microgrids: resilience, reliability, and market structure

Southern Methodist University Southern Methodist University

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Electrical Engineering Theses and Dissertations Electrical Engineering

Spring 5-2018

Microgrids: Resilience, Reliability, and Market Structure Microgrids: Resilience, Reliability, and Market Structure

Saeed Dehghan Manshadi Southern Methodist University, [email protected]

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1

MICROGRIDS: RESILIENCE, RELIABILITY, AND MARKET STRUCTURE

Approved by:

Dr. Mohammad Khodayar

Dr. Jianhui Wang

Dr. Behrouz Peikari

Dr. Khaled Abdelghany

Dr. Eli Olinick

2

MICROGRIDS: RESILIENCE, RELIABILITY, AND MARKET STRUCTURE

A Dissertation Presented to the Graduate Faculty of the

Bobby B. Lyle School of Engineering

Southern Methodist University

in

Partial Fulfillment of the Requirements

for the degree of

Doctor of Philosophy

with a

Major in Electrical Engineering

by

Saeed Dehghan Manshadi

(B.S., University of Tehran, 2012)

(M.S., The State University of New York at Buffalo, 2014)

May 19, 2018

1

© Copyright by

Saeed Dehghan Manshadi

2018

1

TO MY LOVELY WIFE FOR HER UNCONDITIONAL LOVE AND SUPPORT

AND

TO MY DEAREST PARENTS FOR THEIR LIFELONG LOVE AND ENCOURAGEMENT

vi

ACKNOWLEDGMENTS

I would like to acknowledge the guidance of my advisor, Professor Mohammad Khodayar, not

only for his continuous support in guiding me through my research and completing this dissertation, but

also he has been my mentor any time I needed advice. This dissertation would have not been possible

without his encouragement, guidance, and insight and valuable suggestions. I also appreciate the valuable

comments and suggestions of my advisory committee members, Professors Jianhui Wang, Behrouz Peikari,

Khaled Abdelghany, and Eli Olinick.

I was fortunate enough to work alongside a number of wonderful friends and distinguished

current and former students. I’d like to thank them for their invaluable help on many occasions, spirited

discussions about the research, and more importantly for their friendship and support.

As always, my greatest gratitude and appreciation is to my wife, Parya, my parents, Zinat-Sadat

and Hamidreza for their everyday inspiration and encouragement and endless sacrifices without which I

could never stand here. My sister, Zohreh has always been there for me and I cannot thank her enough for

that.

vii

Dehghan Manshadi, Saeed B.S., University of Tehran, 2012

M.S., The State University of New York at Buffalo, 2014

Microgrids: Resilience, Reliability, and Market Structure

Advisor: Mohammad Khodayar

Doctor of Philosophy degree conferred May 19, 2018

Dissertation completed April 3, 2018

This dissertation addresses several challenges corresponding to the operation and planning of

microgrids in distribution networks including the formation, energy supply resilience, the contribution of

microgrids in the short-term and mid-term operation of bulk power networks and the computational

challenges in operation of interdependent infrastructure systems.

An approach to transforming the active distribution network with distributed energy resources

into multiple autonomous microgrids is presented in the first section. The distribution network consists

of several generation resources and demand entities that could be clustered into autonomous microgrids.

Eigen decomposition in the graph spectra of the distribution network is leveraged to determine the

boundaries for microgrids and a mixed-integer programming problem that minimizes the expansion cost

within microgrids. Once microgrids are formed, in the next section, a framework is proposed to identify

the vulnerable components, and ensure the resilient operation of coordinated electricity and natural gas

infrastructures considering deliberate disruptions. The microgrid demands, which consist of electricity

and heat demands, are served by the interdependent electricity and natural gas supplies. The proposed

approach addressed the vulnerability of multiple energy carrier microgrids to deliberate disruptions, in

order to apply preventive reinforcements to improve the resilience of energy supply. The proposed

methodology is formulated as a bi-level optimization problem to address the optimal and secure

operation of multiple energy carrier microgrids.

To investigate the contribution of microgrids in the bulk power network, in the next section, a

hierarchical structure for the electricity market is proposed to facilitate the coordination of energy

markets in distribution and transmission networks. The proposed market structure facilitates the

viii

integration of microgrids in the electricity markets to provide energy and ancillary services. In the

proposed hierarchical structure, microgrids participate in the energy market at the distribution networks

settled by the distribution network operator, and load aggregators interact with microgrids and

generation companies to import/export energy to/from the distribution network electricity markets

from/to the wholesale electricity market. Furthermore, the impact of microgrids in the mid-term

operation of bulk power systems is investigated. A new framework for risk-averse mid-term generation

maintenance scheduling in the power systems is presented. Microgrid aggregators facilitate the

participation of microgrids in the wholesale market. The effect of microgrids as controllable demand

entities on the generation maintenance scheduling practices in the power system is investigated. The

uncertainties in the marginal cost of generation in microgrids, the generation capacity installed within

the microgrids, and the system electricity demand are captured using respective nominal values and

uncertainty intervals. Moreover, the contingencies in transmission network are addressed by introducing

additional variables. A two-stage robust optimization problem is formulated to determine a trade-off

between the performance and conservativeness of the procured solution in the long-term operation

horizon.

