security mechanisms are incorporated into the core of smart grid

solutions so that system components can thwart adverse effects from these insidious attacks or natural

disasters [28].

1.3 Problem Statement and Thesis Objectives

The opportunities and challenges stemming from the smart grid paradigm present many rich and re-

warding research opportunities. As such, in this thesis, methods that enable the seamless plug-and-play

integration of a large number of heterogeneous power components such as consumers with flexible de-

mands and DGs to achieve sustainable, optimal and resilient grid operations at both the distribution

and transmission network levels are investigated. Demand response and economic dispatch are disparate

operations that are traditionally considered separately in a centralized manner. Typical challenges as-

sociated with centrally managing these entities stem from the lack of scalability, single point of failure

issues and sub-optimality. In this thesis, a distributed approach is utilized that enables the coordination

of these active nodes to achieve common goals of sustainability and economical grid operations in a

scalable and robust manner by leveraging on theoretical constructs from convex optimization and game

theory. Thus, following are the objectives of this thesis:

• Extend DR to a large number of residential consumers to coordinate flexible loads so that depen-

dence on peak unsustainable synchronous generation sources can be minimized while also account-

ing for consumer comfort levels.

• Propose economic dispatch for DGs deployed at large numbers in the DN level while accounting

for voltage rise limits in the buses.

• Propose real-time distributed techniques that optimally combines DR and DG dispatch over every

one minute interval to account for high variability in DG generation and demand fluctuations

without being subjected to large error margins from forecast models.

• Theoretically and practically demonstrate strong convergence properties of these proposed tech-

niques to ensure resilience against unexpected or adversarial disturbances.

• Combine DR and DG across multiple distribution networks into a single hierarchical scheme that

coordinates power flow across the transmission network level to minimize network congestion.


1.4 State-of-the-Art

State-of-the-art proposals in the existing literature in the context of DR and economic dispatch are

presented next. These proposals can be broadly classified into centralized, decentralized and distributed

categories that are either real-time (e.g. timescale is in minutes) or used in advanced planning (e.g.

timescale is in hours).

1.4.1 Demand Response

DR proposals represent adjustments made by the EPU to consumer power demand patterns via mecha-

nisms that include controlling thermostatically operating appliances around a threshold, shifting/curtailing

appliance usage and/or providing monetary incentives to influence consumer demand patterns.

Central DR techniques involve a consolidating entity usually a representative of the EPU that cen-

trally coordinates appliance operations (typically thermostatically controlled loads) in participating

residential homes. Central methods are advantageous as EPUs can exercise greater control over the

system. References [29–31] directly control heating, ventilating and air-conditioning (HVAC) systems

via stochastic optimization, queuing theory and simplified forecasting of appliance usage. The com-

putational complexity of solving these DR formulations grows exponentially as the size of system (i.e.

number of DR participants) increases. Hence, sophisticated resources with significant computational

power are required for the practical implementation of these techniques. Moreover, maintaining and

managing consumer preferences that are not consistent in a central repository will result in significant

communication overheads and storage requirements. Furthermore, these techniques do not scale well in

large systems consisting of different types of appliances (both thermostatically and non-thermostatically

controllable loads), as discrete operating points will render the DR formulation intractable. Thus, these

techniques cannot be applied for generalized DLC in the residential sector.

Decentralized and distributed techniques are designed to allow participating entities to capitalize

on the ubiquitous communication capability in the smart grid to iteratively arrive at an optimal DR

solution by exchanging information amongst each other. As a central entity does not directly manage

the operation of end nodes, single-point-of-failure risks are not applicable to these techniques. Moreover,

information about local operational conditions of end nodes is abstracted from the central entity. This

reduces concentrated information flow. Additionally, privacy is preserved as local information is not

exposed to an external entity. These techniques are prone to false data injection attacks in information

exchanges. The consequences entailed due to these are explored in reference [32] which investigates how

malicious false information propagation affects distributed schemes that primarily rely on peer-to-peer

information exchanges. Distributed DR algorithms proposed in the existing literature leverage upon

various theoretical constructs in agent-based optimization. Consensus-based algorithm in reference [33]

engages in DLC via iterative exchanges of local states amongst neighbours. Reference [34] utilizes the

sub-gradient method to arrive at the optimal solution in a distributed manner while taking into account

spatial and temporal appliance constraints. Reference [35] utilizes a hierarchical technique based on dual

decomposition to coordinate demand resources. Another proposal in reference [36] leverages congestion

pricing commonly used in the internet to manage traffic for adaptive electric vehicle charging based

on local preferences. Reference [19] entails direct actuation of inductive and resistive components of

appliances and therefore practicality of this proposal is questionable. Convergence rates to optimality of

the afore-mentioned techniques are dependent on the number of participants in the system. Larger the


system, the greater will be the time required for convergence. Reference [37] utilizes adaptive pricing

based on mathematical models for thermal and deferrable loads which may not accurately reflect the

actual operational patterns of these loads.

Game theoretic techniques are also widely applied in DR schemes. References [38–40] capitalize on the

HEMS present in smart homes to schedule appliances and storage systems in advance based on electricity

prices. These schemes require prior knowledge on what appliances will be operating at which time well

in advance (typically a day-ahead) and are not scalable for a large system with thousands of participants.

Moreover, not all appliances can participate as these schemes focus only on ‘shiftable’ appliances such as

dishwashers, dryers, etc. Reference [41] utilizes non-cooperative and evolutionary game theory to model

how residential users and multiple EPUs interact with one another. Reference [42] utilizes population

games for DR to account for individual consumer comfort budgets, however with no guarantees on the

uniqueness or robustness of the optimal solution. Hence, the ensuing system can be subject to instability

and ringing with minor perturbations.

1.4.2 Economic Dispatch

Next, state-of-the-art in the economic dispatch literature is presented. Demand and supply in today’s

grid is based on a wholesale electricity market that is administered by an independent system operator

(ISO) which seeks to balance consumer demands with available generation in participating energy sources

in an economical manner. Any generator injecting power into the transmission system must participate

in this wholesale electricity market and submit an offer containing details on the amount of electricity

it can supply and the associated price it will levy based on day-ahead demands forecasted and posted

publicly by the ISO. These generators may belong to a vertically integrated system or operate privately as

independent power plants (IPPs). Spinning reserves include generators and/or storage systems that are

operating in standby mode. These are expected to come online in the event of a disruption or unexpected

demand spikes. These ancillary services are paid “operating reserve” payments by the ISO [43]. These

bids can continue to arrive until two hours before the actual dispatch takes place. After this point, the

ISO ranks the submitted bids in the order of increasing price and accepts bids starting from the lowest

price until all of the forecasted demands are met. The market clearing price is uniformly set to the

price specified in the last accepted offer. Security constrained load flow studies are incorporated into

these auctions that are conducted hours in advance. Sensitivity analysis is also incorporated in real-time

dispatch across 5 minute intervals in order to account for inaccuracies in demand forecasts [44].

As mentioned in Sec. 1.2.1, DG sources are typically not dispatched and inject generated power

at maximum possible levels. As DG penetration grows significantly, this model of DG integration

presents great difficulties with the planning processes necessary for the effective operation of wholesale

electricity markets. Load flow analysis conducted hours in advance may not be accurate due to forecast

errors in DG power supply. More standby reserves that are typically expensive will be necessary to

accommodate unexpected changes in DG generation. Moreover, sensitivity analysis conducted in real-

time for transmission lines will not be sufficient to ascertain that all physical limits are heeded as DGs

are increasingly being deployed in low-voltage distribution networks. Existing proposals in the literature

that address these issues are discussed next.

Steady-state power flow constraints consist of sinusoidal components which are non-linear. With

heavy penetration of DGs, these constraints cannot be ignored. Centralized dispatch schemes proposed

in the literature attempt to tackle this difficulty by leveraging on prediction, stochastic optimization,


approximations and heuristics techniques. Moreover, security concerns due to single point of failure

are inevitable in centralized schemes. References [45–49] utilize stochastic, prediction and dynamic

programming methods to dispatch DGs at an advanced time period and this allows the ISO to utilize

ample computing resources to perform security assessment. However, due to the significant error margins

and variabilities present in DGs, a wide margin of risk must be incorporated into the dispatch techniques.

This will not allow the ISO to fully leverage the generation potential of DGs in an economical manner.

Other proposals in references [50–55] utilize heuristic techniques that are quicker to solve but do not

guarantee solution optimality and convergence speed. Hence, these can result in suboptimal solutions

that are more costly. Even if there is some tolerance to sub-optimality by the ISO, not using forecast

models will entail every DG transmitting local generation capacity constraints at a high frequency to

the ISO. This will result in significant storage and communication overheads. Reactive power dispatch

for voltage regulation is considered in reference [56]. References [57, 58] apply approximations such as

linearization and semi-definite programming (SDP) relaxations to non-linear constraints in the power

flow problem. Linearization techniques are typically used in day-ahead scheduling of generation systems.

Although SDP relaxations result in tighter bounds than linearization techniques, these may still result

in infeasible solutions. Applying these for real-time coordination of DGs can result in the infringement

of physical bus voltage and line limits. SDP relaxations can be exact in radial networks such as DNs

with very strict restrictions on system settings that can be impractical.

Distributed and decentralized techniques offload computational and communication intensive oper-

ations from the ISO to individual participating DG sources that are cyber-enabled and equipped with

intelligence. References [59–64] focus on balancing overall power demands with supply without taking

into consideration the underlying physical constraints. These utilize techniques such as sub-gradient

methods, distributed dynamic programming and consensus methods to achieve economic dispatch of

DGs. Convergence of these techniques is proportional to the number of participants in the dispatch

problem. Game theoretic approaches are used in references [65–70] for dispatch. Specifically refer-

ences [69, 70] take into account constraints with respect to power and voltage limits. Scalability is of

main concern with these proposals. Reference [71] proposes using only local state measurements and

no communications to make dispatch decisions that heed physical constraints. This technique does not

guarantee an optimal solution as the overall system is not taken into account in the dispatch. Refer-

ences [72–76] are decentralized techniques based on dual decomposition and alternative direction method

of multipliers (ADMM) which take into account physical grid constraints. References [72–74] apply con-

vex relaxations which are exact in distribution networks under stringent conditions. Reference [75]

applies ADMM to the original non-convex grid constraints with no guarantees on the optimality of the

solution. Furthermore, distributed techniques that depend on information exchanges amongst peers are

prone to false data injection attacks.

1.4.3 Combined DR and DG dispatch

So far, existing proposals in the areas of DR and DG dispatch have been considered separately. Proposals

that combine these two mechanisms to complement one another are presented next. Reference [77]

proposes a method that considers economic dispatch for a small to medium scale system while accounting

for both demand flexibility and physical network limitations. Reference [78] utilizes a primal-dual method

for load-side frequency control. These methods are tailored specifically for frequency control and cannot

be easily generalized to take into account other considerations such as voltage regulation. Reference [79]


proposes optimal frequency regulation by evoking DLC for generators and DR using a linearized power

system model.

1.5 Thesis Outline

In light of the challenges and limitations present in the existing literature with securely managing a large

number of highly fluctuating heterogeneous power components such as flexible consumer appliances and

DGs, this thesis presents a detailed investigation on how these difficulties can be effectively overcome and

proposes novel techniques that can be utilized to enable the seamless integration of these components

while accounting for physical grid constraints. More specifically, this thesis consists of six chapters. First,

a background on the theoretical tools leveraged in this thesis in the areas of convex optimization and game

theory is summarized in Chapter 2. Next, in Chapter 3, an overview of novel DR methods proposed for

coordinating a large number of flexible consumers while accounting for individual consumer preferences

is presented. These techniques are based on principles of water-filling in convex optimization, dual

decomposition and population game theory. Next in Chapter 4, two novel real-time economic dispatch

techniques for DGs are discussed. The first technique utilizes sub-gradient methods and does not account

for physical grid constraints. The second technique, which is based on dual decomposition and population

game theory, accounts for physical constraints. Chapter 5 presents a hierarchical algorithm that combines

DR and DG dispatch in distribution and transmission networks. This method capitalizes on ADMM,

decomposition and population games. Finally, the thesis is concluded in Chapter 6 and future directions

are also presented here. The general logical flows between the chapters are depicted pictorially in Fig.

1.6.

Figure 1.6: Interactions between chapters in the thesis.

The following presents a detailed description of the content in each chapter:

• Chapter 1: The premise of this thesis is presented along with a detailed overview of challenges


and opportunities present in today’s smart grid. A general overview of existing literature in the

areas of DR and DG dispatch is presented. The formal problem statement and contributions of

this thesis are outlined.

• Chapter 2: Main problem formulations commonly constructed and solved in the power grid by

system operators are presented in this chapter. Prior to this, a background on convex sets and

functions is presented which is useful for identifying the tractability of solving an optimization

problem. Then, the optimal power flow problem is introduced that consists of a general objec-

tive which takes into account net power flow balance across buses, bus voltage limits, generation

capacity constraints and flexible demand curtailment ranges. Challenges associated with this for-

mulation are highlighted. Then, various flavours of this formulation such as the DR problem,

economic dispatch problem, linearized power flow problem and voltage regulation problem are pre-

sented. Analysis of various challenges associated with these formulations with respect to convexity

is presented.

• Chapter 3: The DR problem which takes into account discrete load curtailment levels and com-

fort budgets of thousands of residential consumers is addressed in this chapter. First, a distributed

solution based on water-filling in convex optimization is applied to a simplified version of the orig-

inal DR problem formulation without taking into account discrete load curtailment levels. This

technique allows for a simplistic introduction to the distributed solution structure commonly inte-

grated into all proposals presented in this thesis. Then, another proposal that actually takes into

account the discrete load curtailment levels is presented which is based on decomposition and pop-

ulation game theoretic constructs. Theoretical convergence results along with practical simulation

studies using realistic models are presented. The resilience of the solution to perturbations is also

theoretically and practically demonstrated.

• Chapter 4: Dispatch of a large number of DG sources is investigated in this chapter. First,

general economic dispatch is considered which does not take into account the underlying physical

constraints of the grid. A distributed solution based on dual decomposition and the sub-gradient

method is presented to coordinate a large number of highly fluctuating DGs in an optimal manner.

Flux in generation capacities of DGs are accounted for by considering very short optimization

intervals. Next, another dispatch solution based on decomposition and population game theory

is presented that takes into account the underlying physical grid constraints to accommodate a

large number of DGs in the DN. As low-voltage networks such as the DN are sensitive to excessive

power injections, this proposal which takes into account voltage rise considerations is practical

for these systems. Comprehensive practical and theoretical results on various characteristics of

these proposals that include convergence rates, resilience to perturbations and scalability are also

presented.

• Chapter 5: In this chapter, a hierarchical technique is proposed that combines DR and DG

dispatch in multiple DNs situated across the transmission network. This heterogeneous solution

exacts coordination across two tiers. The transmission tier involves completely decentralized co-

ordination amongst buses using ADMM and consensus to achieve optimal linear power flow. This

abstracts details about individual systems such as IPPs participating in energy markets, DGs in

the DN, flexible loads and bulk generation entities from the system operator. Then, at the DN


level, flexible consumers and DGs are coordinated so that the setpoint computed in the transmis-

sion network level can be achieved. This proposal is also accompanied by theoretical and practical

verifications.

• Chapter 6: The thesis is concluded in this chapter. A brief summary is presented and the

contributions of the thesis are identified. Possible future extensions of this work are also presented.

Chapter 2

Power Grid Optimizations

The process of curtailing power demands and dispatching DGs in order to achieve a specific system

objective (e.g. minimizing costs, carbon footprint, etc.) while ensuring that the physical grid and

consumer constraints are met, constitute the formulation and solving of an optimization problem. An

optimization problem is composed of optimization variables, objective(s) and constraints. In the problem

formulations presented in this thesis, associated optimization variables include loads curtailed by flexible

consumers and power injected by DGs into the grid. The grid operator seeks to minimize the system

cost resulting from various configurations of the optimization variables and this is accounted for by the

objective(s) of the problem formulation. The optimization variables are subject to constraints stemming

from underlying physical system properties and/or resource limitations. All points that satisfy these

constraints form a feasible set. This set is composed of various values that the optimization variables

can take without infringing upon any of the imposed constraints. An assorted set of problems that include

optimal load curtailment, economic dispatch, linear power flow and voltage regulation can be formulated

using different combinations of system objectives and physical constraints as detailed later in this chapter.

Certain problem formulations are more suited than the others under particular circumstances. However,

before discussing these, a brief exposition on the notion of convex sets and functions is presented next as

these are imperative in gauging the difficulty of solving an optimization problem. Pertinent information

about the problem formulation that includes scalability and resource requirements can be obtained from

this assessment.

2.1 Convex Sets and Functions

An optimization problem can be classified as either a convex or a non-convex optimization problem. Con-

vex optimization problems can be solved using commercially available solvers [80]. These are tractable

problems and the computational complexity of solving these is typically polynomial with respect to the

size of the problem (i.e. number of optimization variables). On the other hand, non-convex optimization

problems are intractable and the complexity of solving these increases exponentially with the size of

the problem. Hence, non-convex problems are not scalable and impose significant challenges in a large

system. Thus, identifying whether a problem is convex or not provides vital information about the

difficulty entailed in solving the problem. All theoretical constructs listed in this section are based on

the material presented in reference [81].

14

Chapter 2. Power Grid Optimizations 15

2.1.1 Convex Sets

Constraints in an optimization problem define a set of feasible values that can be taken by the opti-

mization variables. Any point not contained within this set will result in the infringement of one or

more constraint(s). A convex optimization problem must have a convex feasible set. The definition of a

convex set C is as follows:

θx+ (1− θ)y ∈ C ∀ x, y ∈ C, θ ∈ [0, 1] (2.1)

Intuitively, this translates to the following. If all points on lines drawn between any randomly selected

points x and y from the set C lies within the same set, then C is a convex set. A set is non-convex if the

definition in Eq. 2.1 does not hold. Simple two-dimensional examples of convex and non-convex sets are

presented in Fig. 2.1.

Figure 2.1: Examples of convex and non-convex sets.

The set depicted in the upper left hand corner is formed by the intersection of five linear inequality

constraints. According to the intuitive description of a convex set presented in the above, it is evident

that this set is indeed convex (lines connecting any two randomly selected points from the polygon lie

within that polygon). The second set located in the upper right hand corner constitutes of points on

a curve. This set is formed by a nonlinear equality constraint and an example of this is a quadratic

equality constraint. It is clear that the points on the curve do not form a convex set as it is possible to

identify more than one line formed by connecting two points on this curve that consist of points lying

outside of this set. Next, in the bottom left hand corner, three discrete points form the set. Evoking the

intuition behind convex sets, it is evident that this is also not a convex set. The final set located in the

bottom right hand corner is formed by a linear equality constraint and all the points on this line form

a convex set. These results can also be derived theoretically by evoking the formal definition presented

in Eq. 2.1 as detailed in reference [81].

2.1.2 Convex Functions

In the objective of an optimization problem, functions are typically used to assign costs to various values

taken by the optimization variables. Like sets, functions f : Rn → R can also be classified as convex or

non-convex. The formal definition of a convex function is listed in Eq. 2.2.

f(θx+ (1− θ)y) ≤ θf(x) + (1− θ)f(y) ∀ x, y ∈ dom(f), θ ∈ [0, 1] (2.2)


This definition of a convex function can also be intuitively interpreted as follows. All points on a line

drawn between any two points on the function must lie above that function within the domain range

specified by the initially selected two points. Fig. 2.2 presents two examples of functions. Evoking this

intuitive definition, it is clear that the quadratic function on the left of the figure is a convex function

while the sinusoidal function on the right is not convex.

Figure 2.2: Examples of convex and non-convex functions.

2.1.3 Convex versus Non-convex Optimization Problems

An optimization problem is composed of objectives (i.e. cost functions) and constraints (i.e. feasible

set(s)). Each constraint constitutes of a set of points and the intersection of all constraint sets in the

problem formulation results in the feasible set. These are points that do not infringe upon any of the

restrictions/limits set in the problem. If any one of the constraints represents a non-convex set, then the

entire feasible set results in being non-convex as well. For instance, optimization problems with discrete

variables are typically known to be difficult to solve [82]. Moreover, if the cost function in the objective

is not convex, then the optimization problem is classified as a non-convex optimization problem. The

difficulty of solving a non-convex problem is typically identified as NP-hard [81,83]. Hence, in a convex

optimization problem, all constraint sets and cost functions must be convex. This notion of convexity is

used to classify the difficulty of solving optimization problems specifically in the context of the electric

grid in the next section.

2.2 Optimization for the Electric Grid

The electric grid consists of a large number of diverse components. Today’s grid primarily consists of

alternating current (AC) power components. Hence, grid states take values in the complex space (i.e.

consists of real and imaginary parts). In this thesis, all symbols representing various complex attributes

of the grid will be listed in capital letters. Vectors will be represented by non-indexed symbols.

2.2.1 Power Components

For the sake of brevity, main components in the power grid that include generators, loads, buses and

lines along with associated constraints and costs are discussed next. Fig. 2.3 presents the single line


Figure 2.3: Simple electric grid.

diagram of a very simple grid for illustrative purposes1. All system configurations considered in this

thesis are based on balanced three-phase circuits.

Generators

Generators output real and reactive power as required by the grid and these include traditional syn-

chronous power plants, DG sources and storage systems. These convert energy from a specific domain

(i.e. hydro, chemical, mechanical, etc.) into electricity to meet consumer power demands. Due to the

AC nature of the grid, there are two types of power flowing in the system and these are real and reactive

power. Real power denoted by the symbol p (measured in Watts (W)) is used to perform actual work

whereas reactive power q (measured in volt-amperes reactive (VAR)) is present when complex voltages

and currents in the AC system are not in phase with one another. Complex power S measured in

volt-amperes (VA) is a combination of real and reactive power: S = p + iq where i =√−1. Apparent

power also measured in VA is the magnitude of the complex power |S|. In Fig. 2.3, there are two power

sources. The generator attached to the bus labelled 3 is associated with a power rating of 800 MVA

and injects 520 MW of real power into the system. Every power source is associated with individual

generation capacity requirements that pose limits on the upper and lower bounds of real and reactive

power generation as follows:

cpg ≤ pg ≤ cpg ∀ g ∈ G (2.3)

cqg ≤ qg ≤ cqg ∀ g ∈ G (2.4)

where G is the set of all generators in the system, g is the index assigned to a generator in the set G, cpg and

cqg represent the lowest possible real and reactive power that can be generated by generator g, cpg and cqg

are the associated upper limits on real and reactive power and pg and qg are the actual real and reactive

power generated by generator g. Traditional power plants are associated with ramping constraints which

are not considered here as this thesis focuses on the dispatch of DGs with negligible ramp up/down

rates [84]. Additional constraints must be taken into consideration when sources generating discrete

levels of power are included in the generation mix. For instance, storage systems at homes may supply

power at finite number of discrete power ratings [10]. In these cases, additional constraints will be

1Image courtesy of reference [3]


imposed on pg as follows:

pg ∈Mg = {M1 . . .Mng} ∀ g ∈ Gd (2.5)

where the set Gd ⊆ G represents all generators with discrete power inputs and the setMg consists of ng

real power levels at which generator g can inject power into the grid. It is important to note that Eq.

2.5 represents a discrete set and therefore is not convex.

Loads

Loads represent entities belonging to the residential, commercial and industrial sectors that consume

power. The real power demand of consumer d is denoted as pd. If the consumer is using inductive loads

then reactive power demands must be taken into consideration as well and this is denoted as qd. Real

power curtailment by a flexible consumer d is denoted as prd. In Fig. 2.3, a set of loads are attached to

Bus 2 and consume 280 MVAR of reactive power and 800 MW of real power. Flexible consumers have

some tolerance to adjustments made to their real power demands. The power reduction by a flexible

consumer d can range between 0 to cd as follows:

0 ≤ prd ≤ cd ∀ d ∈ D (2.6)

where D represents the set of all consumers, the power reduction prd possible for consumer d can range

between 0 (i.e. no curtailment) to cd which is an upper bound on power curtailment imposed by local

preferences and appliance operating conditions which are discussed in Chapter 3. If the consumer is not

flexible then cd can be set to 0 so that the demand cannot be curtailed without infringing upon Eq.

2.6. Typically appliances in the residential sector operate at discrete power demand levels [18]. Fig. 2.4

illustrates the load profile of a home during the winter over a 24 hour period and the discrete nature of

appliance operations is evident in this plot.

0 5 10 15 20 250

500

1000

1500

2000

2500

3000

3500

4000Load Demand Snapshot of a Single Class 1 Home (Winter)

Time (hour)

Po

we

r (W

att

s)

Figure 2.4: Load profile of a home during winter.


Hence, various values that can be taken by real power reduction for each flexible consumer in the

residential sector is further constrained as follows:

prd ∈Md = {M1 . . .Mnd} ∀ d ∈ D (2.7)

where the setMd consists of nd real power levels at which consumer d can curtail local power demands.

Again, it is imperative to note that Eq. 2.7 represents a discrete set and therefore is non-convex.

Next, every flexible consumer has a particular tolerance for the curtailment of individual local loads.

This tolerance is referred to as consumer comfort budget and is captured by imposing a constraint that

ensures that the cumulative demand reductions over a day is well within the energy budget set in advance

by the consumer as follows:

E(prd) ≤ Edbudget ∀ d ∈ D (2.8)

where the function E(.) is an accumulation of all demand reductions so far over a day including prd and

Edbudget is the overall energy reduction budget over a day configured to be tolerated by consumer d.