The last section of this dissertation addresses the coordinated operation of interdependent

electricity and natural gas infrastructure systems to improve the security and reliability measures in both

infrastructure systems and mitigate the risk of demand curtailment. The electricity and natural gas

network operation problems are non-convex mixed-integer nonlinear programming problems that are

hard to solve in polynomial time. The non-convex feasible region is formed by the Weymouth constraint

and the introduced binary commitment decision variables in the natural gas and electricity network

operation problems respectively. A sparse semidefinite programming (SDP) relaxation is utilized to

procure the optimal solution for the coordinated operation of electricity and natural gas networks. The

presented algorithm leverages the sparseness of the natural gas network to construct several small

matrices of lifting variables that are used to form a tight and traceable SDP relaxation. A set of valid

constraints that tighten the relaxation ensures the exactness of the solution procured from the relaxed

problem.

viii

TABLE OF CONTENTS

ACKNOWLEDGMENTS ..............................................................................................................................vi

TABLE OF CONTENTS ............................................................................................................................ viii

LIST OF FIGURES ........................................................................................................................................xi

LIST OF TABLES ...................................................................................................................................... xiii

LIST OF SYMBOLS .................................................................................................................................... xv

1 INTRODUCTION ........................................................................................................................................ 2

1.1 Expansion of Autonomous Microgrids in Active Distribution Networks ........................................... 2

1.2 Resilient Operation of Multiple Energy Carrier Microgrids ............................................................... 4

1.3 A Hierarchical Electricity Market Structure for the Smart Grid Paradigm ......................................... 6

1.4 Risk Averse Generation Maintenance Scheduling with Microgrid Aggregators .............................. 10

1.5 Coordinated Operation of Electricity and Natural Gas Systems: A Convex Relaxation Approach .. 12

2 EXPANSION OF AUTONOMOUS MICROGRIDS IN ACTIVE DISTRIBUTION NETWORKS ....... 14

2.1 Graph spectra and Fiedler theory ...................................................................................................... 14

2.2 Problem Formulation ........................................................................................................................ 16

2.2.1 Network topology problem (Upper-level problem) .................................................... 16

2.2.2 Reliability evaluation problem (Lower-level problem) .............................................. 18

2.2.3 Microgrid topology validation problem ..................................................................... 20

2.2.4 Solution Methodology ................................................................................................ 21

2.3 Case Study ........................................................................................................................................ 24

ix

2.3.1 A Sample Meshed Distribution System ...................................................................... 24

2.3.2 The Modified IEEE 123-bus system .......................................................................... 30

2.4 Summary ........................................................................................................................................... 34

3 RESILIENT OPERATION OF MULTIPLE ENERGY CARRIER MICROGRIDS ................................ 35

3.1 Energy Hub ....................................................................................................................................... 36

3.2 Preventive Reinforcement in Cyber Infrastructure ........................................................................... 37


3.4 Case Study ........................................................................................................................................ 45

3.4.1 Case 1: No preventive reinforcement in the microgrid ............................................... 48

3.4.2 Case 2: First Stage of Preventive Reinforcement ....................................................... 49

3.4.3 Case 3: Second Stage of Preventive Reinforcement ................................................... 51

3.4.4 Third Stage of Preventive Reinforcement .................................................................. 52

3.4.5 Preventive reinforcement procedure ........................................................................... 52

3.5 Summary ........................................................................................................................................... 55

4 A HIERARCHICAL ELECTRICITY MARKET STRUCTURE FOR THE SMART GRID PARADIGM

....................................................................................................................................................................... 57


4.1.1 GENCO’s objective .................................................................................................... 58

4.1.2 Microgrid’s objective ................................................................................................. 58

4.1.3 LA’s objective ............................................................................................................ 59

4.1.4 The WEM clearing process ........................................................................................ 60

x

4.1.5 The DNEM clearing process ...................................................................................... 60

4.2 Multi-level complete information game ........................................................................................... 61

4.3 Case study ......................................................................................................................................... 68

4.3.1 Case 1 ......................................................................................................................... 69

4.3.2 Case 2 ......................................................................................................................... 72

4.3.3 Case 3 ......................................................................................................................... 75

4.4 Summary ........................................................................................................................................... 77

5 RISK AVERSE GENERATION MAINTENANCE SCHEDULING WITH MICROGRID

AGGREGATORS ......................................................................................................................................... 79


5.2 Solution Methodology ...................................................................................................................... 84

5.3 Case Study ........................................................................................................................................ 88

5.3.1 6-Bus Power System .................................................................................................. 88

5.3.2 IEEE 118-bus system ................................................................................................. 94

5.4 Summary ........................................................................................................................................... 97

6 COORDINATED OPERATION OF ELECTRICITY AND NATURAL GAS SYSTEMS: A CONVEX

RELAXATION APPROACH ....................................................................................................................... 99


6.1.1 Short-term Operation of Natural Gas Network .......................................................... 99

6.1.2 Short-term Operation of Electricity Network ........................................................... 101

6.2 Proposed Convex Relaxation .......................................................................................................... 102

xi

6.2.1 Background .............................................................................................................. 102

6.2.2 Convex Relaxation of Natural Gas Network Operation Problem ............................. 103

6.2.3 Convex Relaxation of the Electricity Network Operation Problem ......................... 109

6.2.4 Coordination among Electricity and Natural Gas Networks Operation ................... 111

6.3 Case Study ...................................................................................................................................... 112