In addition to individual constraints imposed on demand curtailment by each consumer, global

coupling constraints that ensure balance between overall real and reactive power demands and supply

are as follows: ∑g∈G

pg − plgrid =∑d∈D

pd (2.9)

∑g∈G

qg − qlgrid =∑d∈D

qd (2.10)

where the first term in the above two constraints represents overall real/reactive power generation, the

second term depicts all real/reactive power losses in the grid encountered during transmission and the

third term is the aggregate real/reactive power demand of all consumers in the system. As highlighted

in the upcoming sections, transmission losses are neglected in some formulations and included in others.

Certain systems are sensitive to bus voltages which are affected by the real and reactive power flows in

the power network. Computation of these values is discussed next.

Buses

Buses serve as connection points in the grid for various power components. All buses in the system are

represented by the set B. There are three types of buses in general and the classification of buses into

these three groups is generally based on two power system measures [87]. One is the complex power

injection Sb into bus b and the other is the complex bus voltage Vb. A slack bus is used as a reference

point for computations. The complex voltage associated with the slack bus is fixed whereas the complex

power injection can vary. A load bus, also referred to as the PQ-bus, has fixed complex power injection

with voltage that can vary. A generator bus b, also referred to as a PV bus, has fixed real power injection

(i.e. pg =Re(Sg)) and voltage magnitude (i.e. |Vg|), however, the reactive power qg and voltage angle

∠Vg can vary. In Fig. 2.3, bus 1 is connected to a power source and bus 2 is connected to loads only.

Bus 3 is connected to both generation and loads. The net power injection into bus 3 is therefore total


generation minus total demand in that bus as captured by the following set of relations:∑g∈Gb

pg −∑d∈Db

pd = Re(Sb) (2.11)

∑g∈Gb

qg −∑d∈Db

qd = Im(Sb) (2.12)

where Gb is a set representing all generators connected to bus b and Db represents all demands attached

to bus b. Buses with varying voltages are subject to the following limits to avoid irreversible equipment

damages:

|Vmin|2 ≤ VbV ∗b ≤ |Vmax|2 ∀ b ∈ B (2.13)

Bus voltage magnitudes are constrained to operate between the thresholds |Vmin| and |Vmax|. As Eq.

2.13 represents a set of quadratic inequalities, these represent convex constraints. Transformers serve

as interfaces between generation/loads to the transmission networks. As power is transmitted at high

voltages over long distances, the voltages are typically stepped up from generator buses and stepped down

at load buses. Constraints associated with the transformers are neglected in this thesis as transmission

and distribution networks are considered separately.

Lines

Lines connect various buses together in the power grid. Set of lines present in the power network that

is under consideration is denoted as L and a line connecting bus a ∈ B and b ∈ B is represented as

a↔ b ∈ L. Various interconnections in the power grid can be succinctly represented as a graph in which

each bus is a node and every line is an edge. Power losses in lines depend on the characteristics of the

corresponding lines and these are captured by line admittances. Hence, the magnitude of power pa,b

flowing across line a↔ b from buses a to b is not the same as that of pb,a which represents power flowing

in the reverse direction. The power flowing across a line at steady-state can be computed by evoking

Kirchhoff’s voltage and current laws as illustrated in Fig. 2.5.

Figure 2.5: Power flow across a line.

First, an assumption about the direction of power flow is necessary. In Fig. 2.5, it is assumed that

power is flowing from bus a to bus b. From the perspective of bus a, the power flow on a line connecting

to bus a via bus b is:

VaV∗a Y∗a,b − VaV ∗b Y ∗a,b

and from the perspective of bus b, the power flow on the line connecting to bus b via bus a is:

VbV∗b Y∗b,a − VbV ∗a Y ∗b,a


Hence, it is evident that the complex power flows Sa,b 6= Sb,a. Suppose that bus a serves as a connection

point for multiple lines, then the net power injected into the bus must be equal to the net power flow

across that bus via lines coinciding with it (i.e. ∀ a↔ b ∈ L ):

pa + iqa =∑

a↔b∈L

(VaV∗a Y∗a,b − VaV ∗b Y ∗a,b)

Separating the real and imaginary parts result in the following set of relations for real and reactive power

injections into bus a in terms of bus voltages and admittances:∑g∈Ga

pg −∑d∈Da

pd = Re( ∑a↔b∈L


)∀ a ∈ B (2.14)

∑g∈Ga

qg −∑d∈Da

qd = Im( ∑a↔b∈L


)∀ a ∈ B (2.15)

The above net power balance equations are quadratic equality constraints in the complex domain which

are not convex. These represent constraints imposed by the underlying physics of the system which are

important especially in low-voltage DN networks that are sensitive to reverse power flows.

Cost Functions

There are numerous objectives that are typically taken into consideration by a grid operator. This

thesis focuses on DG dispatch and DR and therefore the optimization variables constitute of pg ∀ g ∈ Gand pd ∀ d ∈ D. Hence, the system costs incurred will be functions of these optimization variables.

Costs in the power grid are typically designed to be quadratic functions of the associated optimization

variables [88,89].

C(p) =∑g∈G

fg(pg) +∑d∈D

fd(pd) (2.16)

where p is a vector representing the optimization variables, C : R|D|+|G| → R and f : R → R. Every

participating entity i is associated with an individual cost fi. As these costs are summed over individual

participants in the system (generators and flexible consumers) to obtain the final cost, C(.) is referred

to as a separable function. Examples of C(.) in economic dispatch include pTAp where p = [p1 . . . p|G|]

is the vector representing the optimization variables and A ∈ R|G|x|G| is a positive semi-definite matrix

(i.e. A � 0). For separability, A will be a diagonal matrix. Intuitively, this translates to greater costs

for larger levels of dispatch.

2.2.2 Steady-State Power Flow Problem Formulation

So far, a brief introduction to the main power components in the grid and associated properties has

been presented. Next, a generalized formulation referred to as the optimal power flow (OPF) problem

composed of all of these components is introduced in POPF . This formulation is a steady-state prob-

lem. Transients in the system are considered to be absorbed by highly inertial bulk generation entities

commonly present in the grid. In stand-alone microgrid systems (not considered in this thesis), these

perturbations will not be negligible.


POPF : minp

C(p) =∑g∈G

fg(pg) +∑d∈D

fd(pd)

s.t.

cpg ≤ pg ≤ cpg ∀ g ∈ G (C1)OPF

cqg ≤ qg ≤ cqg ∀ g ∈ G (C2)OPF

pg ∈Mg = {M1 . . .Mng} ∀ g ∈ Gd (C3)OPF

0 ≤ prd ≤ cd ∀ d ∈ D (C4)OPF

prd ∈Md = {M1 . . .Mnd} ∀ d ∈ D (C5)OPF

E(prd) ≤ Edbudget ∀ d ∈ D (C6)OPF∑g∈G

pg − plgrid =∑d∈D

pd (C7)OPF∑g∈G

qg − qlgrid =∑d∈D

qd (C8)OPF

|Vmin|2 ≤ VaV ∗a ≤ |Vmax|2 ∀ a ∈ B (C9)OPF∑g∈Ga

pg −∑d∈Da



)∀ a ∈ B (C10)OPF∑

g∈Ga

qg −∑d∈Da



)∀ a ∈ B (C11)OPF

where Gd is a set representing all generators taking discrete generation values. As both DR and economic

dispatch are considered in the OPF, optimization variables of POPF are represented by the vector

p consisting of pd ∀ d ∈ D and pg ∀ g ∈ G. (C1)OPF to (C3)OPF are constraints associated with

the generators. Constraints (C4)OPF to (C6)OPF embody the consumer demand constraints. Global

coupling constraints balancing overall power generated with demands are captured in (C7)OPF and

(C8)OPF . Limits on bus voltages are imposed by constraint (C9)OPF . Net power balance at every

bus introduced by the underlying physics of the system is reflected in (C10)OPF and (C11)OPF . Even

though constraints (C7)OPF and (C8)OPF are implied in (C10)OPF and (C11)OPF , these are included

for ease of formulating certain problems that are presented later in this chapter. Constraints (C1)OPF to

(C11)OPF serve to capture resource limitations and physical grid characteristics. All of these constraints

have already been introduced earlier in Sec. 2.2.1.

Solving this OPF directly will be extremely challenging for the following reasons. First, parameters

corresponding to DG generation capacities and consumer demands (i.e. cpg, cpg,c

dg,c

dg) will be at con-

stant flux. Typically in the literature, forecast models are utilized to fix these parameters for a long

period of time. As these models are associated with high error margins especially over longer prediction

horizons, the optimization interval is reduced to one minute periods in this thesis. Then, much more

accurate forecast models can be leveraged by individual participants (flexible consumers and/or DGs)

to identify these parameters based on local conditions. This, however, imposes significant computation

and communication overheads which are overcome later in this thesis by leveraging upon distributed

and decentralized solution architectures. The second issue is introduced by the presence of non-convex

constraints in POPF . Constraints (C3)OPF and (C5)OPF represent discrete demand and generation


levels. In Sec. 2.1.1, discrete sets have been identified to be non-convex. Other non-convex constraints

in POPF are (C10)OPF and (C11)OPF which are associated with net power flow balance at each bus.

These are complex quadratic equality constraints. When the real and imaginary components are consid-

ered separately, quadratic and sinusoidal terms are present. As mentioned in Sec. 2.1.3, an optimization

problem becomes intractable even when there is one non-convex constraint. In this particular case, there

are |Gd|+ |D|+2|B| such constraints and this is proportional to the total number of nodes in the system.

Hence, directly solving this NP-hard optimization problem over every one minute interval for a large

system is certain to be infeasible.

To resolve these issues of practicality, in this thesis, the OPF is divided into simpler sub-problems

that are suitable representations of the system objectives and limitations under certain circumstances.

Underlying structures of these sub-problems are capitalized to propose more tractable and practical

solutions. Then, these sub-problems are combined into a hierarchical algorithm to coordinate DGs and

flexible consumers in a manner that heeds the constraints listed in POPF while also achieving minimal

costs for the system operator. As such, in the following, the four sub-problems considered later in this

thesis are introduced.

2.2.3 Demand Response Formulation

The DR problem listed in PDR, consists of flexible consumers whose demands are directly adjusted by the

EPU according to individual local operating conditions and comfort budgets so that the overall demand

reduction amounts to SDR at minimal cost. This setpoint SDR is determined by system operator and can

be designed to maintain peak demands around a particular value so that dependence on expensive and

unsustainable peak generating plants is minimized. The EPU provides some compensation to consumers

for participating in the DR program and this is captured in the objective of PDR. The total number

of constraints in this problem is 3|D| + 1. Moreover, many parameters in these constraints such as cd,

Md and Edbudget can vary based on individual consumer preferences. Gathering these parameters from

consumers at a fairly regular basis to accurately capture fluctuations in these will result in significant

communication overheads for the EPU. Moreover, constraint (C2)DR is not convex. Hence, with a large

number of participants, as typical in the residential sector, directly solving PDR becomes intractable.

PDR : minp

CDR(p) =∑d∈D

fd(prd)

s.t.

0 ≤ prd ≤ cd ∀ d ∈ D (C1)DR

prd ∈Md = {M1 . . .Mnd} ∀ d ∈ D (C2)DR

E(prd) ≤ Edbudget ∀ d ∈ D (C3)DR

SDR =∑d∈D

prd (C4)DR

However, problem PDR has a decomposable structure. The objective is a summation of costs incurred

by individual consumers and all constraints with the exception of (C4)DR can be separated into groups

that correspond to every participating consumer. This structure is leveraged in Chapter 3 to design


distributed DR.

2.2.4 Economic Dispatch Formulation

Economic dispatch (ED) problem listed in PED is used by an ISO to balance overall generation with

aggregate demands in a manner that is least expensive. Losses in the transmission lines, bus voltage

limits and net power balance constraints are neglected in this formulation.

PED : minp

C(p) =∑g∈G

fg(pg)

s.t.

cpg ≤ pg ≤ cpg ∀ g ∈ G (C1)ED

pg ∈Mg = {M1 . . .Mng} ∀ g ∈ Gd (C2)ED∑

g∈Gpg −

∑d∈D

pd = 0 (C3)ED

Certain generation sources are more costly than others and these are accounted for in the objective where

individual costs are summed across all participating generation sources. As local generation capacities

can vary significantly for DGs, parameters such as cg, cg and Mg must be obtained from individual

DGs when PED is solved centrally. This can be onerous especially when there is a large number of

active DGs whose generation parameters must be collected frequently in order to effectively capture

fluctuations in generation. Moreover, constraint (C2)ED is not convex. If every DG can intelligently

make its own decision regarding how much power to dispatch based on local operating conditions (i.e.

generation capacity) and general external information conveyed by the ISO, then computational and

communication overheads can be offloaded from the central dispatching entity. This is demonstrated in

Chapter 4 of this thesis by capitalizing on the separable nature of the problem PED.

2.2.5 Linear Power Flow Formulation

The main source of complexity in the OPF formulation in POPF is the set of net power balance equa-

tions that are quadratic equality constraints in the complex domain which are non-convex. In order to

overcome this difficulty, several approximations are applied to these equations so that these result in a

set of linear equations which greatly simplify computations as listed in constraint (C3)FR of PFR. This

linearized model is used for many applications in the power grid.

PFR : minp,θ

C(p, θ)

s.t.

cpg ≤ pg ≤ cpg ∀ g ∈ G (C1)FR

pg ∈M = {M1 . . .Mng} ∀ g ∈ Gd (C2)FR∑

g∈Ga

pg −∑d∈Da

pd =∑

a↔b∈L

−ba,b(θa − θb) ∀ a ∈ B (C3)FR


The optimization variables of PFR consist of the real power pg generated by each generator g and the

bus voltage angle θb at each bus b ∈ B. This reduction in the optimization variables is due to certain

simplifying approximations applied to the power balance equations in POPF as discussed next. Real and

complex power flow across line a↔ b ∈ L is repeated in the following for convenience:

pa,b + iqa,b = Sa,b = VaV∗a Y∗a,b − VaV ∗b Y ∗a,b

This can be equivalently expressed in terms of real terms as follows:

pa,b = |Va|2ga,b − |Va||Vb|ga,bcos(θa − θb)− |Va||Vb|ba,bsin(θa − θb)

qa,b = −|Va|2ba,b + |Va||Vb|ba,bcos(θa − θb)− |Va||Vb|ga,bsin(θa − θb)

where Ya,b = ga,b + iba,b and θ is associated with the complex voltage |V |∠θ. As outlined in [90], the

following assumptions are applied to the above set of equations:

1. |ga,b| << |ba,b|

2. sin(θa − θb) ≈ θa − θb; cos(θa − θb) ≈ 1

3. |Va| ≈ 1, |Vb| ≈ 1 (in per unit system) ∀ a, b ∈ B

which result in the simplification of the power balance equations to:

pa,b = −ba,b(θa − θb)

qa,b = 0

It is important to note that reactive power flows are eliminated by these simplifications. These assump-

tions are somewhat valid in high-voltage networks operating at normal conditions and can be applied to

simplify analysis. These, however, do not apply to low-voltage networks as the above assumptions will

not remain valid and result in bus voltages violating the specified nominal ranges.

2.2.6 Voltage Regulation Formulation

Voltage rise across buses in low-voltage networks is of significant concern for EPUs especially with the

recent proliferation of a large number of DGs in the DNs. DNs are designed to carry power from the EPU

substation to the consumers. It is not designed to handle excessive power flowing back into the grid. One

solution for rectifying this is to perform expensive infrastructure upgrades and allow DGs to operate at

MPP. Another alternative will be to consider bus voltage limits, reactive power and net power balance

to optimally integrate a large number of DGs into the DN by dispatching these as required. The voltage

regulation problem listed in PV R is very similar to the OPF problem. In this problem, the optimization

variables consist of only real power dispatched by DGs (flexible loads are not included). PV R is a more

comprehensive formulation than PED as net power balance at each bus along with voltage limits are

incorporated into the problem formulation.


PV R : minp

C(p) =∑g∈G

fg(pg)

s.t.

cpg ≤ pg ≤ cpg ∀ g ∈ G (C1)V R

cqg ≤ qg ≤ cqg ∀ g ∈ G (C2)V R

pg ∈M = {M1 . . .Mng} ∀ g ∈ Gd (C3)V R∑g∈G

pg =∑d∈D

pd −∑

a↔b∈L

pa,bl (C4)V R∑g∈G

qg =∑d∈D

qd −∑

a↔b∈L

qa,bl (C5)V R

|Vmin|2 ≤ VaV ∗a ≤ |Vmax|2 ∀ a ∈ B (C6)V R∑g∈Ga

pg −∑d∈Da



)∀ a ∈ B (C7)V R∑

g∈Ga

qg −∑d∈Da



)∀ a ∈ B (C8)V R

where pa,bl and qa,bl are real and reactive line losses encountered in line a↔ b ∈ L. Even though this for-

mulation is more comprehensive than PED, there are many more challenges accompanying this problem.

Discrete constraints in (C3)V R and non-convex power balance constraints in (C7)V R and (C8)V R con-

tribute to these difficulties. In the literature, convex relaxations are applied to these constraints which

are exact in radial networks such as the DN under certain stringent conditions. As these conditions

can be restrictive, convex relaxations are not considered for low-voltage DN networks in this thesis. A

distributed solution is introduced in Chapter 4 to overcome these challenges where PV R is divided into

master and secondary problems.

2.3 Remarks

In the above discussion, four problem formulations common in the power grid are introduced which are

suitable for various contexts in the power grid. These are presented in the order of increasing complexity.

In the culmination of this thesis in Chapter 5, a hierarchical distributed solution is presented that

combines various types of networks and problem formulations in a manner that is both practical and

scalable by capitalizing on the techniques introduced in Chapters 3 and 4.

Chapter 3

Demand Response

DR and DG dispatch are each considered separately in this chapter and the following chapter where

various approaches for handling these difficult problems are explored. These are then combined into a

single hierarchical solution in Chapter 5.

In particular, in this chapter, two distributed DR proposals are presented. The first proposal uti-

lizes the water-filling method in convex optimization and the second proposal leverages on theoretical

constructs from population game theory. These proposals are novel departures from the state-of-the-art

as these iterative solutions are designed to meet optimality and scalability goals in real-time (i.e. in the

scale of minutes). The main difference between the two proposals is the manner in which consumer de-

mand curtailment is modelled. In the first proposal, an assumption regarding consumer demand patterns

is made which significantly simplifies the original DR formulation in PDR whereas no such simplifying

assumptions are made in the second proposal, thus rendering it more generally applicable.

3.1 System Model

Details on the distributed and iterative approach utilized to design these two proposals are first presented

in this section. Then, various active components in the model along with assumptions made about the

system settings are introduced.

3.1.1 DR Structure

In a DR program, active participants are flexible consumers whose demands are adjusted by the EPU so

that a particular system goal can be met. The type of demand adjustment considered in this thesis is full

or partial curtailment of flexible loads. Partial curtailment can be achieved in loads consisting of flexible

power-consuming components such as heating elements (e.g. dryers operating at lower heat, heating

water at a lower temperature and so on). These consumers are associated with various comfort budgets

and are compensated for any load curtailment incurred while participating in the DR program. The

general structure of the two DR proposals presented in this chapter is distributed in nature. The EPU

broadcasts common signals to all participating entities at every signalling iteration. Flexible consumers

are each equipped with a HEMS which uses these signals to make appropriate decisions regarding

demand curtailment based on local operating conditions. As decisions are made by participating nodes

rather than a central entity, this solution architecture is considered to be distributed. Each HEMS is a

27

Chapter 3. Demand Response 28

representative of the EPU (like the smart meter) which acts on behalf of the EPU. Fig. 3.1 presents an

illustration of this DR setup.

Figure 3.1: Smart home energy management system.

The EPU capitalizes upon current aggregate grid measurements, which are readily available via

data concentrators, to compute signals that are broadcasted periodically. As these signals are general

and common to all active participants, the EPU will not need to initiate point-to-point communication

links with every active participant and transmit customized signals. This significantly reduces the

communication overhead for the EPU otherwise incurred due to bi-directional communication flows.

These lightweight signals containing a small set of numerical values are broadcast every 0.6 seconds

via the wireless medium. Even though the wireless medium is of broadcast nature, it is possible that

these consist of multiple hops which can result in excessive congestion when thousands of nodes are

exchanging information with the EPU. In our strategies, this is not the case as the EPU broadcasts the

signals in the downlink and capitalizes on the aggregate data made available in data concentrators. As

latencies entailed in transmissions across the wireless channel over a residential neighbourhood are in

the order of milliseconds, this signalling interval is large enough to account for communication delays in

uni-directional flows [21].

HEMS present in each active participant’s residence is equipped with a set of wireless receivers,

transmitters and an intelligent module. In this module, home owner d can program local preferences

regarding how appliances operate and these are translated into the cd parameter which serves as an upper

bound for local power demand curtailment in PDR. The comfort budget Edbudget parameter embodies

the total energy reduction due to demand curtailments that can be tolerated by the consumer over a

day and this can also be configured into the HEMS. The HEMS will be able to actuate local appliances

in the home via one of two methods. The first method leverages upon communication capabilities

in smart appliances to make any actuation decisions. For appliances that are not equipped with any

communication capabilities, smart plugs are used instead to control the operation of the appliance [91].

These plugs can communicate wirelessly and can be turned on or off remotely. The HEMS will be aware

of the current operating statuses of various appliances in the home and will compute actuation signals

using the signals broadcast by the EPU, local consumer preferences and operating conditions.


The HEMS abstracts local operating conditions from the EPU and the EPU is concerned only about

the aggregate behaviour of the system. This decoupling significantly reduces communication and compu-

tational overheads for the following reasons. First, local operating conditions of thousands of consumers

are not transmitted to a central entity. HEMSs residing locally within consumer premises are responsible

for these local constraints. The EPU utilizes real-time aggregate measurements to compute signals that

convey information about the current state of the grid. This information is readily available in data

concentrators that are commonly deployed in DNs [1]. Each HEMS reacts to these signals and makes

appropriate local actuation decisions based on simple computations. Hence, majority of the concen-

trated computations are offloaded to thousands of individual HEMS devices. This distributed approach

parallelizes computations and takes advantage of the small-scale computational resources available in

HEMSs that are deployed in vast numbers throughout the system. The two proposals presented in this

chapter embody two different algorithms for computing these signals and local actuation decisions. In

both of these algorithms, the main design requirement is the guaranteed convergence of the iterative and

distributed decisions made by active participants to the optimal solution of the constructed DR problem

within the allocated optimization interval.

3.1.2 Assumptions

The assumptions made in the system model are summarized as follows:

1. EPU has access to current aggregate demand measurements via data concentrators;

2. EPU is fitted with a wireless transmitter that is used to periodically broadcast a signal containing

numerical values every 0.6s to DR participants;

3. Every flexible consumer participating in DR is equipped with an HEMS;

4. HEMS, also referred to as a DR agent, represents the EPU;

5. HEMS is equipped with intelligence and wireless communication capabilities to facilitate actuation

of local appliances based on consumer preferences and the reception of EPU signals;

6. The optimization interval set by the EPU is one minute in length and therefore demand and

generation in the system is considered to be constant over these one minute periods;

7. A large number of DR participants are present in the system;

8. There exists sufficient overall demands in the system that can satisfy the load curtailment goals of

the EPU; and

9. Compensation provided is a quadratic function with respect to the demand curtailment exacted.

Assumptions 1 to 5 align with the cyber-physical vision of the power grid as detailed in Chapter 1.

Assumption 2 is practical as this is taking place at a less granular scale in today’s grid whereby many

consumers are equipped with smart meters that can exchange information with the EPU at granularity

of minutes. Assumption 3 is now common in building management architectures where lighting and

temperature management technologies are managed via these EMS systems. Assumption 6 is necessary

to ensure that fluctuations in demand parameters are captured at a high granularity. Moreover, very

small error margins result from demand/supply forecast models applied over short time intervals such


as one minute periods [43]. Assumption 7 naturally applies as DR proposals presented in this chapter

target the residential sector which is composed of thousands of participants. As the goal of the EPU is

to curtail some portion of existing demands in the system, assumption 8 is practical. Assumption 9 (also

applied in the existing literature such as reference [88]) offers greater compensation for higher levels of

demand curtailment.

3.1.3 System Model Implementation

In order to ascertain whether the proposed DR algorithms apply in practical systems, realistic models

are implemented in MATLAB to capture various nuances of the system accurately. Household demands

are highly correlated with many factors that include diurnal patterns, seasons, appliance penetration

rates and appliance usage patterns. There are |D| consumers participating in the DR program. Each

consumer represents a residential home. The load profile of each home is generated using parameters

provided in references [18, 92] that include penetration rates, power consumption patterns of common

home appliances and probability of active appliance usage based on the time-of-day and season in various

regions such as Ontario, France, India and Quebec. The appliances considered are washing machine,

dryer, dishwasher, electric stove, oven, freezer, fridge, water heater, space heater and air conditioner. All

load profiles generated in this chapter are based on parameters provided in reference [18] for Ontario. Fig.

3.2 illustrates a sample load profile generated for a home over a 24-hour period using these parameters

in the winter.

0 5 10 15 20 250

500

1000

1500

2000

2500

3000

3500Load Profile of a Home on a Winter Day

Time (hours)

Aggre

gate

Pow

er

(W)

Figure 3.2: Sample load profile of a home.

The periodic load signatures occurring throughout the day can stem from inductive appliances such

as fridge, freezer and air conditioner. Sudden bursts of power consumption occurring in the morning and

afternoon can result from resistive elements used by stoves which have an on/off power demand pattern

over short time intervals or other appliances such as dishwashers, dryers and washing machines.

The consumer can set preferences regarding the operation of various appliances. Demand curtailment

is not limited to completely switching off appliances. Appliances can also operate in conservation mode

where the invocation of resistive heating elements that consume significant amounts of real power during

operation is reduced [18]. For instance, a washing machine can operate using cold water cycles. A dryer


can operate in tumble drying mode. A dishwasher can operate using only cold water. An air conditioner

can operate at a higher setpoint (on-cycles are less frequent) or not operate at all until the comfort budget

is exhausted. On the other hand, when the consumer does not want certain appliances to operate at

reduced modes, the power available for reduction in the home will be adjusted so that demands from

these inflexible loads are not altered. Hence, individual operation parameters of appliances in a home

are lumped into general parameters such as cd by the corresponding HEMS and this prevents details

from detracting from the main problem at hand.

In the numerical studies presented later for each DR proposal, the comfort budget Ebudgetd is set to

1 kWh for each participating flexible consumer. This is equivalent to the continuous reduction of power

consumption by 1 kW over an hour.

3.2 Demand Response via Water-Filling

In this section, the first DR proposal is presented which is based on the work published in reference [93]

by the author of this thesis. Water-filling technique in convex optimization is leveraged to achieve

distributed DR. First, the DR problem formulation (a slight variant of PDR) is presented, after which an

overview of the proposed algorithm is provided. Then, insights from theoretical and numerical studies

of the proposed algorithm are discussed.

3.2.1 Demand Response Formulation

The DR formulation used in the design of the proposed algorithm is listed in PpDR.

PpDR : minpr

C(pr) =1

2(pr)TApr

s.t.∑d∈D

prd = SDR (C1)WF

0 ≤ prd ≤ cd = min(pd, bd) ∀ d ∈ D (C2)WF

where pr is a vector of the form pr = [pr1 . . . pr|D|]

T , A ∈ R|D|×|D| is a diagonal matrix consisting of

strictly positive entries, pd is the current power demand of consumer d and bd is the maximum power

reduction possible for the current interval with respect to Ebudgetd and E(prd) of the consumer from the

previous optimization interval. Although this formulation is similar to PDR presented in Sec. 2.2.3, it is

important to note that there are no constraints discretizing the power reduction levels in PpDR. Hence,

the power reduction by each consumer is a continuous variable. The objective in PpDR is a strictly

convex function as A � 0. Moreover, as (C1)WF is a linear equality constraint and (C2)WF is a linear

inequality constraint consisting of constants, these form a convex feasibility set. Thus, PpDR is a convex

optimization problem. Although, not considering discrete power reduction levels can limit practicality,

this work provides deep insights into the design of real-time DR as discussed later. Moreover, if storage

systems with continuous power output levels are available in homes, then this formulation will be exact

as cd will translate to the state-of-charge (SOC) of the battery. In the second proposal presented later

in this thesis, discrete power reduction levels are considered and therefore is more generally applicable.

As PpDR is a convex optimization problem, off-the-shelf commercial solvers can be used to solve this

problem directly. However, extensive communication overheads will be incurred as parameters related to

the operation of various local appliances and preferences must be exchanged with a central coordinating


entity. In order to avoid these difficulties, a distributed approach is used where DR agents (i.e. HEMSs)

representing the EPU reside at local premises of consumers and make iterative curtailment decisions

based on general information transmitted by the EPU every 0.6 seconds.

3.2.2 Proposed Algorithm

Here, a detailed overview of the proposed algorithm that is derived from PpDR is presented. There are

two important considerations in the proposed algorithm. First is the design of the signals transmitted

by the EPU at every signalling iteration and the second is the manner in which DR agents react to

these signals based on local operating conditions. To facilitate these processes, first the dual of PpDR is

constructed. Then, the feasibility conditions of the primal and dual problems are derived which are then

used to design the EPU signals and actuation actions taken by the DR agents.

3.2.3 Dual of the DR Problem

PpDR is referred to as the primal problem. As PpDR is a strict convex optimization problem, it has a

unique globally optimal solution [81]. The dual problem of PpDR is PdDR:

PdDR : maxλ1,λ2,ν

minpr

L(pr, λ1, λ2, ν)

where L(pr, λ1, λ2, ν) =1

2prTApr − λT1 pr + λT2 (pr − c) + ν(1T pr − SDR)

s.t. λd1 ≥ 0, λd2 ≥ 0 ∀ d ∈ D

where c = [c1 . . . c|D|]T , L(pr, λ1, λ2, ν) is the Lagrangian of PpDR and the dual variables are λ1 ∈ R|D|+ ,

λ2 ∈ R|D|+ and ν ∈ R. All constraints in the primal problem PpDR are moved into the Lagrangian. λ1

is a vector of Lagrangian multipliers associated with the lower bound of 0 imposed on prd in constraint

(C2)WF . λ2 is a vector of Lagrangian multipliers that correspond to the upper bound of cd applied to

prd in constraint (C2)WF . ν is the Lagrangian multiplier associated with the coupling equality constraint

(C1)WF in PpDR.

The Lagrangian multipliers λ1, λ2, ν are referred to as the dual variables. These dual variables can

be interpreted as penalties resulting when primal constraints (C1)WF and (C2)WF are violated. For

all feasible pr in PpDR, the following relation between the primal and dual objectives holds: C(pr) ≥L(pr, λ1, λ2, ν). The optimal value d∗ of PdDR coincides with the optimal value p∗ of PpDR only when

strong duality is satisfied [81]. Strong duality results when the relative interior of the constraint set

formed by (C1)WF and (C2)WF is not empty. Due to assumption 8 listed in Sec. 3.1.2, the Slater’s

condition holds for the system model considered in this proposal. When p∗ = d∗, there is no gap between

the dual and primal solutions. In this case, the dual problem can be solved instead of the primal problem

to obtain p∗.

3.2.4 Optimality Conditions

Optimality criteria for the primal and dual problems are divided into four categories and these are:

primal feasibility, dual feasibility, complementary slackness and stationarity. These form the Karush-


Kuhn-Tucker (KKT) conditions for PpDR and PdDR as follows:

Primal feasibility:

prd∗ ≥ 0, prd

∗ ≤ cd ∀ d ∈ D

1T pr∗ = SDR

Dual feasibility:

λd∗1 ≥ 0, λd∗2 ≥ 0 ∀ d ∈ D

Complementary Slackness:

λd∗1 prd∗ = 0, λd∗2 (prd

∗ − cd) = 0 ∀ d ∈ D

Stationarity:

OprL(pr∗, λ∗1, λ∗2, ν∗) = 0

∂L

∂prd= addp

r∗d − λ1∗

d + λ2∗d + ν∗ = 0 ∀ d ∈ D

where add is the dth diagonal component of the cost matrix A in the objective of the primal problem and

pr∗, λ∗1, λ∗2, ν∗ are optimal primal and dual variables that result in the optimal values p∗ and d∗. Primal

feasibility conditions require that the optimal solution pr∗ satisfies constraints (C1)WF and (C2)WF of

the primal problem PpDR. Dual feasibility conditions require that the optimal solution λ∗1 and λ∗2 satisfy

the feasibility constraints of PdDR. Complementary slackness conditions eliminate the terms in the

Lagrangian that are associated with the primal inequality constraints so that the optimal solution of the

dual problem will be the same as that in the primal problem. Suppose that the ith inequality constraint

is generically represented as fi(pr) ≤ 0. If pr∗ takes value in the interior of the feasible set then the

associated Lagrangian multiplier λi of that constraint is 0. Otherwise, the constraints become activated

and fi(pr∗) = 0. Stationarity conditions, stemming from first-order optimality, require the gradient

of the Lagrangian with respect to the primal variables to be equal to 0. As the Slater’s condition is

satisfied in this system model, these KKT conditions are both necessary and sufficient. Hence, the KKT

conditions can be used to derive the optimal solution pr∗.

First, the complementary slackness KKT conditions are used to divide the values taken by λd1 and

λd2 into the following four groups:

1)λd1 = 0, λd2 = 0, then 0 < prd < cd

2)λd1 > 0, λd2 = 0, then prd = 0

3)λd1 = 0, λd2 > 0, then prd = cd

4)λd1 > 0, λd2 > 0, then prd is undefined

These groups established for λd1 and λd2 are substituted into the stationarity conditions in order to

eliminate the dual variables associated with the inequality constraints. This allows the specification of


the power reduction variables prd only in terms of the variable ν as follows:

prd =

0, if ν ≥ 0,

−ν/add, if − addcd < ν < 0

cd, if ν ≤ −addcd

(3.1)

As ν is associated with the coupling constraint (C1)WF , optimal ν∗ will result when∑ni=1 p

rd = SDR.

Hence, the signals transmitted by the EPU are iterative updates made to ν based on the current offset

in∑ni=1 p

rd − SDR. Updates to ν are continually applied and broadcast by the EPU until the offset

becomes 0 at which point ν is inferred to be optimal. To facilitate the computation of ν, the EPU

acquires current aggregate consumer demand information from data concentrators (i.e. assumption 1).

Upon receiving these signals, every DR agent will revise local demands according to Eq. 3.1.

3.2.5 Theoretical Convergence Properties

Consider the scenario where every DR agent initiates local power reduction to prd = 0 at the beginning of

an optimization interval and then continues to increase reductions until the global coupling constraint is

satisfied. Suppose a function S(ν) =∑d∈Dmin(−ν/add, cd)−SDR is defined. The first term represents

the aggregation of the power reduction decisions made by each DR agent. It is evident that this function

S(.) is increasing with respect to ν. This implies that a unique solution for ν exists.

3.2.6 Analogy to Water-Filling

This method of revising local power reduction to achieve a global balance is akin to the water-filling

method typically utilized for optimal power allocation in multiple-input-multiple-output (MIMO) com-

munication systems [94]. The main difference lies in the orientation of the enclosure representing the

local constraints. Figure 3.3 illustrates how the water-filling analogy applies to the proposed distributed

method.

Figure 3.3: Demonstration of water-filling.


For the purposes of illustration, suppose that add is fixed to 1. This results in a positive definite

matrix which implies that the objective of the primal problem is a strictly convex cost function. If −ν is

steadily increased from 0 and prd is set to min{−ν, cd} in accordance to Eq. 3.1, then plotting the power

reduction values for each consumer at a particular time instant will result in a bar graph depicted in Fig.

3.3. In this water-filling analogy, individual response of DR agents is similar to water levels resulting from

injecting water into an enclosure defined by individual demand curtailment constraints. It is evident

in this illustration that DR agents continue to increase power reductions until either maximum power

reduction levels are reached or until −ν stops changing. −ν reaches equilibrium when∑d∈D p

rd = SDR.

Distributed DR for DR Agent i

Receive signal ν from the EPU:cd ← min(pd, bd)if ν > 0 then

prd ← 0else if 0 > ν > −aiici then

prd ← −ν/addelse

prd ← cdend if

Table 3.1: Distributed algorithm via water-filling for DR agent.

Based on the signal ν received from the EPU at each signalling iteration, each DR agent will respond

according to Table 3.1. The EPU computes and transmits ν to all active DR participants in the system

using aggregate information about the system at every signalling iteration. There are many methods by

which the EPU can compute ν. One option will be to increase −ν steadily from 0 until the global coupling

constraint is satisfied. This information will be made available to the EPU by the data concentrators

(assumption 1). A more efficient method for the EPU will be to execute a binary search algorithm to

find the optimal value of ν by identifying the degree of imbalance in the global coupling constraint. For

instance, if during the current signalling iteration, the overall power reduction in the system is greater

than the setpoint SDR, then ν should be increased and otherwise decreased. Binary search that leverages

on these insights is detailed in Table 3.2. This search algorithm allows the EPU to systematically identify

the optimal ν∗ in an iterative manner whereby the search space is divided into half at every signalling

iteration [94].

In Table. 3.2, the search window for ν is defined by two variables νmax and νmin which represent

the maximum and minimum values ν can take. Initially, νmax is set to 0 and νmin is initialized to

−SDR/mind∈D

(add) as this is the lowest value ν can take to satisfy∑d∈D p

rd = SDR. Then, ν is set to be

the midpoint of νmin and νmax. Each signalling iteration t occurs every δ = 0.6 seconds (in accordance

to assumption 2), during which the EPU measures the deficit or surplus in demand curtailment (i.e.∑d∈D p

rd − SDR). Either νmin or νmax is updated to reduce the search space of ν into half. This is

repeated until the difference between νmin and νmax is below a threshold ε which is set to a very small

value.


Distributed DR for EPU

t← 0a← min

d∈D(add)

νtmin ← −SDR/a, νtmax ← 0, νt ← (νmax −νmin)/2while abs(νtmax − νtmin) ≥ ε do

Broadcast νt to all DR agentsWait for δ secondst← t+ 1if∑d∈D p

rd − SDR > 0 then

νtmin ← νt

elseνtmax ← νt

end ifνt = (νtmax − νtmin)/2 + νtmin

end while

Table 3.2: Distributed algorithm via water-filling for the EPU.

3.2.7 Numerical Results

In order to illustrate the performance of the proposed DR algorithm, numerical studies are conducted

using the residential load models and appliance usage parameters outlined in Sec. 3.1.3. The simulated

system consists of 100 homes in a residential neighbourhood in Ontario over the winter season. Fig. 3.4

illustrates a snapshot of aggregate real power demands from these 100 homes over a 24-hour period. Two

periods of demand peaks are evident in this figure as typical for the winter season. The optimization

interval is set to one minute. Every interval is referred to as a DR cycle. EPU signalling takes place

every δ = 0.6s according to assumption 2 and this translates to 100 signalling iterations per minute. This

period allocated for signalling is practical as the communication capacity of a Zigbee wireless module

is in the order of kilobits per second (kbps) [21]. As each signal transmitted by the EPU consists of a

single value ν, these are lightweight signals which are subjected to latencies in the order of milliseconds.

Hence, this signalling period is practical.

In the first result presented in Fig. 3.5, the proposed DR algorithm is applied over one randomly

selected optimization interval (i.e. DR cycle) and the resulting aggregate demands in the system with and

without the distributed DR program in place are illustrated. According to assumption 6, demands are

considered to be constant over one minute intervals. For the optimization interval under consideration

in Fig. 3.5, the aggregate demand in the system is 10 kW when the proposed DR algorithm is not in

place. The proposed DR algorithm is used by the EPU to maintain aggregate demands in the system

around 9 kW. For this, SDR is set to 1kW. The resulting aggregate reduced demand is shown to converge

smoothly and rapidly to the targetted 9 kW setpoint with the proposed DR algorithm in place. More

specifically, only 8 EPU signalling iterations were necessary in this particular case for all 100 homes with

a heterogeneous mix of appliances to converge to the desired peak demand reduction level in the system.

In the next result depicted in Fig. 3.6a, the DR algorithm is applied over the entire day. In this

case study, the EPU maintains aggregate demands in the system around 20 kW. The dotted line in Fig.

3.6a represents aggregate demands with no DR in place and the red curve represents aggregate demands


0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

4 Aggregate Demand over a Winter Day

Time (hours)

Aggre

gate

Po

wer

(W)

Figure 3.4: Aggregate demand in a neighbourhood containing 100 homes.

0 20 40 60 80 1000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

Aggregate Power for a Single Cycle

Iterations

Ag

gre

ga

te P

ow

er

(W)

Reduced Operation

Regular Operation

Figure 3.5: Reduced power operation for a single iteration.


0 5 10 15 200

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

4

Time (hours)

Aggre

gate

Pow

er

(W)

Aggregate Demand over a Winter Day

Reduced Operation

Regular Operation

(a) Aggregate demand setpoint is 20 kW.

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

4 Aggregate Demand over a Winter Day

Time (hours)

Aggre

gate

Pow

er

(W)

Reduced Operation

Regular Operation

(b) Aggregate demand setpoint is 15 kW.

Figure 3.6: Regular versus reduced aggregate demands over a day.

when the proposed DR algorithm is in place. It is clear that the EPU is able to maintain the aggregate

demands so that these do not exceed 20 kW.

If the EPU aims to reduce aggregate demands to a much lower setpoint such as 15 kW, then the

comfort budgets will become exhausted much more rapidly as illustrated in Fig. 3.6b. When comfort

budgets are exhausted rapidly, the EPU will not be able to maintain aggregate demands around 15 kW

throughout the day and individual consumer demands will eventually return to normal consumption

levels. Thus, demand curtailment cannot be sustained over a long period of time in this case. Fig. 3.6b

illustrates this behaviour as the aggregate demand is maintained around the setpoint from 5 a.m. to 11

a.m. After this point, due to the exhaustion of comfort budgets of certain consumers, aggregate demands

start to surpass this 15 kW setpoint and then the system eventually returns to the typical demand levels

over the remainder of the day. This example highlights the delicate balance between consumer comfort

tolerance levels and the goals set by the EPU. The EPU must be mindful of consumer’s tolerance levels

while designing the setpoint parameters for the DR algorithm and prioritize accordingly. For instance,

suppose that a blackout has occurred in certain regions of the grid. In order to reduce the stress imposed

on the grid, the EPU can set the aggregate demand setpoint to a very low value which can be sustained

over a short period of time during which ancillary services can be activated. On the other hand, if the

EPU desires to minimize dependence on expensive peak generation plants, then the aggregate demand

setpoint should be tuned to a higher value so that peak demand reduction can be sustained throughout

the day.

3.3 Demand Response via Population Games

In the previous section, a DR proposal based on the water-filling technique in convex optimization

has been presented. Although this technique exhibits desirable convergence characteristics, it is fairly

restrictive as load curtailment is considered to take continuous values. In the proposal presented next

(based on author’s work published in reference [95]), discrete power reduction levels are taken into


account into the DR problem formulation, PEDR:

PEDR : minpr

fo(pr)

s.t.

m∑d=1

prd = SDR (C1)EDR

0 ≤ prd ≤ min {pd, bd} ∀ d = 1 . . .m (C2)EDR

prd ∈M = {M1 . . .Mn} ∀ d = 1 . . .m (C3)EDR

where m = |D| is the total number of consumers in the system. The objective function fo(.) is a

quadratic function (in accordance to Assumption 9) that applies a penalty for every power level in Mthat is curtailed. More specifically the structure of fo : Rn → R is:

fo(pr) =

1

2m

∑i∈M

( m∑d=1

Ii(prd))2

Aii

where Ii(prd) is an indicator function that returns 1 if prd = Mi and 0 otherwise and Aii is a strictly

positive value that is associated with the cost of demand curtailment Mi. This cost structure aims to

distribute the demand curtailment across the system so that excessive load reduction is not imposed

on one entity. PEDR is similar to the PDR formulation presented in the earlier chapter. In order to

overcome challenges introduced by discrete optimization variables in PEDR, theoretical constructs from

population games and convex optimization are utilized in the DR algorithm proposed next. The system

model and assumptions presented in Sec. 3.1 apply for this proposal as well. This is another distributed

solution in which the EPU transmits a general signal at every signalling iteration regarding the aggregate

state of the system which is then used by DR agents to make appropriate demand revisions based on

local operating conditions and available consumer budget. However, the method by which the EPU

signals and DR agent actuation are computed is distinct from the previous proposal. Fundamental

challenges present in PEDR due to intractability and highly varying local conditions are transformed

into advantages in this proposal. Discrete strategy space and a system consisting of a large number of

participants are fundamental building blocks in population game theory and therefore naturally applies

to PEDR.

3.3.1 Set of Transformations to the DR Problem

In this proposal, a set of transformations are applied to the DR formulation listed in PEDR. First,

all the local constraints (C2)EDR and (C3)EDR are removed from PEDR. These are then moved into

the corresponding DR agent’s local decision-making process which is discussed later. This results in

the first transformation P ′EDR. With P ′EDR, the EPU no longer needs to be concerned about local

demand constraints of individual participants. This significantly reduces communication overhead which

is otherwise required to maintain up-to-date information on individual demand parameters pd and bd.

P ′EDR : minpr

fo(pr)

s.t.

m∑d=1

prd = SDR (C1)EDR


However, the EPU still requires information about individual power reduction prd of each participating

consumer in the transformed formulation. When there are thousands of participants in the system,

managing and keeping track of prd will require significant storage and computational overhead. In order

to address these concerns, another transformation is applied to P ′EDR with a change of variables by

noting that the main concern of the EPU is only with regards to the aggregate behaviour/costs in the

system and not individual decisions. This change of variables results in P ′′EDR:

P ′′EDR : minx∈4

fo(x)

s.t. m∑i∈M

Mixi = SDR (C1)EDR

where x belongs to a simplex:

x ∈ 4 = {x ∈ Rn |∑i∈M

xi = 1, xi ≥ 0}

The variables pr are replaced by x. The ith component of x represents the proportion of DR agents in

the system selecting power reduction level Mi. x can be computed from pr as follows:

xi =1

m

m∑d=1

Ii(prd)

When m → ∞, x can be considered to be a continuous variable as the discretization error 1m → 0.

Hence, with a large number of participants, x can be treated as a continuous variable which eliminates

the non-convexity issue encountered in P ′EDR due to the discrete nature of pr. Moreover, since n << m,

the optimization variable space of the transformed problem is much smaller. Next, the transformed

objective resulting from the change of variables is:

fo(x) =m

2

∑i∈M

x2iAii

which is a strictly convex cost function when Aii > 0. Strictly convex objective combined with the

linear coupling constraint and simplex conditions render P ′′EDR a strictly convex optimization problem.

Thus, solving P ′′EDR directly to obtain x∗ is straightforward for the EPU. The set of strategies available

to every DR agent is M. x∗ represents the optimal strategy distribution in the system to achieve the

lowest DR cost for the EPU. However, the main challenge arises when deciding how DR agents must be

allocated strategies in M to achieve this optimal x∗. As the EPU is not aware of the local operating

conditions and preferences of the consumers, this cannot be done centrally. It is necessary for the EPU

to transmit general information about the current state of the system repeatedly so that the DR agents

can iteratively revise their local strategies in a manner that achieves this optimal x∗ at equilibrium.


3.3.2 Computation of EPU Signals

The computation of signals by the EPU that are broadcast to all DR agents at every signalling iteration

is presented next. First, the dual of P ′′EDR is constructed in P ′′EDRd:

P ′′EDRd : maxν

minx∈4

L(x, ν)

where L(x, ν) =m

2

∑i∈M

x2iAii + ν(SDR −m

∑i∈M

Mixi)

where ν ∈ R is a single-dimensional Lagrangian variable associated with the global coupling constraint

(C1)EDR. As the Slater’s condition is satisfied due to Assumption 8, the optimal solution of the dual

will be the same as the primal problem. x∗ can be computed directly by the EPU from the primal

formulation in P ′′EDR. This can then be used to solve for ν∗ in P ′′EDRd. The gradient of L(x, ν∗) when

x is not the optimal strategy distribution in the system is:

Fi(x) =∂L(x, ν∗)

∂xi= m(xiAii − ν∗Mi) i ∈M (3.2)

The signal transmitted by the EPU at every signalling iteration is selected to be F (x) = [Fi(x) . . . Fn(x)]T

which is computed using the current strategy distribution x in the system. This aggregate measure x

can be obtained from data concentrators (i.e. Assumption 1). The gradient F (x) can be interpreted

as the potential of the system. The DR agents that perform local strategy revisions use these signals

to react in a manner that reduces the potential of the system and thereby moving the system towards

the optimal strategy distribution. Moreover, it is important to note that the simplex condition on x is

automatically satisfied by the distributed solution model as all participating DR agents make a strategy

selection from one of the n available strategies.

A diagram illustrating this is presented in Fig. 3.7. In this example, there are three strategies (i.e.

|M| = 3) and therefore the simplex4 representing the values taken by x is a plane resembling a triangle.

Level sets associated with L(x, ν∗) are included in this diagram. The optimal solution x∗ lies at the

centre of the level sets and this represents the system state that will result in the lowest cost for the

EPU. Suppose that at a particular time instance, the system state is located at the point marked by

the black tangent line. The gradient at this point is denoted by the purple arrow. This is computed

by the EPU using Eq. 3.2. The DR agents must react so that the system state moves in a direction

that reduces the system potential (i.e. in the opposite direction of the gradient) towards the optimal

equilibrium as indicated by the orange arrow. With these strategy revisions that reduce the potential of

the system while accounting for the local operating conditions, the equilibrium distribution x∗ is in fact

the same as the optimal solution of the original formulation in PEDR as long as m→∞ and Assumption

8 holds. This property is demonstrated both theoretically and practically later in this chapter. It is

important to note that x∗ is unique, however, the optimal solution prd∗ that translates to this x∗ is not

unique. This is mainly due to the fact that the objective function fo(pr) is convex (not strictly convex).

Moreover, there are many non-unique combinations of prd that can be taken by individual consumers

that can result in x∗.


Figure 3.7: Simplex and system potential.

3.3.3 A Game Theoretic Perspective

As per the transformations applied to PEDR detailed in the above, the state of the system is captured

by x, the strategy set available to DR agents is M and the cost assigned to each strategy i ∈ M is

Fi(x). These three elements completely define a game G(F, x,M). The players in the game constitute

of all DR agents participating in the DR program. Due to Assumption 7, as a large number of players

equipped with discrete strategy set are participating in the system, G defines a population game [96].