6.3.1 Scenario 1 – Uncoordinated operation of the electricity and natural gas networks .. 114

6.3.2 Scenario 2 – Coordinated operation of the electricity and natural gas networks ...... 115

6.3.3 Scenario 3 – Scenario 2 with Congestion in the Electricity Network ....................... 117

6.3.4 Scenario 4 – Scenario 3 with Congestion in the Natural Gas Network .................... 118

6.3.5 Discussion ................................................................................................................ 119

6.3.6 IEEE 118-Bus System with 12-junction Natural Gas network ................................. 122

6.4 Summary ......................................................................................................................................... 125

7 SUMMARY ............................................................................................................................................. 126

8 APPENDIX .............................................................................................................................................. 128

BIBLIOGRAPHY ....................................................................................................................................... 130

xi

LIST OF FIGURES

Figure Page

1-1 The hierarchical electricity market structure. ............................................................................................ 8

2-1 The sample connected graph (a). The sample sectioned graph (b). ........................................................ 15

2-2 The spectra of the sample graph. ............................................................................................................ 16

2-3 The proposed electricity distribution network......................................................................................... 27

2-4 Electricity distribution network Case 1 ................................................................................................... 29

2-5 The spectra of the graph associated with distribution network ............................................................... 29

2-6 Electricity distribution network in Case 2 ............................................................................................... 30

2-7 Modified IEEE 123-bus test system ........................................................................................................ 34

3-1 Energy hub with electricity, natural gas and heat ................................................................................... 36

3-2 Disruption of service in multiple energy carrier microgrids ................................................................... 38

3-3 Hyper plane for the Weymouth equation ................................................................................................ 45

3-4 Multiple energy carrier microgrid. .......................................................................................................... 46

3-5 Microgrid operation condition in Case 1 ................................................................................................. 49



3-8 Operation and reinforcement costs of multi-energy carrier microgrid. ................................................... 55

4-1 Competition in the hierarchical electricity market .................................................................................. 66

4-2 Power network topology ......................................................................................................................... 69

4-3 Bidding strategies of market participants, exchanged power, and energy price in Case ......................... 71

4-4 Market clearing results in Case 1 ............................................................................................................ 72

xii

4-5 Market clearing result in Case 2.1 ........................................................................................................... 73

4-6 Market clearing results in Case 2.2 ......................................................................................................... 74

4-7 Market clearing results in Case 3 ............................................................................................................ 76

4-8 Bidding strategies of market participants and energy prices in Case 3 ................................................... 77

5-1 Demand profile for one year ................................................................................................................... 89

5-2 Scheduled capacity on outage in Case 1 ................................................................................................. 90

5-3 Scheduled capacity on maintenance in Case 1 with the increase in the transmission line capacity ........ 90

5-4 Demand profile with and without microgrids ......................................................................................... 91

5-5 Increase in the total operation and maintenance cost because of the increase in the budget of uncertainty

....................................................................................................................................................................... 92

5-6 Demand profile for IEEE 118-bus power system ................................................................................... 94

5-7 Marginal cost of the local generation for the aggregated microgrids ...................................................... 95

5-8 Sensitivity of the total operation and maintenance cost to the installed generation capacity and marginal

cost multiplier (MCM) of microgrids ............................................................................................................ 96

6-1 The algorithm to procure an optimal solution for the natural gas network operation problem. ............ 104

6-2 The 6 bus-system interconnected with 7-junction natural gas .............................................................. 113

6-3 The normalized demand profile for electricity and natural gas ............................................................. 113

6-4 The generation dispatch of G1 in scenarios .......................................................................................... 121

6-5 The generation dispatch of G2 in scenarios .......................................................................................... 121

6-6 The generation dispatch of T1 in scenarios ........................................................................................... 121

6-7 The flow of natural gas in pipeline PL3 in scenarios 3-4. ..................................................................... 122

6-8 The profile for the natural gas volume withdrawn from sources in scenarios 3-4. ............................... 122

6-9 12-junction natural gas network ............................................................................................................ 123

xiii

LIST OF TABLES

Table Page

2-1 Power generation unit characteristics ...................................................................................................... 25

2-2 Required LOEP at each bus .................................................................................................................... 26

2-3 Distribution line characteristics ............................................................................................................... 26

2-4 The LOEP in distribution network with no DER and microgrid ............................................................. 26

2-5 The LOEP of demand at each bus in case 1 ............................................................................................ 28

2-6 The LOEP of demand at each bus in Case 2 ........................................................................................... 30

2-7 Required LOEP of demands in IEEE-123 bus system ............................................................................ 33

3-1 Natural gas power generation unit characteristics ................................................................................... 46

3-2 Electricity and heat demand .................................................................................................................... 46

3-3 Distribution network characteristics (electricity and natural gas) ........................................................... 47

3-4 Outcomes of preventive reinforcement procedure .................................................................................. 54

4-1 Transmission and distribution network data ........................................................................................... 68

4-2 Generation cost curve of microgrids and GENCOs ................................................................................ 68

4-3 Bidding strategy and awarded dispatch in Case 1 ................................................................................... 72

4-4 Bidding strategy and awarded dispatch in case 2 (L1 is congested) ....................................................... 73

4-5 Bidding strategy and awarded dispatch in case 2 (L4 is congested) ....................................................... 74

4-6 Bidding strategy and awarded dispatch in case 3 .................................................................................... 75

5-1 Transmission line characteristics ............................................................................................................ 89