As the EPU exercises control over the cost Fi(x) of each strategy Mi, it dictates the behaviour of DR

agents in the system indirectly through these cost signals. Due to the specific structure of F (x) defined

in Eq. 3.2, G is also a potential game as the following externality symmetry is satisfied by the strategy

costs [96]:∂Fi∂xj

=∂Fj∂xi

, ∀ i, j ∈M (3.3)

The above relation holds for the F (x) defined in Eq. 3.2 for G as:

∂Fi∂xj

= 0,∂Fj∂xi

= 0 ∀ i, j ∈M, i 6= j

and thereby satisfying the externality symmetry conditions in Eq. 3.3. Players adjust individual strate-

gies by factoring into their decision-making process the F (x) transmitted by the EPU. When the players

reach a point where any other strategy revisions incur higher costs, the system has reached Nash equi-

librium (NE) defined as [97]:

xNE = {x ∈ 4|xi > 0→ Fj(x) ≥ Fi(x); ∀ i, j ∈M}

It can be shown that xNE for G is unique and exactly the same as x∗ which is the optimal solution of

P ′′EDR as detailed next. x∗ is a unique and optimal solution as P ′′EDR is a strictly convex optimization


problem. KKT conditions associated with P ′′EDRd

are:

L(x, µ, λ) = fd(x) + µ(1−n∑i=1

xi)−n∑i=1

λixi

(1) Stationarity:∂L

∂xi= 0; Fi = µ∗ + λ∗i ∀ i ∈M

(2) Primal Feasibility:

n∑i=1

x∗i = 1, x∗i ≥ 0 ∀ i ∈M

(3) Dual Feasibility: λ∗i ≥ 0 ∀ i ∈M

(4) Complementary Slackness: λ∗i x∗i = 0 ∀ i ∈M

where fd(x) = L(x, ν∗). It can be shown that any x∗ satisfying the above KKT conditions also satisfies

the necessary and sufficient conditions required for an NE. For all strategies in use (i.e. x∗i > 0), condition

4 requires that the corresponding λ∗i = 0. This results in condition 1 reducing to Fi = µ∗ where i ∈ Mare all strategies in use (i.e. x∗i > 0). This implies that the cost of all of these strategies in use are the

same and is µ∗. For all other strategies j not in use, from condition 1 it is evident that Fj = µ∗+λ∗j and

it can be deduced that µ∗ + λ∗i ≥ µ∗ as λ∗i ≥ 0 due to condition 3. As the cost of these strategies are

higher, these are not in use. This shows that the costs incurred by the incumbent strategies are indeed

minimal and the same at optimality. Hence, the unique x∗ satisfying the KKT conditions is also the NE

of the game G. This establishes an equivalence between the DR optimization formulation and the game

theoretic equilibrium.

3.3.4 Resilience of EPU Signals

As DR agents rely upon cyber signals to make local demand curtailments, irrational behaviour by a

subset ε of these agents should be implicitly sensed by other unaffected agents via the EPU signals and

these will then react in a manner that returns the system to the optimal equilibrium state. The system

state xESS exhibiting this robustness is referred to as the evolutionary stable state (ESS) in population

games. It is next shown that xESS is in fact the same as xNE and therefore equal to x∗. This implies

that the optimal equilibrium in the system is robust to perturbations. As the game G has a strictly

convex potential function, it satisfies the relation [81]:

(y − x)′(F (y)− F (x)) > 0 ∀ x 6= y

which is a condition necessary for a strictly stable game. For x∗ to be an ESS, it must satisfy the

following conditions [98]:

(y − x∗)′F (x∗) ≥ 0 and if (y − x∗)′F (x∗) = 0 then (y − x∗)′F (y) > 0

The first condition dictates that x∗ is an NE according to an alternative definition of NE in reference [97].

It has been demonstrated in the previous section that this is indeed the case. As G is a strictly stable

game, the second condition is also naturally satisfied when (y − x∗)′F (x∗) = 0.


3.3.5 Strategy Revisions by DR Agents

So far, the manner in which the EPU computes the signals that it periodically broadcasts is discussed.

In addition, the equivalence between the optimization and game theoretic framework has been presented

along with interesting insights into the equilibrium properties of the system. All of these static properties

will hold only when DR agents execute strategy revisions that reduce the potential of the DR system.

The strategy revisions made by each DR agent is dictated by the revision protocol ρi,j(x) which is a

function of the most recently received strategy cost F (x). ρi,j(x) defines the probability at which a DR

agent will switch from strategy i to strategy j where i, j ∈M. Prior to making the strategy switch, the

DR agent will ensure that local operating conditions such as consumer comfort budget and appliance

operating ranges are heeded. If these conditions are not satisfied, then the DR agent will not make a

strategy revision. This process ensures that the feasibility set defined in the optimization problem PEDRis always maintained by the distributive revisions.

Table 3.3 summarizes the strategy revisions made by a DR agent according to the afore-mentioned

procedure. In this algorithm, at the beginning of a day, the DR agent d initializes Ed to Edbudget. li is

the sum of reducible power consumption by active appliances that have been given permission by the

consumer for demand curtailment. Each DR agent randomly selects a strategy from y at the beginning

of the first signalling. Then in subsequent signalling iterations, DR agent selects an exponentially

distributed random time τi to revise its current strategy. When the revision time arrives, the DR

agent updates Ed and pd accordingly and proceeds to revise its current strategy using the latest signal

broadcast by the EPU. This is repeated until the current strategy cannot be further revised without

incurring more cost and/or the depletion of comfort budget. The EPU stops broadcasting the cost

signals when the optimal x∗ is achieved by distributive strategy selections in the system. The EPU will

be able to infer this via Assumption 1.

Distributed Algorithm for DR Agent i

Initialization:

• Optimization interval: ti ← 1; Next interval start time: ts ← ti

• Current time: tcurr ← 0, Ei ← Ebudgeti , tcong ← ti

• Current strategy used: sc ← y(1)

Algorithm (during congestion):

1. Compute τi using exponential distribution with rate 1. Settnext ← tcurr + τi which is the next strategy revision time

2. While tcurr < tnext and tcurr < ts:

• Set tcurr to the current internal clock time

3. Update:

• Ei ← Ei − sc min{τi, tcong}• If tcong < τi: ts ← ts + ti, tcong ← ti, τi ← 0

• tcong ← tcong − τi• pi ← Ei/tcong

Use the latest F broadcast by the EPU to select strategy scaccording to conditional switch rate defined by Eq. 3.4

4. sc ← {max{y}|y ≤ min{sc, pi, li}}5. Go to Step 1

Table 3.3: Distributed load curtailment via population game theory.


The specific revision protocol considered in this algorithm leverages on pairwise comparison (PC)

and defined as follows:

ρPCi,j (x) = k[Fi(x)− Fj(x)]+ (3.4)

where k is a constant that normalizes ρPCi,j . During a revision, the DR agent will decide whether or not

to make a switch based on Eq. 3.4. If the cost of the current strategy is greater than the alternative

strategy, then the switching probability from strategy i to strategy j is positive. Otherwise, this is 0.

This switching protocol induces a dynamic in the system state x which is defined as follows:

xi =

n∑j=1

xjρj,i(x)− xin∑j=1

ρi,j(x) (3.5)

where the first term reflects the rate at which agents are switching into strategy i and the second term

reflects the rate at which the agents are switching out of strategy i. As ρ is a probabilistic measure, it

can be expected that the dynamics x of the system will exhibit some stochasticity. However, as there

is a large number of agents in the system, the strong law of large numbers (SLLN) will take effect and

eliminate stochastic effects. For this reason, Eq. 3.5 is referred to as the mean dynamics. This process

is also referred to as evolutionary game theory (EGT) as the system state x evolves due to changes in

F (x) as dictated by the EPU and local strategy revisions. The specific system evolution resulting from

ρPC is referred to as the Smith dynamic [96]:

xi = k

n∑j=1

xj [Fj(x)− Fi(x)]+ − kxin∑j=1

[Fi(x)− Fj(x)]+ (3.6)

The equilibrium of this dynamic should coincide with x∗ and this is discussed in the next section.


The characteristics of the dynamic induced by DR agent strategy revisions listed in Eq. 3.6 determine

whether or not the system converges to the optimal strategy distribution x∗ and remains at that equi-

librium point. This is illustrated in Fig. 3.8. In this particular example, there are three strategies

available to agents. With the distributed strategy revisions, the agents should reach a stable equilibrium

which is also the optimal strategy distribution x∗. State dynamics that exhibit limit cycle behaviour

or consist of multiple equilibrium points are not desirable. With limit cycles, the system will not settle

and will keep on oscillating. With multiple equilibria, the system can converge to a point that is not

x∗. In order to determine whether the Smith dynamic results in desirable convergence characteristics,

concepts associated with negative correlation and (NC) nash stationarity (NS) which are defined below

are evoked [99].

NC: VF (x) 6= 0 implies that VF (x)′F (x) < 0

NS: VF (x) = 0 if and only if x ∈ NE(F )

where VF (x) is the system dynamic (i.e. right hand side of Eq. 3.6). When the first condition is satisfied,

the system behaviour is such that the growth rate of the population state is negatively correlated with the

corresponding system cost. Due to the potential game setup of G, the NC condition is naturally satisfied


Figure 3.8: Illustration of system dynamics.

for the dynamics induced by the proposed algorithm [96]. When the system exhibits no limit cycles and

converges to equilibrium in a guaranteed manner, the following Lyapunov condition is satisfied [100]:

d

dtL(x) ≤ 0,

∂L(x)

∂x

∂x

∂t≤ 0, F (x)x ≤ 0

where L(x) is a Lyapunov function. Setting L(x) to fd(x) (this is the objective of P ′′EDRd

in which

ν = ν∗) and applying this to the above Lyapunov condition results in:

d

dtL(x) =

∂L(x)

∂x

dx

dt= F (x)x = VF (x)′F (x)

The last term is exactly the NC condition. Hence, as the Lyapunov condition holds for this system, the

state dynamic converges to the equilibrium with no limit cycles. The equilibrium of the dynamic defined

in Eq. 3.6 is in fact xNE as VF (x)′F (x) = 0 only when x = xNE and thereby satisfying NS. This implies

that the equilibrium of the dynamic is also the optimal solution x∗. Moreover, since xESS is the same

as x∗, the equilibrium of the distributed revisions is robust to perturbations.


In this section, the theoretical results presented in the discussion above are ascertained via numerical

studies. Appliance penetration and usage parameters defined in Sec. 3.1.3 are leveraged to simulate

the load patterns of 1000 residential homes (i.e. m = 1000). Demand curtailment in each home can be

selected from the set M = {0.00001 0.1 1} kW. The cost aii associated with load curtailment i ∈ M is

set to a random strictly positive value increasing with i. In order to capture various efficient modes that

are made available in appliances by manufacturers, during conservation/demand curtailment periods,

appliances with resistive elements are randomly configured to reduce power consumption up to 90%.

Power consumption by cyclic inductive appliances such as fridges and freezers are not adjusted as the

temperatures maintained by these are critical. During the summer, air conditioners configured to operate


in a flexible manner are completed turned off when required until the local comfort budget is exhausted.

Convergence and Initial States

In the first set of results depicted in Fig. 3.9, the convergence properties of proposed algorithm applied

to a system of 1000 participants are assessed over one DR cycle (optimization interval of 1 minute).

All possible values that can be taken by the three dimensional state x are contained within the simplex

0

0.5

1

0

0.5

1

0

0.2

0.4

0.6

0.8

1

y1=0.00001 kW

Solution Trajectories for PC Revision Protocol

y2=0.1 kW

y3=

1 k

W

Figure 3.9: Pairwise comparison revisions.

depicted by the triangular plane in Fig. 3.9. The costs associated with these states are reflected by the

level sets where the higher costs are associated with warmer colors. The optimal state x∗ lies in the middle

of these level sets. State trajectories ensuing from distributed revisions and iterative signal broadcasts

that begin at randomly selected initial values always converge to the optimal strategy distribution x∗

and it is clear that there are no limit cycles. This reaffirms the theoretical convergence insights gained

from the NC and NS conditions.

Impact of Population Size

An important requirement for obtaining the optimal solution of PEDR via distributive revisions enabled

by the population game construction is that the number of participants in the system must be very

large (Assumption 7). Although this is naturally satisfied by the large number of residential consumers

present in the system, it is possible that restrictions stemming from the exhaustion of comfort budgets or

local appliance operation conditions may limit the total number of participants in the system. In these

cases, the SLLN will not hold and stochastic effects will come into play. In order to illustrate the effects

of this, Fig. 3.10a presents the impact on the aggregate demand curtailment ratio when the number of

participants is varied from 10 agents to 1000 agents. This ratio r =∑i∈D p

rd/S

DDR is varied according to

the number of agents |D| present in the system by adjusting the setpoint SDDR. It is clear that when the

number of agents in the system is low (i.e. 10 agents), there are noticeable oscillatory effects whereas

with larger number of participants, the system smoothly converges to equilibrium. This is expected as

a large system serves to cancel the stochastic effects due to SLLN. In general, the system performs well


when the number of participating agents is large as this renders x to be a continuous variable. The exact

precision of x is dictated by the number of agents in the system. Moreover, every agent contributes to an

incremental change in the system potential. Fig. 3.10b illustrates the overall demand reduction achieved

with respect to Sbudget = 600kW when the proportion of DR agents available is varied in increments of

0.2. It is clear that greater the availability of agents is, the closer will the system be towards achieving

the EPU’s demand reduction goals.

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

1.2

1.4

Time (Seconds)

Ratio

Impact of Number of Agents on Performance

1000 Agents

100 Agents

10 Agents

(a) Aggregate demand curtailment in a limited system.

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1Impact of Availability of DR Agents

Proportion of DR Agents Available

Pro

port

ion o

f A

imed R

eduction A

chie

ved

(b) Availability of DR agents.

Figure 3.10: Performance of EGT based DR scheme under system limitations.

Resilience to Perturbations

Next, the robustness of the proposed algorithm is evaluated in Fig. 3.11 over a single DR cycle. It is

evident from this figure that the system converges to an equilibrium within the first 5 seconds of the

optimization interval. Then, after 25 seconds into the DR cycle, 20% of the agents are forced to choose

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Time (seconds)

x

Proportion of Strategies in Population

y=0kW

y=0.1kW

y=1kW

Figure 3.11: System subjected to perturbation.


a higher load curtailment by a malicious cyber attacker. This act will exhaust comfort budgets faster

and entail in more costs for the EPU. In the results illustrated in Fig. 3.11, it is clear that even after

this disturbance, the system is able to recover soon as other DR agents are able to implicitly infer this

perturbation via the EPU signals and adapt local demand curtailment to offset this perturbation. The

system returns to the equilibrium established prior to the attack soon thereafter. This adaptive resilience

of the system reinforces the theoretical assertion that the optimal equilibrium is ESS and therefore robust

to perturbations such as this.

Aggregate Demands over a Day

Next, the aggregate demands of consumers over a day with and without the proposed DR algorithm in

place is presented next in Fig. 3.12. In Fig. 3.12a, the EPU attempts to maintain aggregate demands

0 2000 4000 6000 8000 10000 12000 140000

50

100

150

200

250

300

Time (Hours)

Aggre

gate

Dem

and (

kW

)

Aggregate Demand with and without DR

No DR

Setpoint

With DR

8 12 24

(a) Setpoint is 200 kW.

0 2000 4000 6000 8000 10000 12000 140000

50

100

150

200

250

300

350Aggregate Demand with and without DR

Time (hours)

Aggre

gate

Dem

and (

kW

)

No DR

Setpoint

With DR

8 12 24

(b) Setpoint is 100 kW.

Figure 3.12: Ability of the EGT based DR algorithm to maintain demands around a setpoint.

around 225 kW. It can be observed that with the proposed algorithm in place, the aggregate demands

are successfully maintained around this setpoint over the whole day. In Fig. 3.12b, the EPU attempts

to maintain aggregate demands at a much lower setpoint of 125 kW. It is evident that the system is

unable to sustain demands around this value throughout the day as observed in the previous proposal

involving the water-filling method. This can be mainly attributed to the rapid exhaustion of the comfort

budgets set by the consumer. Hence, it is important for the EPU to strike a balance between demand

curtailment goals and consumer tolerance levels.

Comparison with the State-of-the-Art

The proposed DR algorithm is compared with a sub-gradient (SG) method similar to that proposed in

reference [34] over one DR cycle. This SG method, based on dual decomposition, consists of parameters

such as α that must be tuned with respect to the current configuration of the system. For the particular

optimization interval considered in Fig. 3.13, α = 0.0024 results in extensive oscillations that can cause

damage to system components whereas α = 0.0001 results in very slow convergence. On the other hand,

PC revisions require no such tuning and exhibits rapid convergence to equilibrium.


0 10 20 30 400

200

400

600

800

1000Comparison between PC and Subgradient Methods

Iterations

Ag

gre

ga

te P

ow

er

(kW

)

PC

Original Demand without DR

SG: α = 0.0001

SG: α = 0.0024

Figure 3.13: Comparison between PC protocol and SG method.

A comparison with proposals in the existing literature that leverage population game theory for DR

is presented next. Work in [32, 39] propose DR schemes that utilize population games for distributed

appliance re-scheduling over an extended period of time (i.e. not real-time). Reference [32] leverages

a replicator dynamic (RD) based revision protocol for appliance scheduling throughout the day. RD is

based on imitative revisions that can result in non-unique equilibrium points. Work in [39], on the other

hand, proposes day-ahead appliance scheduling via best-response dynamic. According to [96], there are

two main issues with this dynamic. First, its convergence characteristics cannot be established using

traditional analysis (similar to that presented in Section 3.3.6) as the trajectories consist of differential

inclusions. Secondly, as the state trajectories are not unique, the system state can cycle in and out

of NE. As robustness and stability are important considerations, best-response dynamic is not ideally

suited for real-time DR.

To draw more general conclusions about the proposed DR strategy with respect to the state-of-the-

art, comparisons are made against four performance metrics as summarized in Table 3.4. DR strategies

in the literature are divided into three general classes and these are centralized real-time, offline and

decentralized real-time. These classes differ from one another with respect to the time horizons over

which computations occur (i.e. hourly or day ahead intervals) and how participating DR entities are

coordinated (i.e. by a central entity or in a distributed manner).

Proposed Centralized Offline DecentralizedDistributed Real-time Real-time

Comm Cost O(1) O(n) O(1) O(n)(n DR Agents)

Forecast No Yes Yes NoError

Solution Yes No Yes YesOptimalityResilience Yes No Yes No

Table 3.4: Comparison of DR methods.


First, communication costs of the DR schemes belonging to these classes are evaluated. Due to the

cyber-physical vision of the smart grid (Assumptions 1-5), EPU and DR agents are able to communicate

with one another to enable efficient DR. Communication overhead is directly proportional to the amount

of information exchanged. In Table 3.4, the O−notation (Big-Oh) is used to capture the limiting

communication costs in terms of n which represents the total number of DR agents that are actively

participating in load curtailment [101]. In the proposed DR technique, communication overhead is

constant (i.e. O(1)) and independent of the size of the DR problem as the EPU broadcasts general costs

F (x) which are the same for all DR agents at every signalling iteration. Hence, the EPU does not forge

individual downlink connections with every DR agent. Moreover, it uses the wireless channel which is

an inherently broadcast medium (Assumption 2) [61]. Thus, a single general broadcast of the cost to

all DR agents at every signalling iteration suffices. Additionally, no information exchanges are required

between the DR agents in the proposed DR technique.

Centralized real-time techniques such as reference [31] utilize daily updates from participating DR

agents about local states (e.g. temperature, appliance operation, etc.) for computations so that fluctu-

ations in demands can be captured accurately. Also, as the central coordinating entity transmits signals

containing information about how to adjust various local configurations of appliances (e.g. temperature

setpoint) for every DR participant, these are specifically tailored for each DR agent. Hence, as com-

munication occurs between the EPU and the DR agents in both the uplink and downlink directions,

at least 2n information exchanges are necessary. This results in the communication cost amounting to

O(n). Offline strategies, such as the day-ahead pricing schemes referenced in [19], use only forecast

models (i.e. no need for information from consumers) to compute the price of electricity. This price is

common to all participating consumers and therefore can be broadcast. Hence, the communication cost

is constant O(1) for these techniques. Finally, decentralized real-time schemes such as the consensus-

based strategy proposed in reference [33] require information exchanges between participating agents,

the communication complexity is listed to be O(n).

Error from forecast is the second performance metric considered in Table 3.4. This error is non-

existent in the proposed and decentralized real-time schemes as prediction models are not employed.

Centralized real-time schemes are typically conducted over hourly intervals and offline schemes are

based on day-ahead forecasts which are associated with prediction errors [102].

Solution optimality is the third performance metric taken under consideration. For the proposed

DR method, this is guaranteed as demonstrated earlier via theoretical proofs and simulations. This

is also the case with decentralized real-time solutions as detailed in reference [33]. Centralized real-

time solutions may not be optimal as overcoming computational overhead which increases exponentially

with the number of participants due to discrete demand curtailment possibilities can be difficult with

time limitations. Heuristics are typically employed to resolve such issues and these have no guarantee on

solution convergence or optimality. On the other hand, as offline schemes typically use facilities equipped

with powerful computational resources with no time deadlines, optimality will be feasible.

Resilience is inherent in the proposed DR strategy as demonstrated via theoretical ESS and simulation

studies. Decentralized systems are vulnerable to cyber attacks as false information can be propagated

by compromised agents to other agents. This can lead to incorrect demand curtailment decisions.

Centralized schemes that depend on bi-directional information transfer are subject to single point of

failure attacks (attack on the central coordinating entity) or false information attacks on measurement

data sent by consumers. In the proposed DR strategy, as the EPU is the only entity initiating all


communications, stringent security mechanisms can be applied that can fortify the EPU from single

point of failure attacks. This is more economical than securing every single participating DR agent.

Offline schemes, on the other hand, depend on forecast models for the computation of pricing signals

and therefore are not dependent on feedback from participants. As pricing information can be made

available from redundant sources, issues stemming from single point of failure can be averted as well.

3.4 Remarks

In this chapter, two DR proposals have been presented. The first is based on water-filling in convex

optimization with a limiting assumption made with regards to the load curtailment values. The second

proposal takes into account discrete demand curtailment levels and utilizes population game theory for

DR. Both proposals are highly scalable and suited for real-time DR in the residential sector. These dis-

tributed solutions abstract local operating conditions from the central coordinating entity and thereby

redirecting simpler computations to local DR agents. Demand curtailment in both of these proposals

factor in consumer comfort requirements. Moreover, robustness and resilience of the system to perturba-

tions such as cyber attacks when the proposed DR algorithm is in place are also demonstrated. The DR

agents are able to adaptively react to various conditions in the system just based on the general signals

transmitted by the EPU. Thus, as demonstrated by the two proposals detailed in this chapter, consumer

flexibility can indeed be leveraged to achieve various system goals that include reducing dependence

on peak generation sources and/or coordinating loads to prevent stress on the grid during blackouts.

These proposals can be easily extended to include deferment of appliance operation to a later time. This

can be achieved by introducing an additional metric that accounts for consumer tolerance to appliance

operation postponement. In order to prevent payback effects from all appliances operating at the same

time, random polling can be introduced prior to initiating the operation of these deferred appliances.

This will be considered in future work.

Chapter 4

Distributed Generation Dispatch

Renewable DGs are sustainable energy sources that are attractive alternatives to traditional synchronous

generation systems as commonly deployed fossil-fuel based plants are associated with high carbon foot-

prints. In addition to being clean energy sources, DGs also aid with the reduction of transmission line

losses as these are typically deployed at close proximity to consumers. This results in an efficient transfer

of power. However, DGs are intermittent energy sources with highly fluctuating generation capacities.

As generation forecast models for these are associated with significant error margins, day-ahead economic

dispatch algorithms used to coordinate traditional synchronous generation plants cannot be directly ap-

plied to manage these. Thus, DGs typically operate at MPP tracking mode in which maximum possible

power available from generation is directly injected into the grid. However, the recent rise of DG pen-

etration in the power grid evident especially in DNs is contributing to line congestion which can lead

to the infringement of physical infrastructure limits. The consequences of this will be irreversible and

expensive damages and outages. Thus, treating DGs as non-dispatchable entities will no longer be viable

in the near future.

In order to overcome these issues, two distributed proposals for enabling real-time DG dispatch are

presented in this chapter. The first proposal solves the classical economical dispatch problem for large-

scale DGs in real-time. Dual decomposition and the sub-gradient method are evoked to decouple the

economic dispatch formulation into a distributed problem which is then solved iteratively. In the second

proposal, a large number of small-scale DGs residing in the DN are coordinated. In addition to balancing

aggregate demand with supply, voltage rise considerations are also taken into account to ensure that

physical network limits are heeded. For this, population game theoretic constructs are leveraged. This

proposal is more generally applicable and practical as the underlying physical grid constraints are taken

into account. Prior to introducing these proposals, the system model, the general distributed solution

structure and assumptions are presented next.

4.1 System Model

The system considered in both proposals include DNs, grid-connected microgrids, and solar and wind

farms as illustrated in Fig. 4.1. This is a cyber-physical system in which every generation source is

equipped with intelligence and the ability to communicate. Moreover, the coordinating entity which can

include the ISO or EPU will have access to real-time measurements made available by data concentrators.