5-2 Thermal units characteristics................................................................................................................... 89

xiv

5-3 Scheduled maintenance in weeks for cases 1-4....................................................................................... 96

5-4 The periods of transmission network contingencies for cases 1-4 .......................................................... 96

6-1 Generation unit charcteristics ................................................................................................................ 114

6-2 Transmission line characteristics .......................................................................................................... 114

6-3 Natural gas resources characteristics..................................................................................................... 114

6-4 Natural gas pipeline characteristics ....................................................................................................... 114

6-5 Hourly unit commitment in Scenario 1 ................................................................................................. 115

6-6 Natural gas pressures in scenario 1 at hour 17 ...................................................................................... 115

6-7 Hourly unit commitment in Scenario 2 ................................................................................................. 116

6-8 Natural gas pressures in scenario 2 at hour 17 ...................................................................................... 116

6-9 The ratio of the two largest eigenvalues in scenario 2 at hour 17 ......................................................... 117

6-10 Natural gas pressures in scenario 3 at hour 17 .................................................................................... 117


6-12 The ratio of the two largest eigenvalues in scenario 3 at hour 17 ....................................................... 118


6-14 The ratio of the two largest eigenvalues in scenario 4 at hour 17 ....................................................... 119

6-15 Hourly unit commitment in Scenario 1 ............................................................................................... 124

6-16 Changes in the hourly commitment in scenario 2 from scenario 1 ..................................................... 125

xv

LIST OF SYMBOLS

sjA Natural gas junction-resource incidence matrix

bB Set of units that are connected to bus b

(.),(.)B Imaginary part of microgrid admittance matrix

,o oj lB B Set of buses o corresponds to the market participants j and l

,j ob Susceptance of the branch between energy hub j and o

pC Pipeline constant

(.) _(.)C Coupling factors in energy conversion matrix

,t sgc Price of electricity at time t in scenario s

,,

g mb t

C Uncertain variable of marginal cost of segment m of aggregated microgrid on bus b at

time t

,0,

gb t

C Marginal cost of the microgrid on bus b at time t

siC Marginal cost for segment s of the cost curve for unit i

E Budget of uncertainty

bEENS Expected energy not supplied at bus b

,b bF F Minimum/maximum limits for power flow in branch b

,c iF Production cost function for thermal unit i

xvi

c,sF Natural gas supply volume cost at source s

,pj of Natural gas flow in pipeline p between energy hubs j and o

,l tf Power flow of the transmission line l at time t

(.),(.)G

Real part of microgrid admittance matrix

jGG Set of natural gas-fired units connected to junction j

bGG Set of units that are connected to bus b

jGS Set of natural gas resources connected to junction j

bGR Set of the utility grid connection to bus b

, ,,j o j og y

Conductance and admittance of the distribution line between energy hub j and o

(.)iH Natural gas consumption function for natural gas generation unit i

jHH Set of non-electric natural gas demands connected to junction j

,i tHF The number of periods in which the unit i was on maintenance at the beginning of period

t

,i tHO The number of periods in which the unit i was available at the beginning of period t

I Identity matrix

,i tI Commitment decision of unit i at hour t ; 1 if the unit is ON and 0 otherwise

k The number of contingencies in each period

bLOEP Loss of energy probability at bus b

xvii

, ,,f j t jL L Set of distribution lines starting from /ending at energy hub j

L Total number of considered contingencies in the operation horizon

(.)M Required resources for interdiction of each component type in microgrid

M Total available resources for interdiction

iMC Maintenance cost of unit i

iMHO Maximum number of periods in which unit i is available

iNHO Minimum number of periods in which unit i is available

(.),d

jP Served demand in energy hub j

h

kP Heat energy produced by heater k

,Qe ei iP Real/reactive generation dispatch of unit i

,inj injj jP Q Real/reactive power injection at energy hub j

,i djP Natural gas consumption of units i connected to energy hub j

,, ,j o j oPL QL Real/reactive power flow between energy hubs j and o

, .,,

,Qd s d sb tb t

P Real/reactive demand served at bus b at time t in scenario s

, ,,Qt s t si iP Real/reactive generation dispatch of unit i at time t in scenario s

, ,,Qt s t sg gP Real/reactive power dispatch of main grid g at time t in scenario s

xviii

, ,, ,

,inj s inj s

b t b tP Q Real/reactive power injection at bus b at time t in scenario s

, ,,

, ,t s t sb b b b

PL QL Real/reactive power flow between buses b and b at time t in scenario s

max max,Qi iP Real/reactive generation capacity of unit i

, ,, ,

,QD s D sb t b t

P Real/reactive power demand at bus b at time t in scenario s

(.), ,,D e D

j jP Q Active/reactive electricity demand at energy hub j

,f jP Set of pipelines starting from energy hub j

,t jP Set of pipelines ending at energy hub j

max max,Qi iP Maximum real/reactive generation of unit i

jP

Generation dispatch of generator j [MW]

mP

Generation dispatch of microgrid m [MW]

exp imp,

l lP P

The power “exported from” / “imported to” DNEM by LA l [MW]

DP Total demand in the distribution network [MW]

DoP The demand on bus o in the WEM [MW]

exp exp,

l lP P

Minimum/maximum power “exported from” the DNEM by LA l [MW]

imp,

impl l

P P

Minimum/maximum power “imported to” the DNEM by LA l [MW]