53

Chapter 4. Distributed Generation Dispatch 54

Figure 4.1: System model for economic dispatch.

Both proposals presented in this chapter are distributed and a general overview of the solution structure

is discussed in the following.

4.1.1 Distributed Dispatch

Like the solution model presented in Chapter 3 for DR, in this chapter, economic dispatch is also

based on a distributed architecture. The local operating conditions of the DGs are abstracted from the

coordinating entity (say the EPU). The EPU will have access to the aggregate state of the system which

is used to design general signals that are then broadcast periodically at every signalling iteration to all

participating DGs in the system. These signals are then used by the DGs to make dispatch decisions

based on the feasibility of local operating conditions (e.g. the current generation capacity, voltage rise,

etc.). This decoupling of local conditions from the main dispatch formulation prevents concentrated

computational and communication overheads as the EPU solves a simplified problem based on aggregate

measures and the DGs will not need to transmit local generation conditions to the EPU. Furthermore,

as the optimization intervals are highly granular, significant flux in demand and supply will be captured

by the proposed dispatch techniques with high accuracy. This implies that there is no need to evoke

forecast models that are associated with high error margins. The convergence and scalability properties

of these signalling and actuation decisions depend on the algorithms designed to compute these.

4.1.2 Assumptions

All assumptions made in the system model are summarized here:

1. The EPU can monitor aggregate behaviour of the system via data concentrators;

2. The EPU is equipped with wireless transmitters that can broadcast signals to all participating

DGs;

3. Each DG is a dispatchable cyber-physical entity equipped with communication capability, access

to local state measurements and intelligence;


4. The demand and supply remains constant over the optimization interval;

5. Cost of generation by sustainable DGs is lower than that of purchasing power from the main grid;

6. Cost of dispatch is quadratic;

7. Steady-state properties of the system are considered;

8. This is a grid-connected system with access to negative spinning reserves, synchronous generation

and storage systems;

9. The system consists of a diesel generator (also a DG) sized for the system; and

10. Three phases in the DN are decoupled into identical single phase network.

The first three assumptions align with the cyber-physical vision of the smart grid. Assumption 4 facili-

tates real-time dispatch goals as the optimization interval is set to 10 minutes in the first proposal and

1 minute in the second proposal presented in this chapter. More accurate forecast models can be evoked

over these short intervals by individual generation systems [103]. Assumption 5 encourages the use of

DGs which are sustainable and renewable power sources in lieu of traditional synchronous generation

associated with high carbon footprint and costs. Assumption 6 is typical in dispatch problems [88].

Assumption 7 implies that transients in the grid are negligible. This is an acceptable assumption due to

the grid-connected nature of the system and the existence of spinning reserves in Assumption 8. This

allows for the computation of economic dispatch in steady-state. Assumption 8 ensures that there is

sufficient generation capacity, inertia and reactive power supply in the system to provide uninterrupted

power supply to consumers. This is practical due to distribution system automation already enabled

in the grid via protective relays and volt/var compensation devices. Assumption 9 allows for complete

independence of the system from traditional synchronous generation systems located in the main grid

for real power supply. The final assumption also made in existing literature like reference [74] allows for

the consideration of the radial DN as a balanced single phase network.

4.1.3 Dispatching DGs

Traditionally, DGs such as solar and wind energy sources are considered to be undispatchable as these

have highly intermittent generation capacities influenced by external factors (e.g., cloud cover, wind

speed, etc.). Computing economic dispatch for renewables in typical day-ahead markets by evoking

prediction models is not common practise as forecast errors introduced by these models, especially

across longer prediction horizons, are not negligible [103]. In this chapter, two distributed algorithms

solve economic dispatch over every optimization interval and this interval is also referred to as the

economic dispatch (ED) cycle. Constraints in the problem formulation consisting of generation capacity

limits are updated every ED cycle and this allows the EPU to capture intermittencies in generation at

a high granularity as dictated by the length of the ED cycle. Power dispatch by DGs at prescribed

levels (not MPP) between zero and the maximum available power is supported by recent advances in

inverter design [84] and other alternative mechanisms. For instance, the power dispatched by solar panels

can be controlled by adjusting the angles of photovoltaic arrays [85]. Novel methods to dispatch wind

generation are actively investigated in the existing literature and one such example is demonstrated

in reference [86] which leverages pitch control and static convertor control to enable dispatch of wind


generation. However, if a system operator desires to not waste power generated, then the DGs can be

deployed in tandem with storage systems which are sized for the DGs now commercially available for

both industrial and residential use in accordance to Assumption 8 [10]. Then, the power output levels

of these storage systems can be varied or ‘dispatched’ as necessary. Although the current technologies

that support the dispatch of DGs and storage systems are still at infancy, significant research efforts in

this area both in the academia and commercial markets will certainly expand the number of available

options in the near future [10].

4.1.4 System Model Implementation

Numerical studies consisting of realistic solar and wind generation models are used to ascertain the

practicality of the proposed dispatch algorithms. A sample system is illustrated in Fig. 4.2. These

Figure 4.2: Sample system model for DG dispatch.

DGs range from roof-top solar panels and micro wind turbines to large-scale energy sources available in

solar/wind farms. These reside at close proximity to consumers.

Solar Generation

Power generation capacities available to solar DGs are governed by external environmental factors that

affect solar irradiance such as cloud cover. In order to realistically simulate generation by these DGs,

hourly solar generation data available in reference [104] is first smoothed so that there are no abrupt

generation changes between every one hour interval. Then, random noise is added to the data to model

irregularities in solar irradiance due to differences in cloud cover in the region.

Wind Generation

Wind generation is affected by the wind speed and the specification of various physical attributes of the

wind turbine. Wind speed is commonly modelled using the Weibull probability density function [61].

The parameters of this function are shape and scale factors which are set to be 1.94 and 4.48 [12]. To

simulate wind power generation, the power curve defined as follows is utilized:

Pwind =1

2Aρθv3


where A is the cross-sectional area of the turbine rotors, ρ is the air density, θ is the efficiency of the

turbine and v is the wind speed computed using the Weibull probability density function. The specific

parameters used in the power curve are detailed in numerical studies conducted later in this chapter for

each one of the proposals presented.

Power Demands

Consumer demands are simulated using models and parameters presented in Sec. 3.1.3. Specifics

regarding the number of homes implemented in the simulations will be listed in the numerical studies

conducted for each proposal later in this chapter.

4.2 Distributed Dispatch via Dual Decomposition

The first proposal presented in this chapter is based on the author’s work published in reference [106].

This is a distributed economic dispatch solution based on dual decomposition and the sub-gradient

method which is a classic optimization technique (as presented in reference [107]) applied in the context

of coordinating DGs in real-time in our paper listed in reference [106]. This proposal does not take into

account voltage rise considerations as common in high-level dispatch algorithms in the literature [45,50,

51]. Moreover, the possibility of discrete generation dispatch levels is ignored. However, the presence of

storage devices coupled with DGs that can actuate continuous values of real power, will eliminate the

need for this relaxation. The second proposal presented in this chapter is more practically and generally

applicable as it takes into account the underlying physical grid constraints and the possibility of discrete

dispatch. The main challenge overcome by the current proposal introduced next is the management of

DGs with highly varying generation capacities over short optimization intervals.

4.2.1 Economic Dispatch Formulation

The economic dispatch problem formulation used in the design of the proposed algorithm is listed in

PpDD.

PpDD : minp

C(p) =∑g∈G

2Cgp2g

s.t.∑g∈G

pg −∑d∈D

pd = 0 (C1)dd

0 ≤ pg ≤ cg ∀ g ∈ G (C2)dd

where p = [p1 . . . p|G|] is a vector consisting of power generation values of all generation systems, Cg > 0

is the cost of generation by DG g, (C1)dd is the global coupling constraint balancing aggregate power

demand with generation supply, (C2)dd ensures that the dispatch pg by each generator remains within the

available generation capacity. pd and cg parameters are updated at the beginning of every ED cycle. As

DGs are primarily considered, ramping constraints and specific lower bounds on generation capacity are

not included. Moreover, discrete generation dispatch is also not considered in this problem formulation.

Hence, as the objective and constraints are convex, this is a convex optimization problem. Although

this problem can be solved directly analytically or via commercially available solvers, the main issue

with a central approach is due to the communication overhead incurred when transferring fluctuating


generation capacity information by individual DGs to the EPU at a regular basis. This will result in

extensive communication overhead and resource requirements.

4.2.2 Master and Agent Problems

In order to overcome these issues, a two-step transformation process is applied to PpDD. First, the dual

of PpDD is constructed:

PdDD : maxλi1≥0,λi

2≥0,ν;minpL(λ1, λ2, ν, p)

where L(λ1, λ2, ν, p) =∑g∈G

2Cgp2g −

∑g∈G

λg1pg +∑g∈G

λg2(pg − cg) + ν(∑g∈G

pg −∑d∈D

pd)

where λ1, λ2 and ν are Lagrangian multipliers (i.e. dual variables) associated with the upper and lower

bounds of (C2)dd and the global coupling constraint (C1)dd respectively. The optimal solution of PdDDwill be same as that of PpDD when strong duality holds. This is indeed the case due to Assumption 9.

As PdDD is separable, this is decomposed into the master problem which is solved by the EPU and sub-

problems that are each solved by individual agents residing in DGs. First, local constraints are moved

from PdDD into the sub-problem PgDD that is solved by DG agent g. This eliminates the Lagrangian

multipliers λ1 and λ2 associated with the moved local constraints. Then, all terms in the objective of

PdDD that correspond to generator g are grouped together and these terms form the objective of PgDD.

The resulting sub-problem for DG g is:

PgDD : gg(ν) = minpg

2Cgp2g + pgν

s.t. 0 ≤ pg ≤ cg

where ν is a common constant that is computed and broadcasted to all DG agents by the EPU. Solving

PgDD will be straight-forward for each DG agent as the variable space is one-dimensional (i.e. pg) and

this is a convex problem that can be directly solved by a quadratic solver. The slack variable ν is key to

ensuring that there is a balance between overall demand and supply in the system. To compute ν, the

EPU will need to solve the master problem PmDD which is obtained from PdDD:

PmDD : maxν

M(ν) =∑g∈G

gg(ν)− v∑d∈D

pd

gg(ν) is the optimal solution of PgDD for fixed ν. This problem, however, cannot be solved directly by

the EPU for the following reasons. First, gg(ν) is the optimal value of individual PgDD and these sub-

problems are dependent on local operating conditions. Although M(ν) is concave, it is not differentiable

due to the local constraints in each gg(ν). Moreover, as the EPU is abstracted from the local generation

constraints, it cannot directly compute gg(ν). In order to overcome these issues, the sub-gradient method

is used by the EPU to iteratively compute ν as follows:

νk+1 = νk + αk+1q(νk) (4.1)

where k is the current updating/signalling iteration, q(νk) is the sub-gradient of M(ν) and αk is the

step-size at iteration k. The sub-gradient q(νk) of a non-differentiable function is not unique but must


satisfy the following relation [107]:

M(vk+1) ≤M(vk) + q(vk)(vk+1 − vk) (4.2)

One possibility of such a sub-gradient is:

q(νk) =∑g∈G

p∗g(νk)−

∑d∈D

pd (4.3)

where the first term reflects an aggregation of the power dispatch decisions made by individual DGs

based on the νk last broadcasted by the EPU which is used to solve PgDD. This is indeed a sub-gradient

as Eq. 4.3 satisfies Eq. 4.2 as follows:

M(vk+1) = infp∈X

(∑g∈G

2Cgp2g + νk+1(

∑g∈G

pg −∑d∈D

pd))

≤∑g∈G

2Cgp∗2g (νk) + νk+1(

∑g∈G

p∗g(νk)−

∑d∈D

pd)

=∑g∈G

2Cgp∗2g (νk) + νk(

∑g∈G

p∗g(νk)−

∑d∈D

pd) + (νk+1 − νk)(∑g∈G

p∗g(νk)−

∑d∈D

pd)

= M(νk) + (νk+1 − νk)(∑d∈D

p∗d(νk)−

∑d∈D

pd)

where X is the feasible set representing values that can be taken by p. The sub-gradient q(νk) can be

easily measured by the EPU by subtracting overall demands from current aggregate generation (which

are optimal dispatch decisions made by the DG agents based on νk that was last broadcast by the EPU)

at the kth iteration. Once equipped with the current value of the sub-gradient, to compute the next

update of νk which is νk+1, the EPU will need to define the step-size αk+1. Three step-size methods

are considered and these are defined as follows [107,108].

Constant step-size

With constant step-size, αk is fixed to the constant c > 0. Hence, αk will not change across the iterations.

Square-Summable-but-not-Summable step-size

In this specific method of assigning values to αk, the infinite summation of the square of αk should

be finite (i.e.∑∞k=1(αk)2 < ∞). However, this is not the case for the infinite summation of αk (i.e.∑∞

k=1 αk =∞). One possiblity of αk that satisfies this is αk = c/(d+ k) where c > 0, d ≥ 0.

Dynamic step-size

The main idea behind the dynamic step-size method is that νk+1 must be closer to the optimal ν∗ than

νk. Thus, substituting Equation 4.1 into ||v∗ − vk+1||22 < ||v∗ − vk||22 and rearranging results in:

αk = β(Mk −M(vk))/||q(vk)||22 (4.4)


where 0 < β < 2 and Mk is an estimate of the optimal value of M . The dynamic step-size algorithm

iteratively updates αk to reach the target level Mk. In the literature (i.e. reference [109]), the target is

set to Mk = maxi∈1...k

M(vi) + εk and εk is updated according to the following:

εk+1 =

ρ εk, if M(vk+1) ≥Mk.

max{γεk, ε}, otherwise.

where ε0 = ε, γ < 1, ρ > 1 are constants. The update algorithm at the kth iteration attempts to improve

the former value by εk. If the result is greater than the current iteration, the value of εk is increased by

a factor of ρ. Otherwise, εk is decreased by a factor of γ as long as εk ≥ ε.

An improvement to this computation is the path-based incremental target level algorithm detailed in

reference [110] that computes Mk by accounting for possible oscillations. An additional variable ωk is

maintained to detect oscillations. This variable is updated as follows:

ωk+1 = ωk + αkmQ

where Q is an upper bound on |q(vk)| and m is a scaling constant. Eq. 4.4 is altered where q(vk) is

replaced with mQ. The target level value is set to Mk = Mreck + εk where Mrec

k and εk are updated as

follows:

if M(vk) ≥Mreck +

εk2

;

εk = εk−1

Mreck = M(vk)

ωk = ωk−1

otherwise if ωk > b;

εk = εk−1

2 ,

Mreck = max

i∈1...kM(vi)

ωk = 0

else ;

εk = εk−1

Mreck = Mrec

k−1

ωk = ωk−1

where b is a path length bound on ωk, and Mk and εk are updated when the function value reaches the

target level. At this point, Mreck is set to the latest value of the function or is updated when the path

length exceeds the upper bound b. This implies that εk is too high and therefore reduced by half. When

these conditions are not met, all variables remain unchanged.

4.2.3 Termination Criterion

In an ED cycle representing one optimization interval, iterative updates using one of the step-size update

methods presented in the above and sub-gradient measurements obtained from data concentrators are

continued until the pre-defined termination criterion is met. This termination criterion for PpDD is set

to q(vk) = 0. This implies that there is a balance between aggregate supply and demand in the system

at which point optimality is attained and the algorithm can terminate.


4.2.4 Summary of Proposed Dispatch Algorithm

Table 4.1 summarizes the proposed distributed dispatch algorithm based on dual decomposition executed

during a single ED cycle (one optimization interval).

Distributed Dispatch for a single ED cycle via DualDecomposition

Initialization: EPU set νk = 0 for k = 1

1. EPU broadcasts νk to all participating DGs

2. DG g computes dispatch strategy pg(νk) by solving Pgdd

3. EPU implicitly infers sub-gradient q(νk) by measuringthe surplus or deficit power supplied by the utility tothe system

4. EPU updates αk+1 and νk+1 according to a step-sizeupdate method and the measured sub-gradient

5. If termination criterion is met (i.e. q(vk) = 0), thealgorithm ends. Else, set k ← k + 1 and go to Step 1

Table 4.1: Distributed dispatch algorithm via dual decomposition.

4.2.5 Convergence Analysis

The iterative updates that are broadcast by the EPU must converge to the optimal dispatch solution.

This convergence is mainly dictated by the step-size update method evoked to compute νk at every

signalling iteration. It is important to note that the smallest difference dk between the optimal value

M∗ of the cost function in PMdd and the current value of M(vk) evaluated up to iteration k is bounded

by:

dk = M∗ − maxi∈1...k

(M(vi) ≤

R2 +∑ki=1(αi)2Q2

2∑ki=1 α

i

)where ||v1 − v∗||22 ≤ R and ||q(vi)||22 ≤ Q. If R < ∞ and Q < ∞, then dk is an upper bound on the

difference between the current and optimal value of the dual cost function. This is mainly due to the

following set of derivations [111]:

||vk+1 − v∗||22 = ||vk + αkq(vk)− v∗||22 =

||vk − v∗||22 + 2αkq(vk)(vk − v∗) + (αk)2||q(vk))||22≤ ||vk − v∗||22 − 2αk(M∗ −M(vk)) + (αk)2||q(vk)||22

where the condition M∗ −M(vk) ≤ q(vk)(v∗ − vk) which is true for concave functions is applied to

obtain the final inequality in the above. Applying this relation recursively and using the facts that

||v∗ − vk+1||22 ≥ 0 and∑ki=1 α

i(M∗ −M(vi)) ≥ (M∗ − maxi∈1...k

M(vi))∑ki=1 α

i will result in dk.

For constant step-size updates, dk reduces to Q2h/2 as k → ∞. Hence, the optimal solution lies

within this threshold of the computed solution. Square-summable-but-not-summable step-size rule

converges to the optimal solution. Since∑∞i=1 α

2i < ∞ and

∑∞i=1 αi = ∞, dk → 0, this step-size

method converges to the optimal solution as k → ∞. A necessary condition for convergence of the

dynamic step-size computation is M∗ < ∞. The sub-gradient considered in this work is bounded as


Q = max{∑d∈D pd,

∑g∈G cg−

∑d∈D pd}. Since Q is bounded, the sub-gradient is also bounded. There-

fore, a necessary condition for dynamic step-size update convergence is met [112].


In order to verify the theoretical conclusions presented earlier, the proposed algorithm is implemented

via MATLAB in simulations conducted using models presented in Sec. 4.1.4. The power rating of

the solar DGs is based on data supplemented for 3.08 kW panels [104]. The wind turbines considered

in these studies are assumed to have 50 kW power rating, 25m height, 13 m rotor diameter and 0.4

efficiency [105]. The generation sources supplement power for 200 homes. The diesel generator is sized

at 4 MW and is treated as a DG. The cost of wind, PV and diesel power generation is 0.135$/kWh, 0.802

$/kWh and 2.08$/kWh, respectively [43]. The cost of power supplied by the main grid is assumed to

be the same as that of diesel generation. Each ED cycle is 10 minutes in length. There are 30 signalling

iterations during this 10 minute period.

Comparison of Step-sizes

In the first set of results, the DG mix included in the simulations are 2 wind turbines, 20 solar panels and

one active diesel generator. The convergence behaviour of the system for the three step-size methods in-

troduced earlier is presented in Fig. 4.3 over three consecutive ED cycles. Various parameters associated

with these step-sizes are altered and the results are plotted for constant, square summable not summable

and dynamic step-size methods in Figs. 4.3a, 4.3b and 4.3c respectively. The solid blue line represents

aggregate demand in the system, all other curves represent aggregate real power dispatched by DGs

based on the proposed dual decomposition method for various combinations of step-size parameters.

From these results, it is evident that these methods converge to optimality. However, fine-tuning of var-

0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12x 10

4 Constant Step Size

Time (hours)

Po

we

r (k

W)

c=1

c=2

c=10

Actual Demand

(a) Constant step-size.

0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12x 10

4Square Summable but not Summable Step Size

Time (hours)

Po

we

r (k

W)

c=10, d=0

c=10, d=2

c=10, d=10

Actual Demand

(b) Square summable not summable.

0 0.1 0.2 0.3 0.4 0.50

5

10

15x 10

4 Dynamic Step Size

Time (hours)

Po

we

r (k

W)

b=6

b=7

b=8

Actual Demand

(c) Dynamic step-size.

Figure 4.3: Power dispatched by DGs for the three types of α computations.

ious parameters associated with these step-size computations is necessary to achieve rapid convergence

to optimality.

Aggregate Dispatch

Next, the ability of the proposed DG dispatch algorithm to distributively meet aggregate demands over

a day while accounting for generation fluctuations is investigated in Fig. 4.4. The dynamic step-size


update method is used in the updates of ν for this set of results. In Fig. 4.4a, the aggregate generation

capacity of the DGs (not including diesel generation) is plotted over a 24 hour period. It is clear that

the generation capacities are significantly fluctuating over the day. In Fig. 4.4b, the aggregate dispatch

of DG generation coordinated by the proposed algorithm is plotted alongside with overall demand in

the system. It is evident that the distributed dispatch of DGs closely matches the aggregate demands

in the system as expected.

0 5 10 15 200

0.5

1

1.5

2

2.5x 10

5

Time (hours)

Pow

er

(kW

)

Dispatch by 2 Wind Turbines and 20 PVs

Power Generated

Actual Demand

(a) DG generation without diesel generator.

0 5 10 15 200

0.5

1

1.5

2

2.5x 10

5

Time (hours)

Pow

er

(kW

)

Dispatch by 2 Wind Turbines, 20 PVs

Power Generated

Actual Demand

(b) DG generation with diesel generator.

Figure 4.4: Real-time dispatch over a day.

Sudden spikes in DG dispatch present in Fig. 4.4b are caused by transitions between one ED cycle

to another. Changes in generation capacities and aggregate power demands in the system can cause

the proposed algorithm to over-compensate and thus lead to the surges observable in Fig. 4.4b. This

is expected especially for the dynamic step-size update method as this method is associated with fast

convergence accompanied by initial oscillatory behaviour as illustrated in Fig. 4.3c.

Impact of Generation Mix

In the final set of results presented in Fig. 4.5, the impact of varying DG generation mix on the overall

power saved (i.e. power demand supplemented by not depending on the main grid or diesel generator)

is investigated. In Fig. 4.5a, various generation mix of solar panels and wind turbines are presented as

tuples in the x-axis. The overall power supplementing potential of these generation mixes are plotted

in this figure. It is clear that as the total number of generation in the DG mix increases, the greater

is the aggregate generation capacity of the system over the day. Next, in Fig. 4.5b, the regression of

power saved over the day for various generation mix combinations is explored. Solar panels, also denoted

as PVs, contribute to the highest generation around noon and wind turbines contribute more over the

evening. It is clear that significant rise in generation is evident during mid-day. Expensive peak power

can be supplemented by these sustainable generation sources. This regression analysis will be useful to

policy-makers when designing the integration of renewable generation into the grid.


(2,20) (2,40) (6,20) (6,40) (10,20) (10,40)0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6x 10

14 Generation Mix Impact

Generation Mix

Overa

ll P

ow

er

Saved (

kW

)

(a) Generation Mix Impact on Power Savings.

0 5 10 15 20 250

0.5

1

1.5

2

2.5x 10

8 Impact of Generation Mix

Time (hours)

Regre

ssio

n o

f P

ow

er

Saved (

kW

)

2WT, 20PV

2WT, 40PV

6WT, 20PV

6WT 40 PV

10WT, 20PV

10WT, 40 PV

(b) Regression of power savings over a day.

Figure 4.5: Power Savings.

4.3 Distributed Dispatch via Population Games

So far, a distributed economic dispatch solution has been proposed for optimally balancing aggregate

load and generation across short optimization intervals. Next, another distributed proposal is presented

which, in addition to being highly scalable with rapid convergence properties, also accounts for discrete

generation and underlying non-convex physical grid constraints. It is important to take into account

physical grid limitations as existing DNs are designed to accommodate power flowing from the main

grid to the consumers and not the other way around [113]. Due to significant penetration of DGs

which are located at close proximity to consumers located within a DN, physical network limitations

must be accounted for while integrating these. Otherwise, expensive infrastructure upgrades will be

necessary to prevent equipment damages. The economic dispatch proposal presented next allows for

safe coordination of large number of DGs present in radial networks such as the DN which are sensitive

to voltage rises resulting from excessive DG generation. Thus, this proposal generalizes the scope of the

dispatch formulation by leveraging on theoretical constructs from convex optimization and population

game theory. Details presented next are based on the author’s work published in references [114,115].

4.3.1 Dispatch Formulation

The problem formulation PEDV considered in this proposal is similar to that presented in PV R from

Chapter 2. This particular proposal is designed for DNs that are grid-connected (Assumption 8) and

focuses on real power dispatch by DGs. Hence, reactive power demands and line losses in the DN are


supplemented for by the main grid.