(.) (.),P P

Minimum/maximum generation dispatch [MW]

xix

,,g m

b tP Generation of segment m of aggregated microgrid connected to bus b at period t

,i tP Generation dispatch of unit i at period t

,d

b tP Demand on bus b at time t

,0,d

b tP Demand on bus b at time t

,

h

k tP Non-electric natural gas demand k at hour t

,D

b tP Electricity demand in bus b at hour t

,f jP Set of pipelines starting from junction j

,t jP Set of pipelines ending at junction j

,Qe dj Served reactive electricity demand of energy hub j

jR Payoff function of GENCO j [$]

mR Payoff function of microgrid m [$]

,l lR R Payoff function of LA l in the DNEM/WEM [$]

iRHF Required number of periods for maintenance of unit i

iRU Maximum ramping up of unit i

iRD Maximum ramping down of unit i

,b br Resistance of the line between buses b and b

xx

,,

t sb b

SL

Apparent power flow between buses b and b at time t in scenario s

max,b bSL Maximum capacity of line connecting buses b and b

blSF Shift factor of line l with respective to bus b

o

bSF Shift factor of branch b with respect to injection at bus o

,l bS Element of transmission line-bus incidence matrix

,offon

i iT T Minimum up/down time of unit i

,b bu Decision variable for the connectivity between buses b and b

, ,,d db t b tu v Binary variables representing the uncertainties in demand

, ,,b t b tu v Binary variables representing the uncertainties in generation capacity of microgrid

, ,,

g gb t b t

u v Binary variables representing the uncertainties in marginal cost of microgrid

,l tu Binary variable representing the uncertainties in availability of components

,t siUX Generator’s outage status at time t in scenario s, 1 if available, otherwise 0

iUX Availability status of generator unit i , 1 if interdicted, otherwise 0

,,

t sb b

UY Distribution line’s outage status at time t in scenario s, 1 if available, otherwise 0

,lj oUY Availability status of transmission line l between energy hubs j and o , 1 if interdicted,

otherwise 0

,t sgUZ The main grid connection's outage status at time t in scenario s, 1 if available, otherwise 0

pUZ Availability status of natural gas pipeline p , 1 if interdicted, otherwise 0

xxi

max min,s sv v Max/min natural gas supply volume of resource s

sv Natural gas supply volume of resource s

,t sb

V Voltage magnitude at bus b at time t in scenario s

max minV ,V Maximum and minimum voltage magnitude

jV Voltage magnitude at energy hub j

(.)VOLL Value of lost load

, ,,j o j ox r Line reactance and resistance between energy hubs j and o

lX Inductive reactance of the transmission line l

,i tX Binary variable representing the transition to outage stage for maintenance of unit i at

period t

,oni tX On time of unit i at hour t

,offi tX Off time of unit i at hour t

,b by Admittance of the line between buses b and b

,i tY Binary variable representing the transition from outage to available mode for unit i at

period t

, ,i i i Fuel cost coefficients of generation unit i

,t sb

Voltage angle at bus b at time t in scenario s

xxii

s Probability of scenario s

t Duration of time period in load duration curve

.b b Auxiliary parameter

j Natural gas pressure at energy hub j

j Voltage angle at energy hub j

max min, Max/min natural gas pressure at each energy hub

max min, Max/min voltage angle at each energy hub

j Initial natural gas pressure at energy hub j

m lρ ,ρ Bidding vector of microgrid m/ LA l in the DNEM [$/MW]

,j l ρ ρ Bidding vector of GENCO j/ LA l in the WEM [$/MW]

m lψ ,ψ

Bidding strategy vector of microgrid m/LA l in the DNEM

j l ψ ,ψ

Bidding strategy vector of unit j of GENCO i/LA l in the WEM

τ ,φ Vector of slack variables

, , , θ π θ π Vector of Lagrangian multipliers

0 0, , ,b b Lagrange multipliers

(.) (.)ψ ,ψ

Upper/lower limit of the bidding strategy vector in the DNEM

xxiii

(.) (.)ψ ,ψ Upper/lower limit of the bidding strategy vector in the WEM

,b t Uncertain variable for generation capacity of microgrid on bus b at time t

(.) (.)(.) (.)

, Lagrangian Multipliers

0,b t Generation capacity of microgrid on bus b at time t

Convergence tolerance

,gb t

C Deviation in marginal cost of the microgrid on bus b at time t

,d

b tP Deviation of demand on bus b at time t

,b t Deviation of generation capacity of microgrid on bus b at time t

,pj o Maximum compression ratio for natural gas pipeline p

s Cost of natural gas supply at source s

, Minimum/Maximum natural gas pressure

,s sv v Max/min natural gas withdrawal of resource s

2

Chapter 1

1INTRODUCTION

1.1 Expansion of Autonomous Microgrids in Active Distribution Networks

Reliability of energy supply in distribution networks is dependent on the availability of the distribution

feeders, as they have radial topology [Bar89]. A fault or failure in the line or feeder within the radial

network will lead to demand curtailment. These failures are caused randomly by component outages in the

network [DOE10]. The increase in the installed capacity of DERs requires distributed and efficient control

of such resources, which is facilitated by forming microgrids [Nak13]. Microgrids are composed of DERs

and demands with distinct boundaries that are connected to the utility grid through the point of common

coupling (PCC) or operated in island mode [Iag13]. Forming microgrids can help to improve the restoration

capability of the distribution networks [Li14], [Gao16]. Islanding in microgrids reduces the adverse effects

of outages and power quality deterioration on local demands. Moreover, developing efficient control

schemes in microgrids facilitates the trade of electricity and ancillary services in distribution market

[Man16].

While increasing the number of microgrids promotes the service reliability, facilitates distributed control

over the generation and demand resources and reduces the investment cost to improve the reliability and

resilience; defining the boundaries for microgrids in distribution networks to serve the customers is a

challenging task. In this context, partitioning the electricity network into several sub-networks using

graphical and Eigenvalue sensitivity-based approaches is discussed in [Roy00]. The graph spectra are

assessed using a sensitivity-based approach in order to partition the network and analyze the dynamic

properties of the power system in the procured sub-networks. However, the base for network partitioning

does not capture the power flow constraints and the reliability of the power system.

While earlier research is focused on maintaining the supply adequacy in contingencies, chapter addresses

the supply adequacy capturing the reliability requirements of the demands and the quality of service at

demand buses is ensured by forming microgrids and expansion of the distribution network. As a

3

consequence, the developed microgrids provide heterogeneous energy supply reliability at demand buses

considering the outages in generation and distribution assets. Sectionalizing the distribution network to

provide boundaries for the DERs to serve the consumers will increase the reliability of energy supply in

contingencies as DERs are dispatched as backup generation resources in the distribution network.

Moreover, sectionalizing the distribution network to form microgrids will improve the robustness in the

control of generation and demand assets. In this context, DERs operate within the microgrids to ensure the

reliability of energy supply. A fundamental challenge in controlling DERs is the significant number of

control variables in the distribution network [Las11]. When DERs serve the customers in contingencies, a

failure in a complex control system can shut down the entire system considering the fact that the generation

resources and their inertia are limited in the distribution network. In order to mitigate the disturbance

propagation and reduce the vulnerability of the distribution networks to voltage sags, swells, faults, and the

uncertainties in demand and supply balance, microgrids are formed and islanded in emergency conditions.

Islanding improves the quality of energy supply during disturbances in the network. Minimizing the

number of generation, distribution and demand assets in each microgrid – to provide more robust control –

will maximize the number of microgrids formed in the network. However, the reliability requirement of the

demand will limit the size of the microgrids. Larger microgrids with more DERs provide higher service

reliability. Moreover, expanding the distribution network within a microgrid improves the service

reliability. Therefore, the size of the microgrids is determined by a trade-off between the controllability of

microgrids and the reliability of service and network expansion within microgrids improve the reliability

further.

Service restoration and reconfiguration performed by distribution management systems ensure the

reliability of energy supply for the consumers; however, managing large-scale distribution automation

systems with large number of DERs, switches, and controllable demands is a challenging task. Developing

a robust real-time energy management framework with a large number of control variables is

overwhelming for a central controller in the distribution network. Moreover, distributed energy

management solutions to control the energy flow between the interconnected microgrids may not provide

robust and reliable solutions [Saa12]. Hence, sectionalizing the distribution networks, and islanding the

4

microgrids will reduce the control processing burden and mitigates the disturbance and uncertainty

propagation.

Allocating the generated energy to demand buses in each microgrid based on the ownership of the

generation assets could be another criterion to determine the PCCs in the distribution networks. DER owner

may offer to sell electricity to nearby demand buses within the microgrid [Rah07]. The distribution network

operator or distribution company may own the DERs in the system [Imi15]-[Act15]. In such cases, the

presented formulation could be used to assign certain consumers to the DERs and form islanded microgrids

in contingencies within the distribution network.

1.2 Resilient Operation of Multiple Energy Carrier Microgrids

The increase in penetration level of distributed energy resources (DERs) supplied by natural gas in

electricity networks warrants the coordinated operation of electricity and natural gas distribution networks.

The rising concerns over the reliability and quality of service in energy distribution networks promotes the

concept of multiple energy carrier microgrids. Multiple energy carrier microgrid is referred to as an

interconnected electricity and natural gas distribution network in which the electrical network is composed

of a group of interconnected electrical demands and DERs that represent a single controllable entity within

the electricity grid with the ability to operate in grid-connected and island modes. The natural gas

distribution network within the microgrid is composed of several source points which are connected to the

gas distribution network and a group of load points which represent electrical generation units or heat

demands. Several literatures have addressed the interdependence of electricity and natural gas

infrastructures in power systems. The model in [Que07] presents a fundamental understanding of the

interdependent energy systems including coal, natural gas and electricity while ignoring the network

constraints. The interdependence of natural gas and electricity infrastructures is discussed in [Sha05],

where the optimal operation of the interconnected natural gas-electricity network is affected by several

factors including, physical characteristics of natural gas pipeline and electricity networks, operational

procedures in electrical and natural gas distribution networks, types of electricity generating plants,

availability of natural gas and electricity supply, transmission and delivery constraints in natural gas and

electricity networks, and their respective volatile market prices. In [Li08], switching between fuel supplies

5

in electricity generation is proposed as an effective approach to perform peak shaving on natural gas

demand while enhancing the power system operation security. Several factors including the natural gas

market price, natural gas pipeline pressure loss, and simultaneous demand peak in electricity and natural

gas which causes price spikes in severe weather conditions and possible outages in natural gas pipelines;

affect the power system operation by increasing congestions, price of electricity or even demand

curtailment in severe conditions. A security-constrained short term generation scheduling in electrical

networks is developed in [Liu09]. The slow transient process in natural gas transmission networks was

considered by a set of partial differential and algebraic equations in [Liu11]. The proposed approaches

addressed the effects of natural gas transmission network on power system security by implementing

determined scenarios for possible contingencies in both energy infrastructures.