PEDV : minp

C(p) =∑g∈G

fg(pg)

s.t.∑g∈G

pg −∑d∈D

pd = 0 (C1)EDV

0 ≤ pg ≤ cpg ∀ g ∈ G (C2)EDV

pg ∈M = {M1 . . .Mng} ∀ g ∈ Gd (C3)EDV

|Vmin|2 ≤ VaV ∗a ≤ |Vmax|2 ∀ a ∈ B\0 (C4)EDV

Va = |Vref |∠0 a = 0 (C5)EDV∑g∈Ga

pg −∑d∈Da

pd + i(∑g∈Ga

qg −∑d∈Da

qd) =∑

a↔b∈L

(VaV∗a Y∗a,b − VaV ∗b Y ∗a,b) ∀ a ∈ B\0 (C6)EDV

pm −∑d∈Da

pd + i(qm −∑d∈Da

qd) =∑

a↔b∈L

(VaV∗a Y∗a,b − VaV ∗b Y ∗a,b) a = 0 (C7)EDV

The slack bus is set to be bus 0 and this is used as a point of reference. The substation bus where the

DN connects to the main grid is typically considered to be the slack bus. Voltage at the slack bus is

held to be constant value |Vref | with bus angle that is set to 0 (i.e. (C5)EDV ). (C1)EDV is a global

coupling constraint that enforces balance between overall distributed generation and demands. (C2)EDVto (C7)EDV are individual constraints that are local to every participating DG and buses in the DN.

(C7)EDV consists of pm and qm which are real and reactive power supplied by the main grid to meet

real line losses and reactive power demands in the system. In this formulation, individual DGs are not

modelled separately. These are represented by the main bus in the DN to which these physically connect

to (i.e. (C6)EDV and (C7)EDV ). This is a similar approach taken by other proposals in the literature

like references [74,113].

The main idea behind this proposal is to abstract local operating conditions from the EPU. This

process is illustrated in Fig. 4.6. The EPU will transmit general signals representing the aggregate state

of the power grid and the agents representing the DGs (Assumption 3) will respond to these signals

based on local operating conditions. Hence, the formulation presented in PEDV is decomposed into

master and secondary tiers. The EPU computes signals at the master tier and the DG agents compute

dispatch based on the received signals and local feasibility checks in the secondary tier. It is possible for

every DG to be represented by more than one agent and the reason for this will be discussed later.

In Fig. 4.6, a data concentrator aggregates all of the current strategies Si in use by DG agents and

overall power demands wi by consumers. The aggregated values are then transmitted to the EPU which

then uses these to compute signals that are then broadcasted to the DG agents.

4.3.2 Master Tier

In order to compute the signals broadcast by the EPU, a master problem is constructed by first moving

local constraints (C2)EDV to (C7)EDV in PEDV to the individual dispatch decisions made by the DG

agents. The resulting problem Pm′EDV is still associated with individual power dispatch values that can


Figure 4.6: System model for economic dispatch via master and secondary tiers.

be discrete.

Pm′

EDV : minp

C(p) =∑

pg∈Mg,g∈Gfg(pg)

s.t.∑g∈G

pg −∑d∈D

pd = 0 (C1)EDV

The EPU is only concerned about the aggregate behaviour of the system. A change of variables x ∈4 = {x ∈ Rn|xi ≥ 0,

∑ni=1 xi = 1} is applied to Pm′EDV . There are n components in x which are

associated with n power dispatch levels available to DG agents. These levels or strategies are denoted

as Y = [y1 . . . yn]. The ith component in x represents the fraction or proportion of agents in the system

using strategy yi. It is possible that the generation capacity of a DG is much higher than the largest

strategy level in Y. To prevent wasting the available generation capacity, the number of agents in a DG

is set to be mg = bcg/max(Y)c. Hence, the total number of agents in the system will be m =∑g∈Gmg.

The cost of power dispatch is represented by the function C(p). Since the cost of dispatch is typically

quadratic and higher penalty is levied for greater dispatch, C(p) is set to:

C(p) =

n∑i=1

(∑g∈G

mg∑j=1

Ii(pgj )pgj

)2

Ci

where pgj ∈ Y is the power dispatch selected by the jth agent in DG g, Ii(pgj ) is an indicator function that

returns 1 if pgj = yi and 0 otherwise and Ci > 0 is the cost assigned to power level yi. This cost structure

aims to distribute the power generation across the system so that excessive generation does not occur

in one part of the DN as this can result in over-voltage conditions. With this change of variables, the


resulting optimization problem for the EPU is PmEDV .

PmEDV : minx∈4

f(x) =

n∑i=1

(myixi)2Ci

s.t. m

n∑i=1

xiyi −∑d∈D

pd = 0 (C1)EDV

The dual of PmEDV is PdEDV :

PdEDV : minx∈4

L(x, ν) =

n∑i=1

(myixi)2Ci + ν(m

n∑i=1

xiyi −∑d∈D

pd)

where ν is the Lagrangian variable associated with the global coupling constraint (C1)EDV . The optimal

solution of PmEDV is identical to that of PEDV when m→∞ and the Slater’s condition is satisfied (due

to Assumptions 8 and 9). The EPU can directly compute x∗ and ν∗ as Pm′EDV is a strictly convex

problem with optimization variables x that are continuous as m → ∞ and the optimization variable

space is much smaller n << m. Although the EPU can directly compute x∗, the main difficulty arises

in assigning strategies to DG agents in the system to achieve this system distribution while ensuring

that local feasibility constraints such as generation capacity and bus voltage limits are met. From these

insights, the EPU signals are computed as follows:

Fi(x) = Kmyi(2Cimyixi + ν∗)

where F (x) = [F1(x) . . . Fn(x)]T is the cost of each strategy in Y that are available to the DG agents.

F (x) is in fact the gradient of L(x, ν∗). ν∗ is the optimal solution of PdEDV for fixed x∗ and K is a

constant used for normalizing as presented later on. The system state x, strategy set Y and strategy

cost F (x) define a game G(x, y, F ). Moreover, since a large number of DGs are considered, this is a

population game. The EPU will compute F (x) at every signalling iteration and broadcast this to all the

DG agents. The response by the DG agents is discussed next.

4.3.3 Secondary Tier

Each DG agent selects a random time to revise its current strategy which is selected from the set Y.

Dispatch Revision Protocol

Strategy revision by a DG agent is dictated by the revision protocol ρi,j(x) which is the probability of

switching from strategy yi to yj . The revision protocol defines the probability of a DG agent switching

from one strategy to another. Projection revision protocol listed in reference [96] is considered in this

proposal which is defined as:

ρi,j(F (x), x) =[Fi − Fj ]+

nxi(4.5)

This is a projection of the gradient of L(x, ν∗) onto the simplex 4. General mean dynamics is obtained

by subtracting the rate at which agents leave strategy i from the rate at which agents enter strategy i


as follows:

xi =

n∑j=1

ρj,ixj − xin∑j=1

ρi,j (4.6)

Substituting the projection revision protocol defined in Eq. 4.5 into Eq. 4.6 results in:

xi =1

n

n∑j=1

Fj(x)− Fi(x) (4.7)

For rapid reduction in the potential of the system, the system dynamic of x must evolve in a direction

opposite to this gradient and this is indeed the case with ρi,j .

Local Feasibility Voltage Checks

Prior to making a strategy switch, the DG agent ensures that local feasibility conditions hold. The

method by which this feasibility check is made is discussed next and illustrated in Fig. 4.7. There are

Figure 4.7: Local feasibility checks with voltage rise considerations.

two main feasibility checks that are conducted by the DG agent. First ensures that the DG’s generation

capacity limit is heeded and as this is a local check (i.e. (C2)EDV ). The discrete generation dispatch

level constraints in (C3)EDV are already handled by the discrete homogeneous strategy set Y made

available to each DG agent. The second check is to ensure that the current change in dispatch will not

cause voltage rise in the DN that will result in the violation of bus voltage limits (i.e. (C4)EDV and

(C5)EDV ). The bus voltages are governed by the power balance equations in (C6)EDV and (C7)EDVwhich constitute a set of non-convex quadratic equality constraints.

In order to ascertain that the voltage rise (if any) is within acceptable bounds, two main insights are

leveraged. First is that every DG agent selects a random time to revise its strategy. As the probability


of another agent selecting exactly the same time for revision is very low, it is possible to conclude that

each agent revises its strategy one at a time (i.e. not simultaneously). The second insight is that the DN

is a radial network which has a tree structure. Increase in generation will, therefore, only affect the local

feeder branch in which the revising DG agent resides. Based on these observations, the local feasibility

check conducted by each revising agent is discussed next and summarized in Tables 4.2 and 4.3.

Strategy Selection Process by DG Agent i

Initialization:

• Initialize time: tnext ← 0, t← 0.

• Initiate strategy: sc ← rand(y).

Start Algorithm: (repeat the following):

1. Time for next revision τi is computed via exponentialprobability distribution with rate µ. Next revision willoccur at tnext ← t+ τi.

2. When t > tnext,

• Utilizing the latest F and x broadcast by theEPU, select strategy si according to ρi,j(x).

• Set: sc ← si only if pi − sc + si ≤ ci.– Check if sc is feasible according to Table 4.3.

– If not, set sc ← si−1 and repeat previousstep.

– Else, go to Step (1).

Table 4.2: Strategy revisions by DG agents with voltage rise considerations.

The revising agent will need to ascertain whether the change in power dispatched ∆p = yi − yj

accompanying the strategy switch from yi to yj will have any impact on bus voltages of the local feeder

branch in the DN. Let the bus that the revising agent is currently residing at be denoted as b and the

parent bus as a (i.e. a ↔ b ∈ L and bus a is one line closer to the substation). Agents in bus a can

approximate the power flow pnewa,b in line a↔ b due to this change in dispatch using current power flow

denoted by polda,b obtained from local measurements (Assumption 3) as follows:

pnewa,b ≈ polda,b + ∆p (4.8)

As power loss in the line is not accounted for in this update, this is an approximation that allows

for a conservative estimation of voltage rise and thereby incorporates margins that can accommodate

unexpected transients. This impending change in power ∆p is communicated up the feeder branch until

all ancestors of bus b (up to the EPU/substation) update local power flows according to Eq. 4.8. This

upward propagation is referred to as the backward sweep as illustrated in Fig. 4.7. In the worst case,

the total number of information exchanges that can take place in the backward sweep step is the height

of the feeder branch when the revising agent is physically located in the last level of the feeder.

Then, a forward sweep process as illustrated in Fig. 4.7 is initiated starting with the substation node

(labelled a) and its immediate child (labelled b) that is an ancestor of the bus at which the revising

DG agent is residing. The voltage of the substation node is typically held constant. Voltage at bus b is

updated using the approximated power flow computed previously in the backward sweep step and qa,b

obtained from local measurements. If pa,b > 0, a voltage drop will occur across line a ↔ b and Vb is


updated by solving:

Va(Va − Vb)∗Y ∗a,b = pa,b + jqa,b (4.9)

Otherwise, there is a voltage rise and Vb is updated by solving:

Vb(Vb − Va)∗Y ∗a,b = −(pa,b + jqa,b) (4.10)

As Va is fixed at the substation, Vb is the only unknown variable. After updating the voltage, bus b

checks whether the local voltage constraint met (i.e., constraints (C4)EDV and (C5)EDV in PEDV ).

When this fails, an alarm is broadcast through this feeder to alert the revising agent that the dispatch

change is not feasible. Otherwise, the descendants of bus b repeat this until either an alarm is evoked

or the leafs of the updating feeder branch are reached. In the first case, the revising DG agent will infer

that the feasibility check has failed and in the second case feasibility is confirmed. This algorithm is

summarized in Table 4.3.

Feasibility Check by DG Agent i of Dispatch Change ∆P

• Backward sweep: Let b ← i, where i is the bus in which DGagent i resides in. Repeat the following until parent node isthe substation:

– Set a← parent(b) and send ∆p to node a.

– Update pa,b according to Eq. 4.8. Set b← a.

• Forward sweep: Let a ← substation bus and b ← child(a) inthe updating feeder branch. Repeat until descendent bus is aleaf or alarm activation:

– If pa,b > 0, then solve for Vb using Eq. 4.9 otherwise useEq. 4.10.

– If (C4)EDV or (C5)EDV are violated, activate alarmacross the updating branch

– Otherwise, set a ← b and b ← child(a) and send Va tonode b.

Table 4.3: Voltage feasibility check by DG agents.

Next, the computational complexity of the forward and backward sweep method is analyzed. The

communication complexity of the feasibility check in the worst case is 3hn where h is the height of

the tree representing the DN, the constant 3 denotes three sets of information flow across the updating

branch (i.e. the forward, backward and alarm information exchanges) and n denotes the number of

strategies available for each DG agent. For instance, consider the low-voltage Danish DN consisting of

34 buses supplying power to 75 homes detailed in reference [116] amd depicted in Fig. 4.8 in which,

the height of the deepest feeder tree is 7 and the number of strategies available to agents is n = 3.

As the latency incurred by information exchanges is in the order of microseconds due to the physical

proximity of the nodes, the time required to check voltage constraints will be much less than 1 second.

Also, computational complexity at each agent is constant as the backward sweep involves one addition

and the forward sweep requires the solving of a quadratic equation in the worst case. In general, the

computations performed by each DG agent for the secondary tier involves determining ρi,j(x) from

the signals broadcast by the EPU, estimating local power flows and comparing dispatch strategies (to


ensure that generation capacity limits are heeded). All of these secondary tier computations entail simple

mathematical operations and thus are straightforward.

4.3.4 Equivalence between Game Characterization and Optimization

A population game G is defined by individual active players selecting a strategy from the set Y based

on the allocated cost F (x). x represents the proportion of players in the population selecting strategies

in Y. Suppose, there exists a function f : Rn → R such that:

F (x) = Of(x) ∀ x ∈ 4

then f is defined to be a full potential function for the game G. Moreover, if the following full symmetric

condition is met:∂Fi(x)

∂xj=∂Fj(x)

∂xi∀ x ∈ 4, yi ∈ Y, yj ∈ Y

G is a fully potential game in which revisions made by each player reduce the potential of the system.

Nash equilibrium x∗NE of this system is the same as the optimal solution of the following problem when

m→∞ [96]:

Pp : minf(x)

pgj ∈ y ∀ j = 1 . . .m

xi =1

m

m∑j=1

Iyi(pgj ) ∀ i = 1 . . . n

Due to Assumptions 8 and 9, reactive and real power losses in the lines are considered to be supple-

mented by the main grid where qm =∑d∈D qd−

∑a↔∈E q

la,b−

∑g∈G qg and pm =

∑a↔b∈E p

la,b thereby

simplifying PV R to PEDV reiterated in the following for convenience of reference:

PEDV : minp

C(p) =∑g∈G

fg(pg)

s.t.∑g∈G

pg −∑d∈D

pd = 0 (C1)EDV

0 ≤ pg ≤ cpg ∀ g ∈ G (C2)EDV

pg ∈M = {M1 . . .Mng} ∀ g ∈ Gd (C3)EDV

|Vmin|2 ≤ VaV ∗a ≤ |Vmax|2 ∀ a ∈ B\0 (C4)EDV

Va = |Vref |∠0 a = 0 (C5)EDV∑g∈Ga

pg −∑d∈Da

pd + i(∑g∈Ga

qg −∑d∈Da

qd) =∑

a↔b∈L

(VaV∗a Y∗a,b − VaV ∗b Y ∗a,b) ∀ a ∈ B\0 (C6)EDV

pm −∑d∈Da

pd + i(qm −∑d∈Da

qd) =∑

a↔b∈L

(VaV∗a Y∗a,b − VaV ∗b Y ∗a,b) a = 0 (C7)EDV


With the change of variables x and the notion of multiple agents per DG, PEDV can be equivalently

posed as:

P ′EDV : minx

n∑i=1

Ci(m.yi.xi)2 − ν∗(

∑d∈D

pd −mn∑i=1

yi.xi)

m =∑g∈Gbcg/max(Y)c (C1’)EDV

pjg ∈ {yi ∈ y|1Cj (yi) = 1} ∀ j = 1 . . .m (C2’)EDV

xi =1

m

m∑j=1

1yi(pgj ) ∀ i = 1 . . . n (C3’)EDV

where the coupling constraint (C1)EDV from PEDV is moved to the objective and this term will be 0 at

optimality as it is assumed that there exists sufficient generation capacity to meet overall demand. ν∗ is

a constant computed from PdEDV . The fully symmetric condition is satisfied by the objective of P ′EDV .

The indicator function ICj (yi) determines the feasibility of strategy yi based on constraints (C2)EDV

to (C7)EDV for DG agent j. Problems PEDV and P ′EDV are equivalent due to the change of variables

and feasibility checks in (C2’)EDV . m will depend on the availability of generation capacity during the

current optimization interval. When m→∞, x can be considered to be continuous. It is important to

note that C(p) in PEDV results in non-unique optimal solutions with the same optimal value. To see

this, consider∑ni=1 Ci(m.yi.xi)

2 which is equivalent to C(p) in terms of x. xi is computed by counting

the total number of agents in the population using strategy i divided by m. Suppose, x = {0.2, 0.3, 0.5}and m = 1000, then the total number of combinations of pig that result in this x is

(1000200

)(800300

). A subset

of these vast number of combinations result in the feasibility of ICj (yi), given that there exists sufficient

generation capacity in the system. Hence, there are many ways of arriving at x∗. Although x∗ is unique,

PEDV can lead to the same optimal x∗ with non-unique combinations of pig. Thus, with the projection

revision protocol and feasibility checks, DG agents arrive at a configuration not necessarily unique but

results in this x∗. Convergence characteristics and bounds to ensure that the feasibility checks are not

overly conservative are provided in the following.


The distributed strategy selections by the DG agents should be such that the state dynamics converge

to the optimal solution of PEDV without any limit cycles. For this, the dynamic introduced by these

revisions listed in Eq. 4.7 must have a strict Lyapunov function. For rapid convergence to optimality, it

is necessary to demonstrate that the system converges to optimality exponentially fast. Moreover, the

approximations applied in the voltage checks must not be too conservative.

Exponential Convergence due to Projection Revisions

In order to show that the projection dynamic converges exponentially fast to the equilibrium x∗ (i.e.

||xt−x∗|| ≤ e−αt/2||x0−x∗|| for α > 0) where xt is the system state at time t and x0 is the initial system

state, it is necessary to show that there exists a Lyapunov function L(x) such that L(x) ≤ −αL(x) for

α > 0. The candidate Lyapunov function is selected to be L(x) = (f(x) − f(x∗)) + 12 ||x − x

∗||2 where


f(x) is the abbreviated form of the potential function L(x, v∗) and x∗ is the optimal solution of PmEDV .

Prior to discussing the proof, it is important to note that the cost F (x) = [F1(x) . . . Fn(x)]T is strongly

monotone as the following holds:

(F (x)− F (x∗))T (x− x∗) ≥ m(min(Ciyi)||x− x∗||2)

Let µ = m(min(Ciyi)) and it is evident that µ > 0 as Ci, yi and m are strictly positive. Moreover, as

the function f(x) is convex, the following holds as well:

(F (x)− F (x∗))T (x− x∗) ≥ f(x)− f(x∗)

Equipped with these properties, the following is a proof of exponential convergence of the projection

dynamic:

d

dt(L(x)) = OL(x)T x = (F (x) + (x− x∗))T x

= (F (x) + (x− x∗))T (1

n(

n∑j=1

Fj(x))1− F (x))

= −||F (x)− (1

n

n∑j=1

Fj(x))1||2 − (F (x)− F (x∗))T (x− x∗)

≤ −(F (x)− F (x∗))T (x− x∗)

where 1 is an n-dimensional vector of ones. The first term in the third line of the above proof is ≤ 0

and removing it has resulted in the last inequality. The fact that F (x∗) = 0 is also used to obtain the

third line. Combining the strong monotone condition of F (x) and the convexity of f(x), the following

inequality results:

(F (x)− F (x∗))T (x− x∗) ≥ 1

2(f(x)− f(x∗)) +

µ

2||x− x∗||2

≥ min(1

2, µ)((f(x)− f(x∗)) +

1

2||x− x∗||2)

= min(1

2, µ)L(x)

Hence substituting the above inequality into the last line of the initial proof, we are able to show that

L(x) ≤ −min(1

2, µ)L(x)

where α = min( 12 , µ) > 0 and thus the projection dynamic will converge to x∗ exponentially fast.

Bound on Voltage Rise

Here, a proof is presented on the upper bound of voltage increase due to increase in power dispatch by

∆P at bus b. Suppose that bus a is the direct ancestor of bus b (i.e. a ↔ b ∈ E). The relationship

between bus voltages and power flow across the lines is as follows:

Vb(Vb − Va) = (pb,a + iqb,a + ∆P )za,b


Vb and Va are the voltages resulting from the back flow of power in the line (i.e. pb,a + iqb,a + ∆P ) and

∆P is the increase in power dispatch across the line b↔ a. Since the above relation consists of complex

variables (i.e. Va, Vb, pb,a + iqb,a), the magnitude of voltage difference across the line a↔ b is:

|Vb − Va| =|(pb,a + iqb,a + ∆P )za,b|

|Vb|

By evoking the triangular inequality, the above is bounded by:

|Vb − Va| ≤|(pb,a + iqb,a)za,b|+ ∆P |za,b|

|Vb|

The first term on the right side of this relation is |(pb,a + iqb,a)za,b| = |V ′b ||V ′b − V ′a| where V ′b and V ′a

voltages at buses a and b prior to the change in dispatch. Substituting this, the following inequality is

obtained:

|Vb − Va| ≤|V ′b ||V ′b − V ′a|

|Vb|+

∆P |za,b||Vb|

As the dispatch increase will result in a voltage rise,|V ′b ||Vb| ≤ 1. Moreover, as the fixed substation voltage

is the point of reference and |Vb| ≥ |Vs|, it can be concluded that ∆P|Vb| ≤

∆P|Vs| . From these observations,

the above bound is further generalized to:

|Vb − Va| ≤ |V ′b − V ′a|+∆P |za,b||Vs|

Hence, the first term in the above inequality represents the voltage difference across the bus prior to

the increase in dispatch. The second term represents an upper bound on voltage rise contributed by the

increase in dispatch. As ∆P is small, the margin of error is also small.

4.3.6 Resilience

The system dynamic has been shown theoretically to have strong convergence characteristics due to

exponential stability. This property is also important in identifying how the system will react to per-

turbations introduces by cyber attacks or other disturbances. Due to exponential stability, even with

perturbations, the system state trajectory will always be attracted to the stable equilibrium point which

is also the optimal state. Hence, as long as there exists sufficient capacity in the system (i.e. generation

capacity and bus voltage limits), the DG agents that are not compromised will be able to sense these

changes implicitly through the EPU signals and react to offset the disturbances.


Simulation studies are conducted using MATLAB to verify whether the theoretical results presented

earlier also apply in practical settings. The DN implemented is a low-voltage radial network illustrated

in Fig. 4.8 consisting of 34 buses and 75 homes.

Line impedance parameters are obtained from reference [116]. Bus voltage and reactive power flow

measurements used by individual agents are computed using Gauss-Seidel load flow analysis method

[117]. As the main intent of this proposal is to coordinate a large number of DGs, every home is


Figure 4.8: Low-voltage distribution network.

considered to contain a roof-top solar panel and a micro wind turbine. Power demands of homes in the

DN and generation by the DGs are modelled as outlined in Sec. 4.1.4. The specific parameters used

to model wind turbines are obtained for a power rating of 1.9kW as listed in reference [118]. The set

of strategies available to the DG agents is Y = [0.0000001 0.01 0.02] kW in these simulations. Costs

of these strategies are set to strictly positive random values which are increasing with respect to the

power dispatch levels (i.e. Ci < Ci+1). The voltage magnitudes are bounded by ±10% of the nominal

voltage of 1 p.u. (per unit). The forward and backward sweep steps require communication between

Figure 4.9: Topology of cyber-physical DN.

nodes along the local feeder branch. Hence, in this proposal, in addition to EPU broadcasting signals


at regular intervals, DG agents exchange information with a subset of other nodes in the system. The

overlay of the communication network with the physical DN is illustrated in Fig. 4.9.

Convergence over One ED Cycle

The convergence characteristics of the proposed dispatch algorithm is first assessed over one optimization

interval in Fig. 4.10. Various initial states of x at the beginning of an optimization interval are randomly

00.5

1

0

0.5

1

0

0.2

0.4

0.6

0.8

1

y1=0.000001 kW

State Trajectories for Single Dispatch Cycle

y2=0.01 kW

y3=

0.0

2 k

W

1500

2000

2500

3000

3500

4000

4500

Level Set

Optimal Solution of P

M

A State Trajectory

Figure 4.10: Projection revisions.

generated and the state trajectories induced by distributed revisions are captured in Fig. 4.10. The level

sets of L(x, ν∗) are included in this figure. The lowest possible value of the potential function L(.) is

located at the centre of these level sets as indicated by the teal circle. It is clear that the state trajectories

rapidly converge to the optimal state regardless of the initial state. Moreover, the state trajectories are

more or less orthogonal to the cost level sets. This indicates that the convergence is exponentially fast.

Next the impact of DG agents’ availability on the convergence properties of the system is examined

in Fig. 4.11. An important assumption made in this proposal is the presence of a large number of

DG agents which ascertains optimality and validates the use of population game theoretic constructs.

The plot located at the top left-hand corner of Fig. 4.11 shows that when only 20% of the agents

are available, there are significant oscillations in the demand and supply ratio which are introduced by

active stochastic effects. This is not the case for larger proportions of availability such as 50% and 100%.

Fluctuations in overall costs are not significantly evident due to the considerably large cost coefficients

that are in effect. The minimum and maximum voltages across the 34 buses in the system are plotted

in the bottom figure. It is clear that the bus voltages are maintained well within ±10% of the nominal

voltages limits.

Resilience to Perturbations

The resilience and recovery speed of the system in the event of perturbations to system state due to

cyber attacks is investigated in Fig. 4.12. It is possible for malicious entities to perpetrate attacks


0 0.005 0.01 0.0150

0.5

1

x 10−3Ratio of Real Power

Time (hours)

Dem

and/S

upply

Ratio

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.0161.026

1.028

1.03

1.032

1.034

Min and Max Voltage in the DN during a Day

Time (hours)

Norm

aliz

ed V

oltage

Max Volt 20%

Min Volt 20%

Max Volt 50%

Min Volt 50%

Max Volt 100%

Min Volt 100%

0 0.005 0.01 0.0150

0.5

1

Cost

Time (hours)

Cost R

atio

Availability: 20%

Availability: 50%

Availability: 100%

Figure 4.11: Impact of population size.

0 0.005 0.01 0.0150

10

20Impact of Disturbance

Real P

ow

er

(p)

0 0.005 0.01 0.0150.9

1

1.1

Time (hours)

Voltage (

p.u

.)

Dispatch

Demand

Max VoltageMin Voltage

Figure 4.12: Recovery of DG Agents from Disturbance.


such as forcing DG agents to choose more expensive strategies (higher dispatch level) that can cause

unnecessary expenses and voltage rise issues. In Fig. 4.12, 20% of agents (i.e. 200 DG agents) have

been forced to select the highest power dispatch level possible 30 seconds into a dispatch cycle. At this

point, a spike in aggregate real power dispatch and maximum bus voltage is evident. However, other

DG agents immediately respond to this disturbance and adjust their local dispatch levels to bring the

system back to the optimal steady state operation. Since this is a distributed dispatch strategy with

agents making independent decisions, it is difficult for an adversary to compromise a large number of

agents.

Dispatch over a Day

The performance of the proposed algorithm over a period of 24 hours is evaluated next. In Fig. 4.13,

0 5 10 15 200

20

40

60

80

100

120

140

160Aggregate DG Capacity

Time (hours)

Real P

ow

er

(kW

)

(a) DG Capacity.

0 5 10 15 20

0.9

1

1.1

1.2

1.3

1.4Voltage Profile over a Day with no Dispatch

Time (hours)

No

rma

lize

d V

olta

ge


Voltage Limits

(b) Voltage Profile Over 34 Buses.

Figure 4.13: System behaviour (real power and bus voltages) without DG dispatch in place.

no dispatch coordination mechanisms are in place and all power generated is directly injected into the

grid. The aggregate real power generation is plotted in Fig. 4.13a and the minimum and maximum

bus voltages are plotted in Fig. 4.13b over a period of one day. It is clear that the bus voltages are

exceeding the maximum voltage threshold at many points throughout the day. These over-voltages will

result in adverse equipment effects. Fig. 4.14 illustrates the behaviour of the system after applying

the proposed dispatch technique in the system. It is clear that the aggregate demands are closely met

by the aggregate generation in the system as depicted in Fig. 4.14a. Furthermore, as illustrated in

Fig. 4.14b bus voltages are maintained well within the ±10% threshold. These results demonstrate the

effectiveness of the proposed dispatch algorithm in coordinating a large number of DGs located in a DN

while accounting for physical infrastructure limitations. No fine-tuning of any parameters is required

and the technique is highly scalable, adaptive and flexible.

Comparison with State-of-the-Art

Next, the proposed dispatch strategy is compared with similar existing literature. More specifically, in

Fig. 4.15, a comparison is presented in which the proposed algorithm is compared with a sub-gradient


0 5 10 15 200

2000

4000

6000

8000

10000

12000

14000

16000

18000Aggregate Real Power Demand and Dispatch

Time (hours)

Real P

ow

er

(W)

DG Dispatch

Consumer Demand

(a) Real Power.

0 5 10 15 20

0.9

0.95

1

1.05

1.1

Min and Max Voltage Profile over a Day with Dispatch

Time (hours)

No

rma

lize

d V

olta

ge


Voltage Limits


Figure 4.14: Real-time dispatch (real power and bus voltages) over a day with dispatch solution in place.

method based on dual decomposition. For sub-gradient methods, it is necessary to fine-tune various

parameters. Specifically in the implementation presented in Fig. 4.15, α is the parameter being tuned.

For large values of α, significant oscillations are present in the dispatch of real power and the bus voltages

as illustrated in Fig. 4.15a and 4.15b. These oscillations are significant and render the system conducive

to equipment damages. Moreover, if the oscillations are too high, the bus voltage limits can be violated

as well. With lower values of α, the convergence speed is too slow. The proposed algorithm, on the

other hand, requires no such fine-tuning and exhibits extremely fast convergence to optimality with no

oscillations.

0 20 40 60 80 1000

2

4

6

8

10

12

14

Signalling Iterations

Re

al P

ow

er

Dis

pa

tch

ed

(kW

)

Comparison of Distributed Dispatch Methods

Our Proposal

SG with α: 0.025

SG with α: 0.0005

(a) Real Power Dispatch.

0 20 40 60 80 100

1.02

1.025

1.03

1.035

1.04

1.045

1.05

1.055

1.06Min/Max Voltage Over a Day for Various Algorithms

Signalling Iterations

Vo

lta

ge

(p

.u.)

Min Voltage for Our Proposal

Min Voltage for SG α: 0.025

Min Voltage for SG α: 0.0005

Max Voltage for Our Proposal

Max Voltage for SG α: 0.025

Max Voltage for SG α: 0.0005


Figure 4.15: Comparison of Dispatch Methods.

Next, general characteristics of the proposed and existing dispatch strategies are evaluated against

four performance metrics (communication costs, information overhead, error from forecast and solution

optimality) as summarized in Table 4.4. The O-notation is used for quantifying communication costs and


information overhead with respect to the size of the systemm (i.e. number of participants). The proposed

distributed dispatch strategy is compared with three general classes of dispatch strategies in the existing

literature and these are centralized real-time (i.e. online), offline and decentralized online solutions.

Centralized online solutions use a central coordinating entity to compute dispatch in real-time (over

short optimization intervals). Offline solutions compute dispatch using generation prediction/forecast

models well in advance. Decentralized online solutions use peer-to-peer communications to compute

dispatch in real-time.

Proposed Centralized Centralized/ DecentralizedDistributed Online Distributed Online

Offline

Communication Costs O(hn) O(m) 0 O(m2)

Information Overhead O(hn) O(m) 0 O(m2)

Forecast Errors No Yes Yes No

Solution Optimality Yes No Yes Yes

Table 4.4: Comparison of dispatch methods.

Communication cost refers to the number of connection links forged by nodes in a particular strategy

at every dispatch cycle. For the proposed strategy, the utility does not make point to point connections

with the DGs but broadcasts a single transmission. During feasibility checks, the number of information

exchanges in the worst case is 3hn. Hence, the communication complexity is O(hn) where hn << m.

For offline strategies, as the entire dispatch schedule will be communicated in advance, communication

cost is negligible. For centralized online strategies, all m DG agents will need to communicate individual

generation capacities to the EPU that will then communicate m dispatch values to the DGs resulting in

a communication cost of 2m [43]. For decentralized algorithms, according to [61], communication cost is

m(m− 1) where m represents the total number of relays in the system which is assumed to be at least

as large as the number of DGs coordinated.

Information overhead accounts for the total units of data exchanged between the nodes participating

in dispatch. This is proportional to the communication costs detailed in the above for the proposed

dispatch strategy. The information overhead for offline strategies is ignored since exchange occurs in

advance. For centralized online strategies, DGs transmit current generation capacity and the central

controller transmits the computed dispatch vector to all DGs. Given that there are m DGs in the

system, this results in 2m units of information exchanged during a dispatch cycle. For decentralized

strategies, nodes exchange load and generation information with one another resulting in 2m(m − 1)

data exchanges.

Forecast model inefficiencies can inject error into the computed dispatch vector due to generation

prediction inaccuracies. As the proposed distributed and existing decentralized strategies rely on real-

time generation data for dispatch and do not use forecast models over long time horizons, the error

introduced is close to null. Centralized online strategies can use short term (e.g. 3-hour) generation

models and offline strategies can use a-day-ahead prediction models. Errors introduced in these models

are provided in [11].

Global optimality of DG dispatch is an important consideration. The theoretical studies presented


earlier for the proposed technique guarantee optimality under certain non-restrictive conditions. Cen-

tralized online strategies may not guarantee an optimal solution due to the sheer size of the problem and

imposed time limitations (short optimization intervals). Moreover, distributed strategies such as that

outlined in reference [60] may not converge to optimality within the short optimization interval used for

real-time dispatch.

From the comparisons presented in the above, it can be concluded that the proposed dispatch strategy

provides a good balance amongst a variety of critical performance metrics.

4.4 Remarks

In this chapter, two dispatch proposals have been presented. The first solves economic dispatch so

that consumer demands can be balanced with available DG generation capacity at optimal costs. The

underlying physical constraints and discrete optimization variables are not accounted for in this proposal.

The second proposal is more generally applicable as this takes into account physical grid limitations while

balancing available generation with demands in a resilient manner. These proposals are shown to have

highly desirable convergence properties and scalability. In the next chapter, the proposals presented in

this chapter and the previous chapter are combined into a hierarchical algorithm in order to coordinate

both flexible demands and sustainable generation across both transmission and distribution networks.

Chapter 5

Hierarchical Optimization of

Distributed Generation and Demand

In this chapter, a hierarchical technique is proposed for the adaptive coordination of flexible demands

and DG generation spanning across both the transmission and distribution network levels so that the

overall operation of the power grid is sustainable, economical and resilient. This is a hybrid scheme that

taps onto the potential available in heterogeneous end nodes such as flexible consumers and distributed

energy sources (i.e. DGs and storage). In the proposed hierarchy, there are two main tiers and these are

referred to as transmission and distribution tiers.

The topmost level in the hierarchy is the transmission tier. Coordination in this tier takes place

amongst buses situated in the high-voltage transmission network level. This tier involves completely

decentralized optimization which is ideally suited to accommodate both bulk generation entities and

competitive energy markets stemming from deregulations. Traditionally, system operators managed

power generation operations in a central manner and this provided operators with unprecedented access

to these entities. With recent deregulations, central authorities will not have deep insights into the inner-

workings of private energy and service providers. Moreover, it will not be tractable to maintain details

about each and every transient private energy market participant. Furthermore, as competitive energy

markets are not yet fully embraced by all jurisdictions, coordinating surplus and deficits in generation

across various inter-ties and also accounting for unpredictable transmission line congestions will be chal-

lenging to accommodate centrally. The proposed decentralized process involves every EPU representing

a DN, characterized by local generation mix and flexible loads, coordinating with neighbouring EPUs

from other DNs to guide local participants according to global conditions spanning across the entire

transmission network. This allows for the seamless integration of a large number of heterogeneous enti-

ties such as flexible loads and DGs to meet system-wide goals. This approach leverages upon by convex

optimization based theoretical constructs such as alternating direction method of multipliers (ADMM)

and consensus protocols.

The second tier is referred to as the distribution tier as cooperation is driven by the EPU amongst

flexible consumers and DGs which are residing within the low-voltage DN managed by that EPU. In this

tier, the flexible loads adjust local demands and DGs dispatch power based on local generation capacities

to balance fluctuations in power consumption patterns while ensuring that the overall power injection

into the main grid computed in the transmission tier is maintained over the allocated optimization

82

Chapter 5. Hierarchical Optimization of Distributed Generation and Demand 83

interval. In the distribution tier, a central coordinating entity (which will typically be the EPU) transmits

general signals reflecting the current state of the DN which are then used by flexible end nodes in

that DN to make local adjustments to achieve the targetted power injection into the main grid in an

economical manner. In Chapters 3 and 4, distributed coordination techniques via population games have

been presented for both DR and DG dispatch across highly granular optimization intervals which can

effectively capture fluctuations in demand and generation levels. These are utilized in the distribution

tier for the coordination of local flexible consumers and DGs in this chapter.

This hierarchical approach allows for the decoupling and abstraction of various restrictions and

operating conditions associated with the DNs and competing power entities from one another. This allows

for the seamless plug-and-play integration of a wide variety of heterogeneous components into various

tiers of the hierarchy. Since only aggregate information is transferred between tiers and no individual data

is exchanged, privacy is enhanced. This method also allows for the safe injection of surplus generation in

DNs stemming from significant DG penetration in a manner that heeds voltage limitations in the DNs

while also ensuring that this surplus power contributes to reducing the deficit in sustainable generation

in other DNs. Moreover, flexible consumers are also leveraged to provision for dire conditions in which

generation capacity across the entire transmission network is simply not sufficient to sustainably and

economically meet demands in various DNs. This hierarchical technique promotes sustainable generation

and demand balance in the grid while also encouraging private sustainable generation competitors to

participate in energy markets. It can also be applied to hierarchical structures naturally present within

sub-systems contained within the power grid (e.g. microgrids or DNs).

5.1 System Model

The proposed hierarchical technique spans across transmission and distribution networks illustrated in

Fig 5.1. The buses in the transmission network are represented by EPUs managing individual DNs

Figure 5.1: System model for hierarchical topology.


(located in the transmission tier of Fig. 5.1). These buses can communicate with one another and are

equipped with agents that can make intelligent decisions. These bus agents communicate with only

neighbours in an iterative manner until a consensus on the optimal solution in the transmission tier is

reached. Unlike previous chapters, there are no central authorities broadcasting general information to

participating bus agents and therefore the optimization performed in the transmission tier is completely

decentralized involving peer-to-peer communications only. The transmission network typically has a

mesh topology consisting of lines that form cycles unlike the DNs which commonly have a tree structure.

This mesh network topology consisting of possibly multiple cycles will not be problematic in the proposed

solution as DC power flow formulation is used to derive the transmission tier algorithm. For economic

dispatch in high-voltage networks, the DC power flow formulation is typically utilized for frequency

control in the grid [78]. Each bus is associated with minimum and maximum values associated with

power injection which are computed based on the availability of local generation capacity and flexible

loads in the corresponding DNs. The optimization interval considered for the transmission tier is ten

minutes in length and therefore fluctuations in generation and dispatch can be closely captured by the

proposed algorithm.

Once coordination of power injection between bus agents in the transmission network is complete

over one optimization interval, the optimal setpoints for power injection is established at each bus and

these are used by EPUs to coordinate local flexible loads and DGs to achieve the precomputed power

injection setpoint. This distribution network level coordination (as illustrated in the distribution tier of

Fig. 5.1) occurs at a more granular level in order to ensure that variations in local consumer demands

and generation capacity flux are accounted for more accurately. Thus, the optimization interval at the

distribution tier is set to one minute. Coordination in DNs occur independently as these are decoupled

from one another by the decentralized optimization performed in the transmission tier.

5.1.1 Assumptions

Here, the assumptions made in the hierarchical algorithm are presented. In addition to the assumptions

made in Chapters 3 and 4, the following are further assumptions made for the hierarchical system which

now includes the transmission network:

1. There exists sufficient DG generation and flexible consumer demands to meet system goals;

2. Every bus in the transmission network is equipped with an intelligent agent (referred to as the bus

agent) which can also communicate with one another and if there is one DN attached to a bus in

the transmission network then the corresponding EPU will be the bus agent;

3. The optimization interval at the transmission tier is every 10 minutes;

4. The optimization interval at the distribution tier is every 1 minute;

5. Steady-state power grid conditions are considered in both tiers; and

6. Forecast of local demands and supply are available to every bus agent.

The first assumption is necessary for strong duality to hold and is also practical as ancillary generation

and storage systems will be available in standby mode in the event of deficit in power supply [106].

Assumption 2 coincides with the cyber-physical vision of the smart grid. Assumption 3 is reasonable


as economic dispatch over transmission networks take place over much larger timescales such as 24

hours [88]. The optimization interval being set to 10 minutes allows for the prevention of large error

margins introduced by forecast models over longer prediction horizons. Assumption 4 allows for the EPUs

to account for significant fluctuations in demand and supply within DNs. Assumption 5 is typically

the case in economic dispatch problem formulations [72]. The final assumption can be achieved by

DGs communicating to the EPU local generation capacities expected over 10 minute intervals. Demand

patterns are fairly consistent and are already available to the EPUs. Another option will be for individual

consumers to send local demands expected over the optimization intervals to data concentrators which

will then be aggregated and transmitted to the EPU.

5.2 Transmission Network Coordination

In Chapters 3 and 4, distributed algorithms consisting of a central entity that transmits general in-

formation about the state of the grid are used by participating agents to make local decisions. This

distributed solution structure with a central coordinating entity will not be practical in the transmission

network level due to the lack of homogeneity resulting from current trends in deregulations and the sheer

number of actuating nodes (i.e. thousands of flexible consumers and small/large scale DGs belonging to

hundreds of DNs) in the system. Hence, a completely decentralized algorithm is necessary for this. This

is facilitated by ADMM and consensus constructs from convex optimization and these are introduced

next.

5.2.1 Introduction to ADMM

Combining ADMM and consensus methods allow for completely decentralized decision-making that

involves communication with only adjacent neighbours. Suppose that the original formulation is a

convex optimization problem with the following general structure:

PA : minx∈Cx,y∈Cy

f(x) + g(y)

Ax+By = D (C1)A

where x and y are the optimization variables, f : R|x| → R and g : R|y| → R are convex functions, Cxand Cy are separable convex constraints associated with x and y, A ∈ Rc×|x| and B ∈ Rc×|y|, c is the

number of equality constraints in PA and D ∈ Rc is a vector of constants. The coupling constraint in

(C1)A defines a set of linear equality relationships between x and y. Without this constraint, PA will be

a completely separable and decomposable problem that can be solved independently by every agent in

the system. In order to overcome the difficulty caused by (C1)A, PA is relaxed to the following problem:

PdA : maxν

minx∈Cx,y∈Cy

Lρ(x, y, ν) = f(x) + g(y) + νT (Ax+By −D) +ρ

2||Ax+By −D||22

where ν ∈ Rc is a set of Lagrangian multipliers associated with the c equality constraints in (C1)A and

ρ > 0. Lρ(x, y, ν) is referred to as the augmented Lagrangian. The dual problem PdA can be solved


iteratively via ADMM as follows [121]:

xk+1 = argminx

Lρ(x, yk, νk)

yk+1 = argminy

Lρ(xk+1, y, νk)

νk+1 = νk + ρ(Axk+1 +Byk+1 −D)

where k is an index denoting the current updating iteration. The first two steps in ADMM minimize

the Lagrangian with respect to the primal variables x and y separately. This separability allows for

decentralized computation enabled by information exchanges. While minimizing Lρ with respect to x, y

and ν are held constant and are set to the values yk and νk computed in the previous iteration. When Lρ

is minimized with respect to y, x and ν are set to xk+1 computed immediately before and νk computed

in the last iteration. Finally, ν is updated using xk+1 and yk+1 computed in the current iteration and

the step-size is set to ρ which is the Lagrangian parameter in PdA. Information exchanges are necessary

during each one of these sequential update steps and these contain the precomputed parameters necessary

for the current update. Primal variables involve minimization and dual variables are associated with

maximization. Hence, these updates take place in alternating directions and thus referred to as ADMM.

This method will converge to optimality if strong duality holds and this is indeed the case due to

Assumption 1. As outlined in reference [119] linear convergence of distributed ADMM is guaranteed

under mild conditions. The convergence speed of the algorithm has been observed to be fast to attain

moderate accuracy which suffices for most applications [121]. Attaining finer accuracy via ADMM results

in much slower convergence. The degree of accuracy can be inferred by computing the residual r:

rk = ||Axk +Byk −D||2 (5.1)

where rk → 0 as k →∞ and optimality is achieved.

5.2.2 Problem Formulation

In order to take advantage of the decentralized solution structure and desireable convergence character-

istics enabled by ADMM, the optimization problem considered at the transmission level is transformed

using consensus-based techniques into a form to which ADMM can be applied. First, consider the

problem formulation presented in PT :

PT : minp

f(p) =∑n∈B

αn(pgn)2

pgn − pdn = −∑m∈Bn

bn,m(θn − θm) ∀ n ∈ B (C1)T

0 ≤ pgn ≤ cgn ∀ n ∈ B (C2)T

where pgn is the real power generation in bus n, pdn is the aggregate consumer demand from bus n,

Bn represents the set of buses adjacent to bus n and n ↔ m is a line adjacent to bus n (i.e. m is

a neighbour of n). This formulation is similar to the linear power flow problem presented in PFR of

Chapter 2. This consists of optimization variables that are real power generation p = [pg1 . . . pg|B|] and

the voltage angles θ = [θ1 . . . θ|B|]. The normalized voltage magnitudes in high-voltage transmission


networks are close to unity and conductances gn,m are much smaller than susceptances bn,m in lines

(n↔ m) ∈ L. These assumptions render reactive power flows negligible. It is evident that the objective

in PT and constraints in (C2)T are separable. However the power balance constraints in (C1)T consists

of variables from multiple buses (specifically neighbouring buses). This particular constraint prevents

the straightforward decomposition of PT into individual sub-problems.

5.2.3 Transformations to ADMM Form

Next, a set of transformations are applied to PT so that the resulting formulation is in a form similar

to PA to which ADMM can be directly applied. Constraint (C1)T represents the net power balance at

each bus. This power balance is governed by bus angles of neighbouring buses. In order to decouple the

net power balance equations, new variables are introduced. Each bus uses a subset of these variables

to keep track of its perspective of what the primary variables are of neighbouring buses. In particular,

the primary variables that are of main concern to each bus are the bus angles θ. For instance, consider

bus n and its neighbouring bus m where m ∈ Bn. If an agent residing in bus n is to compute the

local net power flow balance in (C1)T , it will require θm from all buses that are its neighbours. Bus n

can maintain a new variable θm,n which is the value of θm inferred from the perspective of bus n. At

optimality, this perspective variable θm,n must be equal to the original variable θm (i.e. θm,n = θm).

This equality enforces consensus between the new perspective variable and the original variable. Let all

of these perspective variables form the set y = [y1 . . . y|B|] and all the original variables form the set x

where each element in these sets is a tuple as follows:

yn = {pgn,n, θn,n, {θm,n| m ∈ Bn}} ∀ n ∈ B,

xn = {pgn, θn} ∀ n ∈ B

where yn contains all the perspective variables maintained by bus n and xn contains all of the original

variables local to bus n. In the set yn, new variables such as pgn,m where m ∈ Bn are not required as the

net power balance equation associated with a bus in (C1)T does not include real power generated from

any adjacent buses. The constraint sets Cx = {C1x ∪ . . . ∪ C

|B|x } and Cy = {C1

y ∪ . . . ∪ C|B|y } are defined as:

Cnx :

0 ≤ pgn ≤ cgn−∞ ≤ θn ≤ ∞

Cny :

pgn,n − pdn = −∑m∈Bn

bn,m(θn,n − θm,n)

It is important to note that the constraints sets Cnx and Cny are completely separable from one another as

these are based on local primary and perspective variables respectively. Equipped with these definitions,


the problem formulation in PT can be equivalently transformed into:

P ′T : minp

∑n∈B

fn(xn)

xn ∈ Cnx ∀ n ∈ B (C1)’T

yn ∈ Cny ∀ n ∈ B (C2)’T

xn = yn,m ∀ m ∈ {Bn, n}, n ∈ B (C3)’T

where yn,m is the perspective variable associated with the primary variable in bus m from the point of

view of bus n and the third constraint establishes consensus with the perspective and original variables.

This enforces the primary variables xn and the associated perspective variables in yn,m to be the same.

For example, θm is a primary variable of bus m and θm,n is the perspective variable of θm maintained

by bus n which is a neighbour of bus m. These variables must be the same (i.e. θm = θm,n) for the

feasibility of problem P ′T . P ′T is in essence equivalent to the original problem PT with the consensus

constraint in effect. The only difference is that more variables are maintained by each bus agent to allow

for separability. It is also important to note that P ′T is now of the same form as PA presented earlier in

this chapter to which ADMM can be applied. The dual of P ′T using the augmented Lagrangian is:

P ′Td

: maxν

minx∈Cx,y∈Cy

Lρ(x, y, ν) =∑n∈B

fn(xn) +∑n∈B

∑m∈{Bn,n}

νn,m(xn − yn,m) +ρ

2

∑n∈B

∑m∈{Bn,n}

(xn − yn,m)2

where ν ∈ R|y| with a slight overloading of notation where xn is referencing a single primary variable

not a set. Grouping terms that are associated with each bus, the Lagrangian can be decomposed into

sub-problems. When applying ADMM to solve P ′Td

iteratively, three sets of updates are made at every

iteration.

First, x (i.e. primary variables) are solved for by fixing yk and νk from the previous iteration.