A framework for the comprehensive modeling of multiple energy delivery systems in which energy is

converted within an energy hub is presented in [Kra11]. Energy hub provides the features of input, output,

storage, and conversion of multiple energy carriers. An integrated optimization framework which

incorporates a simplified model for the network flow of the interdependent energy carriers is proposed in

[Gei07]. An approach for optimal scheduling of multiple energy networks including electricity, natural gas,

and district heat is introduced in [Gei06]. A linear formulation is proposed in [Alm11] to solve the large-

scale optimal energy flow in multiple energy delivery networks. An expansion planning framework within

the electricity and natural gas distribution networks with high penetration of gas-fired distributed

generation is presented in [Sal13], which incorporates heuristic methodologies to find the optimal

expansion planning strategies. Discussion in [Uns10] expressed the savings gained by optimal operation of

integrated electricity and natural gas networks. In [Awa09], an integrated formulation of electricity and

heat distribution networks to procure the optimal electricity and heat flow within a microgrid is presented.

Chapter 3 addressed the coupling between electricity and natural gas energy infrastructures within the

microgrids which can affect the resilience of energy supply. The resilience of energy infrastructure is

addressed by identifying vulnerable components in electricity and natural gas infrastructures. Deliberate

interruptions do not occur frequently, but, when they do, they can be tragic. Quantitative probabilistic

evaluation of such low-probability-high-consequence contingencies is very challenging, resource-

6

demanding, and subjected to inaccuracies. Moreover, the outcome of a disruption strongly depends on the

system operation conditions prior to the attack, as well as the available resources to the offenders, both of

which are also highly uncertain. There are a number of publications addressed the contingencies as a

consequence of threat in power systems [Tep12],[Sal04],[Bie07][Rom12],[Yao07]. Contingencies as a

result of natural disasters, physical or cyber-attacks would extinguish the key elements of the electricity

infrastructure and may lead to blackouts. The disruptions of energy supply in electric power grids caused

by physical attacks are addressed in [Sal04] which is presented as a bi-level mixed integer programming

problem and solved by a heuristic approach. A model to protect the power network against a range of

scenarios which lead to disruption in energy supply and blackouts is developed in [Bie07]. The three-level

attacker-defender-planner model presented in [Rom12] as a multi-level mixed integer programming

problem finds the optimal defense strategy in power system. Here, the interdependence between energy

networks was pointed out as a potential improvement in finding the more realistic defense strategies. In

[Yao07], a tri-level optimization model is proposed for power network defense which is solved iteratively

by forming nested bi-level optimization problem with budget constraints associated with defender and

attacker assuming that the defended components are infallible once they are hardened. The physical

disruption, modeled as a mixed-integer bi-level programming (MIBLP) problem in [Mot05] ignores the

interdependence of electricity and natural gas infrastructures. Here, the power system under attack is

represented by dc power flow approximation, without offering any preventive guidelines for reinforcing the

system. Other publications emphasized the importance of considering the interdependencies between

natural gas and electricity to improve resilience and reliability measures without quantifying the outcomes

[Zim09],[Rin04]. Since microgrids are developed to provide higher power quality and reliability for

consumers, the resilience of energy supply is an important concern within microgrids.

1.3 A Hierarchical Electricity Market Structure for the Smart Grid Paradigm

Developing the hierarchal market structure composed of generation entities and demand aggregators

facilitates the participation of small demand entities and end customers in the electricity market. With the

advent of active distribution networks with distributed energy resources, responsive demands, and

intelligent controls, there is a concern on the increased number of intelligent supply and demand entities in

7

the distribution network that affects the wholesale electricity market operation. Scheduling of such entities

by ISO will increase the scale of the day-ahead and real-time scheduling problem and corresponding

computational burden. Moreover, the competition among these entities are not fully addressed as a result

technical limitations and lack of effective business model in this paradigm. Chapter 4 proposed a

framework to facilitate competition among larger number of market participants in the WEM and the

DNEM. The presented framework is further extendable to support the participation of microgrids in the

WEMs. LAs were considered as intermediate agents between the utility operator and consumers to perform

Demand Response (DR) in [Gka13],[Par13],[Zug13]. While the proposed structures addressed the

interaction between the customers and aggregators who respond to the price of electricity, the pricing

scheme does not reflect the synergy between the utility grid, aggregator, and the customers. As microgrids

are deployed in low or medium voltage distribution networks, they exchange energy with the main grid

through an aggregator which represents the middle agent who interacts with the microgrid and the WEM.

Distribution Companies (DISCOs) participate in the WEM and the DNEM and interact with GENCOs,

distributed generations (DG) and interruptible loads (IL). DISCOs bid for energy and ancillary services in

the WEMs [Mas11]. DGs and ILs controlled by DISCOs will benefit from participating in electricity

markets while DISCOs are considered as agents with no financial benefits in [Hag12], [Doo12]. A multi-

period framework for participating DISCO in day-ahead electricity market is proposed in [Li07] using bi-

level optimization technique to maximize the DISCO’s profit while minimizing the operation cost of the

system. A framework to facilitate the participation of DISCOs in a day-ahead bi-lateral and pool energy

markets is presented in [Pal05]. In these publications, the interaction between DGs, ILs and DISCOs

through competition was not addressed. In [Akb11] the synergy between DISCOs and GENCOs is

considered in the WEM where DISCOs bid based on the available capacities of DGs and ILs, and GENCOs

bid based on the marginal cost of the generators owned. DISCO can also participate in the DNEM operated

by the DNO [Lop11]. Here the interaction between DISCOs and GENCOs in the WEM is not considered.