By fixing y and ν into constant values, the above Lagrangian becomes completely separable. Ignoring

constant terms in Lρ(x, y, ν), each bus agent solves the following to obtain xk+1:

Pxn : xk+1 = [pgnk+1, θk+1

n ] = argmin(pgn,θn)∈Cnx

(αi + ρ)(pgn)2 + (νpn,nk − pgn,n

kρ)pgn+

+∑

m∈{Bn,n}

(ρθ2n + νθn,m

kθn − θn,mkθnρ)

where the optimization variables pgn and θn are highlighted in blue, νpn,nk is the Lagrangian variable

already computed from the last iteration k that is associated with the constraint pgn−pgn,n = 0, similarly

νθn,mk

is the Lagrangian variable also computed from the last iteration k which is associated with the con-

straint θn−θn,m = 0, pgn,nk is a y variable computed from the previous iteration k which is the perspective

from bus n the real power generation at the same bus (this variable is included to allow for separability)

and θkn,m is another y variable computed by the agent residing at bus m to obtain a perspective of θn

from bus m. Prior to solving the above optimization problem, every bus agent must gather precomputed

variables νθn,mk

and θkn,m from neighbours and locally computed variables pgn,nk, νpn,n

k and νθn,nk

in the

previous iteration. The transfer of these values is facilitated by communications established between

buses that are physical neighbours in the transmission network. Next, a set of updates are applied to


the y variables at each bus as follows:

Pyn : yk+1 = [pgn,n, θm,n| ∀ m ∈ {Bn, n}] = min(pgn,n,θm,n| ∀ m∈{Bn,n})∈Cny

(pgn,n)2ρ− (νpn,nk + pgn

k+1ρ)pgn,n

+∑

m∈{Bn,n}

(θ2m,nρ− (νθm,n

k+ θm

k+1ρ)θm,n)

where once again the optimization variables [pgn,n, θn,m| ∀ m ∈ {Bn, n}] are highlighted in blue. These

variables reflect bus n’s perspective of variables belonging to local and neighbouring buses that are

used in the coupling constraint (C1)T . The x and ν variables used as parameters in this optimiza-

tion problem that have been solved immediately prior to this update or in the previous iteration k

are pgnk+1, θm

k+1, νθm,nk

and νpn,nk where θm

k+1 and νθm,nk

are exchanged between neighbours. After

updating the y-variables, the final step in this ADMM iteration involves updating the dual variable ν

associated with the consensus constraints as follows:

νpn,nk+1 = νpn,n

k + ρ(pgnk+1 − pgn,n

k+1)

νθn,mk+1

= νθn,mk

+ ρ(θnk+1 − θn,mk+1) ∀ m ∈ {Bn, n}

which improves the Lagrangian variable ν via dual ascent. To compute these variable updates, x and

y variables that have been computed in the previous two steps are necessary in a manner similar to

the primary and perspective variable updates conducted earlier. For this, another set of information

exchanges are executed between neighbouring buses.

5.2.4 Summary of Decentralized Algorithm

Table 5.1 presents a summary of the decentralized algorithm used by bus agents to coordinate power

injections to achieve the optimal solution of PT .

In reference [74], ADMM and consensus protocol has been applied to a radial network (i.e. distribu-

tion network) to address a relaxed version of optimal power flow which is a semi-definite program (i.e.

convex optimization). This formulation is exact for the radial network under certain conditions [122].

The proposal presented in this chapter, on the other hand, applies ADMM and consensus to a mesh

network (i.e. transmission network) to achieve optimal linearized power flow. Economic dispatch in the

transmission level is typically based on the DC formulation and therefore additional complexities intro-

duced by the original OPF with non-linear constraints is not necessary. Furthermore, as the DC problem

is convex, the mesh structure of the transmission network will not be problematic in the application of

the proposed algorithm. If the problem is non-convex, then convergence to the optimal solution will not

be guaranteed. There are other proposals such as reference [75] that apply ADMM to the original OPF

without any relaxations. As the original OPF is non-convex, the proposed technique in reference [75]

has no strong guarantees of convergence.


In this section, the transmission level coordination algorithm proposed via ADMM is implemented in

two transmission network systems using MATLAB. One is the WECC 3 generator 9 bus system and

the other is IEEE 10 generator 39 bus system which are illustrated in Figs. 5.2 and 5.3 respectively.


Decentralized Algorithm at the Transmission Tier

Initialization:

• Set xn ← 0, ym,n ← 0 ∀ m ∈ {Bn, n}, νn ← 0 andk ← 0

• Set cgn and pdn reflecting local DN’s generation and de-mands over the next 10 minutes

While r > ε:

1. Set k ← k + 1

2. Solve Pxn to obtain xkn using the initially computed

yk−1n and νk−1

n

3. Broadcast xkn to all neighbours m ∈ Bn4. Solve Py

n to obtain ykn using the initially computed xknand νk−1

n

5. Broadcast ykn to all neighbours m ∈ Bn

6. Compute νkn using previously computed νk−1n , xkn and

ykn

7. Broadcast νkn to all neighbours m ∈ Bn8. Compute r = ||x− y||2

Table 5.1: Decentralized algorithm for bus agent n in the transmission tier.

Although the WECC 3 generator 9 bus system is small in comparison to the IEEE 10 generator 39 bus

system, it provides means for comparison of the convergence properties of the ADMM algorithm in a

small versus large system. All parameters have been obtained using MATPOWER [124] and the base

MVA is 100. In these networks, every bus maintains its own set of variables and exchanges already

computed information amongst physically adjacent neighbours only.

Residual

The convergence properties of the ADMM algorithm for the WECC 3 generator 9 bus system is examined

first. Fig. 5.4 illustrates the residual (i.e. Eq. 5.1) computed over 1000 iterations in the WECC system

with the proposed algorithm in place. It is clear, that the residual stabilizes and approaches 0 within

500 iterations. Moreover, at iteration k = 114, the residual approaches 0 very fast and then resumes

to oscillate until k = 413. This steep descent of the algorithm reinforces the observation made earlier

regarding how the ADMM algorithm reaches moderate accuracy very quickly. The residual across the

iterations are not monotonously decreasing as the sub-problems constructed for each bus agent are not

differentiable due to local bus constraints.

Next, the convergence properties of the ADMM algorithm for the IEEE 10 generator 39 bus system is

depicted in Fig. 5.5. The initial value taken by each variable is 0. It is clear that initially, the algorithm

drives the system from a high residual to a residual of 5 × 10−3 very rapidly. However, as expected,

obtaining a fine accuracy in which the residual is very low requires a significant number of iterations.

Hence, if the system operator has some tolerance with respect to the accuracy of the solution, then it is

not necessary to fine-tune the solution.

The impact of the Lagrangian parameter ρ on the convergence speed of the algorithm is investigated

next. The smaller the step-size is, the more can the solution be fine-tuned but over larger number of

iterations. Fig. 5.6 illustrates the impact of varying ρ on the aggregate dispatch at k = 100 in the


Figure 5.2: WECC 3 bus line diagram, adapted from reference [123].

Figure 5.3: IEEE 39 bus line diagram, adapted from reference [27].


0 200 400 600 800 10000

0.5

1

1.5

2

2.5

Residual versus Number of Iterations

Iterations

Resid

ual

Figure 5.4: Residual for WECC 9 bus system.

0 2000 4000 6000 8000 100000

0.005

0.01

0.015

0.02

0.025

Residual versus Number of Iterations

Iterations

Resid

ual

Figure 5.5: Residual for IEEE 39 bus system.

IEEE 39 bus system. The x-axis is based on a logarithmic scale and represents the step-size α. It is

evident that larger the step-size is, the faster is the solution approaching optimality. However, the speed

of convergence tapers off as α increases above 106. Additionally, the larger the value of ρ is, the less

fine-tuning can be applied to the resulting solution. Hence, the system operator will need to define ρ in

a manner that strikes a fine balance between speed and accuracy.

Comparison of Central and decentralized Methods

Next, the results of applying the proposed algorithm for the WECC 9 bus system depicted in Fig. 5.2

is listed in Table 5.2. In this table, the decentralized ADMM algorithm is compared with the values


103

104

105

106

107

0

0.5

1

1.5

2

2.5

3

Impact of α on Convergence Speed of ADMM

Step Size (α)

Ag

gre

gate

So

luti

on

Figure 5.6: Impact of Lagrangian step-size on ADMM convergence.

obtained from applying a central quadratic solver in MATLAB to the DC formulation. The central

solution is the exact solution whereas the decentralized solution is obtained by terminating the algorithm

when the residual is less than 9 × 10−4. The power demand at each bus is randomly assigned. There

are 18 primary variables in this particular example and when the perspective variables are included, the

optimization space becomes much larger. It can be observed that the optimal generation at each bus and

the bus angles obtained via the decentralized algorithm is very close the exact solution obtained using

a central solver. Hence, this illustrates that if the system operator has some tolerance to the accuracy

of the solution, then the proposed algorithm is suitable for decentralized computation within practical

number of iterations.

Bus Bus Bus Generation Bus Generation Bus Angle Bus AngleNumber Demand (Central) (decentralized) (Central) (decentralized)

1 0 38.3333 38.3467 0.0000 02 20 38.3333 38.3617 -0.6447 -0.64383 10 38.3333 38.3621 0.3157 0.31674 30 0.0000 0 -2.2080 -2.20875 10 0.0000 0 -2.5295 -2.53026 15 0.0000 0 -1.3446 -1.34497 10 0.0000 0 -2.0306 -2.03148 10 0.0000 0 -1.7906 -1.79119 10 0.0000 0 -2.6350 -2.6358

Table 5.2: Comparison of results from central versus decentralized algorithms WECC 9 bus system.


The results obtained for the IEEE 39 bus system illustrated in Fig. 5.3 are listed in Table 5.3.

Once again, the problem is solved using a central quadratic solver and these results represent the exact

solution. The decentralized ADMM algorithm is also applied to the system. There are 78 primary

variables maintained in the system in addition to the perspective variables amongst which consensus

must be reached. The values listed in Table 5.3 are associated with a residual of 2× 10−8. These values

are fairly close to the exact solution obtained via a central solver. These results demonstrate the efficacy

of the proposed decentralized solution for the transmission tier.

Bus Bus Bus Generation Bus Generation Bus Angle Bus AngleNumber Demand (Central) (decentralized) (Central) (decentralized)

1 0 0.0000 0 0.0000 02 0 0.0000 0 0.0691 0.06943 1 0.0000 0 -0.0431 -0.04304 1 0.0000 0 -0.1827 -0.18375 4 0.0000 0 -0.2557 -0.25736 5 0.0000 0 -0.2556 -0.25727 2 0.0000 0 -0.2749 -0.27658 4 0.0000 0 -0.2753 -0.27699 4 0.0000 0 -0.1963 -0.197010 0 0.0000 0 -0.2433 -0.244911 3 0.0000 0 -0.2584 -0.260112 2 0.0000 0 -0.2945 -0.296013 5 0.0000 0 -0.2434 -0.244914 0 0.0000 0 -0.1808 -0.181815 2 0.0000 0 -0.0432 -0.043216 2 0.0000 0 0.0355 0.035917 2 0.0000 0 -0.0066 -0.006118 4 0.0000 0 -0.0409 -0.040619 1 0.0000 0 0.1730 0.173120 4 0.0000 0 0.1799 0.180021 2 0.0000 0 0.1210 0.121422 0 0.0000 0 0.2377 0.238123 1 0.0000 0 0.2455 0.245824 4 0.0000 0 0.0455 0.046025 2 0.0000 0 0.1340 0.134626 0 0.0000 0 0.0701 0.071027 2 0.0000 0 0.0188 0.019628 3 0.0000 0 0.0886 0.089929 1 0.0000 0 0.1402 0.141630 4 8.5000 8.5025 0.1505 0.150931 1 8.5000 8.5021 -0.0681 -0.069732 5 8.5000 8.5007 -0.1733 -0.174933 1 8.5000 8.4906 0.2797 0.279834 4 8.5000 8.4897 0.2611 0.261035 1 8.5000 8.4903 0.3450 0.345236 1 8.5000 8.4899 0.4496 0.449637 0 8.5000 8.5003 0.3313 0.331938 3 8.5000 8.4946 0.2263 0.227639 4 8.5000 8.5136 -0.0418 -0.0420

Table 5.3: Comparison of results from central versus decentralized algorithms IEEE 39 bus system.


5.2.6 Comparison with State-of-the-art Decentralized Techniques

Generally, the performance of ADMM is shown to be linear with the size of the system under some

mild conditions [74, 119]. Other decentralized algorithms used commonly in the state-of-the-art include

average consensus and dual decomposition via sub-gradient methods. The convergence rate of average

consensus is dependent on the second smallest eigenvalue of the Laplacian matrix representing the

underlying network topology. In the worst case, this is O(n3) [120]. Convergence of dual decomposition

via sub-gradient method depends on the step-size used to update the dual variables. In addition to

performance, another advantage of ADMM is that it naturally applies to the underlying decentralized

structure of the optimization problem at hand [106].

5.3 Distribution Network Level Coordination

Next, the coordination of flexible demands and DGs in the distribution tier is presented. It is important

to note that curtailment of flexible demand can be considered to be positive generation. At the beginning

of each optimization interval, the EPU coordinates the dispatch of DGs to achieve the pgn computed in the

transmission tier according to the algorithm presented in Table 4.2 in Chapter 4. In this computation,

the offset between demand forecast used at the transmission level denoted as∑d∈D p

′d and current

demands∑d∈D pd must also be accounted for. When the overall dispatch poverallg at the end of the DG

coordination is not sufficient to meet the targeted pgn, then the EPU coordinates demand curtailment

in flexible consumers in order to meet the targeted power generation in accordance to the algorithm

presented in Chapter 3 in Table 3.3. For this, the setpoint for demand curtailment is set to:

SDR = pgn − poverallg +∑d∈D

pd −∑d∈D

p′d (5.2)

This distribution tier coordination in summarized in Table 5.4. Separation between DR and DG

dispatch is necessary as the demand curtailment levels will not necessarily coincide with the generation

dispatch levels. Moreover, demand curtailment should be a last resort option as consumers will naturally

prefer to operate appliances as usual. All participating entities (i.e. flexible consumers or DGs) will

respond to the signals transmitted by the EPU as summarized in Tables 3.3 and 4.2 in Chapters 3 and 4.

The DR coordination curtails loads in accordance to local operating conditions and consumer comfort

levels. The DG coordination takes into account voltage rise considerations which are imperative in DNs.

This combined approach used to achieve the targeted power injection computed in the transmission tier

within the short optimization interval of one minute is feasible as the convergence of these algorithms

is exponential when projection revisions are used (as demonstrated in Chapter 4). Hence, convergence

of both of these coordination algorithms operating sequentially can be achieved within the one minute

optimization interval allocated to the distribution tier level optimization.

5.3.1 Resilience

In the coordination of vast number of cyber-enabled heterogeneous entities that rely upon communica-

tion broadcasts to make local revisions, resilience is an important factor. As demonstrated in Chapters

3 and 4, the proposed distributed DR and DG dispatch algorithms are associated with strong resilience


Distributed Coordination by the EPU at the Distri-bution Tier

For every optimization interval:

1. Coordinate DGs residing in the local DN using thealgorithm in 4.2 where

∑d∈D pd is replaced with∑

d∈D pd −∑

d∈D p′d + pgn until |xk − xk−1| < ε

2. Coordinate DR participants residing in the local DNusing the algorithm listed in 3.3 where SDR is set tothe value obtained from Eq. 5.2. During every revision,perform voltage feasibility check

Table 5.4: Decentralized algorithm for bus agent n at the transmission tier.

properties. Hence, in the event of perturbations, other agents in the system will be able to infer this

implicitly through broadcast signals and act appropriately. As these algorithms are utilized for distri-

bution tier coordination in this chapter, all properties associated with resilience are also inherited in

the distribution tier coordination. Thus, resilience is built into the core of the design of the proposed

hierarchical optimization algorithm.

5.4 Remarks

The hierarchical technique presented in this chapter allows for the decentralized and distributed co-

ordination of heterogeneous power grid components. Superior convergence characteristics obtained by

applying population game theory constructs are tapped upon to enable the seamless coexistence of flex-

ible loads and DGs with extremely different local operational conditions using a central coordinating

entity like the EPU in the distribution tier. Moreover, resilience innate in these algorithms is inherited

in the distribution tier level. In order to abstract operational details of individual DNs, a completely de-

centralized algorithm with fast convergence properties obtained by leveraging on ADMM and consensus

theoretical constructs is proposed to manage the transmission network tier. This decentralized approach

is ideal for managing independent entities such as IPP, bulk generation and power supply from DNs in

an effective manner.

Chapter 6

Conclusions

Emerging trends in the power grid are introducing many advances and changes to the traditional energy

infrastructure and operational processes. Power components are no longer passive as these can now

communicate, learn, monitor and make intelligent decisions. This integration of the cyber domain and

intelligence with the physical grid gives rise to self-healing, resilient and adaptive decision-making ability

by many actuating grid components. The evolution of the power grid into a smart grid also, inevitably,

introduces many challenges due to the high penetration of fluctuating generation systems, deregulations,

cyber-security and so on. In this thesis, it is demonstrated that these challenges can be effectively

overcome by tapping onto the connectivity and intelligence enabled by the cyber-physical smart grid.

Self-adaptive responses by individual power components can be designed by bridging mathematical

theoretical constructs with the underlying physics of the power grid.

6.1 Summary

As such, a basic overview of the evolving landscape of the power grid is presented first in this thesis.

Here, the advantages and challenges due to these changes are introduced and these set the premise for

the work presented in this thesis. Then, various optimization problems encountered in the power grid

along with the method for gauging the difficulty of solving these are also introduced. This exposition

serves to highlight the need for alternative methods to overcome difficulties caused by non-convexity

and in-scalability of the traditional optimization formulations. The general theme in these problems

is the presence of thousands of actuating nodes each with individual operating limitations which are

bound together by a global constraint. These nodes must be directed in an effective and economical

manner so that a particular goal (i.e. minimize cost, carbon footprint, etc.) can be achieved in the

system. Using traditional approaches such as central optimization for directing these nodes in an optimal

manner is mathematically intractable and imposes extensive resource overheads. Hence, distributed and

decentralized approaches are alternative solutions whereby individual nodes are presented with some

general information about the global state of the system at periodic intervals which are then used by these

nodes to make individual decisions based on local operating conditions. This distributed/decentralized

solution structure capitalizes on the individual monitoring, communication and actuation capabilities of

active power components. Moreover, recent developments in data aggregation and wireless broadcast

capabilities are also levied to keep individual nodes informed about the current state of the system.

97

Chapter 6. Conclusions 98

This approach of offloading decision-making to a large number of small-scale actuating devices es-

sentially parallelizes computations and taps onto the vast potential of the cyber-physical amalgamation

in the power grid. However, in order to ensure that these individual decision-making entities converge

to the optimal operating state, it is necessary to demonstrate both theoretically and practically that the

proposed distributed algorithms are indeed associated with desirable convergence traits. Proposals pre-

sented in this thesis utilize convex optimization and population game theoretic constructs to construct

distributed and decentralized algorithms to coordinate various power grid components such as demand

curtailment of flexible consumers and DG dispatch. Dynamics induced by these distributed decisions are

proven to have superior convergence properties (asymptotic and exponential) to optimality regardless of

the initial state or the significant number of participants in the system. Moreover, it is demonstrated

that the system is robust and resilient to perturbations that can be caused by fluctuating local operating

conditions of the grid and/or cyber attacks. These results are verified in practical MATLAB simulations

with realistic models of consumer demands and DG generation capacities. In the culmination of the

thesis, a hierarchical solution is presented for the coordination of heterogeneous components such as

flexible consumers and DGs across multiple tiers in the power grid.

Thus, in this thesis, principles of abstraction and decoupling are applied to non-intrusively and

seamlessly integrate diverse components in the power grid to promote economical and sustainable power

grid operations.

6.2 Contributions

In this thesis, novel distributed and decentralized algorithms have been presented to allow individual

actuating components with local operating constraints and global operational constraints to make their

own actuation decisions based on external information obtained through either a central coordinating

entity or peer-to-peer information sharing. These proposals are highly scalable with strong convergence

characteristics. Hence, in this thesis, the seamless integration of a large number (thousands) of hetero-

geneous power components to achieve the common goal of economical and sustainable grid operations

in real-time has been investigated. Main contributions of this thesis are summarized in the following:

1. Novel approach to DR is presented that is: 1) Distributed: computational processing is offloaded

to cyber-physical DR agents located at consumer premises; 2) Consumer-centric: end-users

configure load operation preferences on their respective DR agents based on comfort requirements;

3) Real-time: the EPU broadcasts unidirectional communication signals employed by DR agents

to elicit load shedding decisions based on current local conditions and these decisions rapidly

converge to optimal aggregate peak reduction; 4) Resilient: it is possible to rapidly recover from

system perturbations when a subset of DR agents are corrupted; 5) Scalable: better convergence

is achieved with a larger number of participants.

2. Novel approach to dispatch of a large number of distributed energy sources is presented that is: 1)

Distributed: Computational efforts are offloaded to local cyber-physical DG agents that compute

their own dispatch via lightweight communications from the utility; 2) Real-time: DGs use

current local generation conditions for dispatch instead of long-term forecast models; 3) Scalable:

DG addition has little impact on the complexity and convergence properties of the algorithm; 4)

Efficient: Only unidirectional utility broadcast communication is required; signals transmitted


by the utility enable rapid dispatch convergence. Moreover, the proposed mechanism does not

require communication of individual DG state information, but merely an aggregate providing

inherent privacy and resilience to denial-of-service attacks; 5) Resilient: As the system dynamic

is exponentially stable, the DG agents are able to recover from perturbations rapidly and restore

the optimal equilibrium.

3. Hierarchical approach combining DR and dispatch is presented that is: 1) Decentralized at

the transmission network tier and distributed at the DN level; 2) Real-time to accommodate

fluctuations in demand and generation; 3) Scalable: large number of nodes across the power grid

can be coordinated in real-time; 4) Accommodates heterogeneous devices and thereby supports

diversity in actuating devices; 5) Resilient: As algorithms with strong resilience characteristics

are deployed in the distribution tier, these properties are inherited by the proposed hierarchical

algorithm.

4. All of the convergence characteristics have been verified via theoretical studies and realistic simu-

lations using practical models and real data.

These algorithms are practical and can be applied by the grid operators to enable the sustainable, eco-

nomical and resilient grid operations. Moreover, these solutions effectively overcome many challenges

associated with high penetration of DGs, large power consuming devices, deregulations and existing vul-

nerabilities in the communication protocols. These solutions have been obtained by applying theoretical

constructs in optimization and game theory.

6.3 Future Directions

This thesis has explored how decentralized and distributed methods can be designed to enable intelligent

actuating nodes to respond to various changes in the power grid in an optimal manner. These efforts

have specifically focussed on flexible demands and sustainable generation sources. These applications

demonstrate the power of self-healing and adaptive actions of power grid components which have been

mainly enabled by the cyber-physical smart grid. This automated distributed decision-making overcomes

many challenges due to resource overheads, security and tractability. As such, the following are some

future extensions to this work:

• Voltage control study should be extended to unbalanced three phase radial DN for completeness.

• Microgrids and DNs can be leveraged in grid-connected mode to promote resiliency in the power

grid. These systems can be configured to automatically come online based on the overall grid

conditions and have the potential of being more economical than traditional ancillary services.

• Investigate how convex optimization techniques can be applied for optimal primary control in

microgrid settings. Control in microgrid components has been traditionally challenging due to the

extremely granular time margins available for achieving stability in the system (in the range of

nanoseconds). Using communication mechanisms such as large-scale broadcasts result in latencies

that are not acceptable in these time-scales. Hence, a completely decentralized solution with

minimal communications is necessary to address primary control in the microgrid level.


• With self-actuating devices, it is necessary to ensure that safe operating limits and additional

protection mechanisms are integrated into these actuating nodes in order to prevent malicious

cyber attacks that can undermine the underlying stability of the system.

• Validations of proposed techniques in this thesis have been implemented using realistic models

and parameters in MATLAB. In order to ensure that these have acceptable performance in real

systems, advanced simulation systems such as Opal-RT or RTDS should be utilized. These can

also incorporate advance models to take into account transients and changes resulting from the

underlying physical attributes of the grid.

• In the hierarchical technique presented in this thesis, DC problem formulation has been used which

ignores the impact of reactive power flow. Extending this work to examine the effects of reactive

power flow in the transmission network on line loadability and other effects will provide interesting

insights into how system protection devices can be deployed to protect the grid.

• Investigate the impact when the system does not converge within the allocated optimization interval

(i.e. parameters change prior to convergence) via multi-period optimal power flow with a receding

horizon.

Hence, this particular area of research in the smart grid is rich and rewarding. There are many research

opportunities that can serve to enhance the security, sustainability and economical operations of the

power grid.

Appendix: List of Acronyms

DN Distribution Network EGT Evolutionary Game Theory

AGC Automatic Generator Control OPF Optimal Power Flow

DG Distributed Generator ED Economic Dispatch

MPP Maximum Power Point VR Voltage Regulation

IoT Internet of Things FR Frequency Regulation

DR Demand Response SOC State-of-Charge

DLC Direct Load Control KKT Karush Kuhn Tucker

EPU Electric Power Utility NS Nash Stationarity

TOU Time of Use NE Nash Equilibrium

HEMS Home Energy Management ESS Evolutionary Stable State

Systems AC Alternating Current

NIST National Institute of Standards NP Non-deterministic

and Technology Polynomial time

HVAC Heating, Ventilating PC Pairwise Comparison

and Air-Conditioning SLLN Strong Law of

ISO Independent System Operator Large Numbers

IPP Independent Power Plant NC Negative Correlation

ADMM Alternative Direction MIMO Multiple Input

Method of Multipliers Multiple Output

SG Sub-Gradient RD Replicator Dynamic

IEEE Institute of Electrical and WECC Western Electricity

Electronics Engineers Coordinating Council

101

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