The synergy between DERs and demand resources within the microgrids with the energy service providers

(ESP), who participate in the energy market is addressed in [Asi13], while the interaction between the

WEM and local DNEM is ignored. Several research efforts presented frameworks for the participation of

microgrids and DISCOs in a competitive electricity market by a two-stage hierarchical optimization

8

framework for the day-ahead and real-time electricity markets [Kim11],[Alg09],[Saf13]. However, the

impact of the microgrids on the locational marginal prices (LMPs) of electricity in the WEM was not

addressed.

The synergy between the microgrids and the WEM to maximize the social welfare is presented in

[Zha14]. The electricity market is represented by randomize auction framework ignoring the role of

“microgrid aggregators” as a broker with respective bidding strategies. Direct participation of microgrids in

the WEM is not applicable for the following reasons: 1) technical limitations on voltage levels, generation

and demand capacity, 2) lack of an efficient business model and technical infrastructure to provide

microgrids with access to Open Access Same-Time Information System (OASIS) and Open Access Non-

discriminatory Transmission Services, 3) shortage in an efficient framework to evaluate the potential

services provided by microgrids in the WEM, and 4) increase in the number of microgrids, with diverse

capacities and coverage areas; and the technical restraints on transmitting electricity to the bulk power

network including congestion and voltage constraints [Sha12], [Koe04].

A system of system (SoS) framework is proposed in [Kar14] to address the interaction between the

Figure 1-1 The hierarchical electricity market structure.

Wholesale Electricity Market (WEM)

GENCO j

LA l

MG m

Distribution Network Electricity Market (DNEM)

9

DISCOs and microgrids as independent systems, which exchange energy. The proposed approach

addressed the optimal operation of the distribution network and does not consider any interaction between

DGs and DISCOs to maximize the agent’s payoff in the WEM or the DNEM. In the coordinated energy

management structure presented in [Wan15], microgrids and the DNO exchange electricity to minimize

their operation cost, while the DNO exchanges electricity with the WEM. However, the electricity prices

are considered as parameters in the coordinated energy management structure. Hence, no mechanism is

proposed to represent the simultaneous active participation of LAs in the DNEM and the WEM. In chapter

4, LAs are considered as brokers who participate in the DNEM, which is cleared by the DNO as well as the

WEM settled by the ISO. Introducing LAs in the proposed framework reduces the number of market

participants in the WEM and promotes the competition among participants in the WEM and the DNEM.

Here DGs and ILs are merged to form microgrids and the new market paradigm facilitates the participation

of multiple microgrids in the WEM. In the proposed hierarchical market structure, each market participant

will categorize the unknown strategies and information correspond to other market participants by realizing

several “types”, which eventually would transform the incomplete information game into a complete

information game with imperfect information using the joint probability distribution function to address the

uncertainties associate with the “types” of market participants. In this chapter, the complete information

game between the market participants is presented and the incomplete information game is considered as an

extension to the proposed approach. The main contributions of this chapter are summarized as follows: 1)

The synergies between the WEM and the DNEM are presented. In this paradigm, each LA is an

intermediate agent that participates in the WEM and the DNEM. Hence the strategy chosen in one market

would impact the strategy taken in the other. 2) The proposed hierarchical electricity market structure

provides the required infrastructure for microgrids to participate in energy market in the distribution and

transmission networks, without increasing the computational burden on the ISO associated with the

increased number of market participants. 3) The proposed hierarchical market structure is an application of

dynamic game to facilitate the synergy between multiple electricity markets, where the WEM interacts with

the DNEM [Bas95]-[Rom97]. Figure 1 shows the proposed hierarchical market structure that composed of

these markets. Here, each market participant bids in associated electricity market, while each LA is an

intermediate agent that participates in both markets to address the synergy of the two markets. The LA bids

10

in the WEM and presents the awarded dispatch in the DNEM as demand on its bus in the WEM. The ISO

clears the market and provide the awarded dispatch for the LA and LMP at the LA’s bus. The LA bids in

the DNEM and presents the awarded dispatch in the WEM as demand on its bus in the DNEM. The DNO

clears the market and provide the awarded dispatch for the LA and the market clearing price (MCP) of the

market.

1.4 Risk Averse Generation Maintenance Scheduling with Microgrid Aggregators

Generation maintenance scheduling determines the most effective periods for planned generation

outages in order to maintain the reserve capacity margin and avoid costly demand curtailments in the power

systems. Such practices could be coordinated with transmission maintenance scheduling [Fu07]. The long-

term generation and transmission maintenance scheduling is coordinated with short-term generation

scheduling in [Fu09] to improve the security of the power system. The coordinated generation and

transmission maintenance scheduling that captures the degradation of generation and transmission assets

and equipment malfunction because of loading or ambient temperature and weather condition is addressed

in [Wu10]. The uncertainties in the long-term and short-term operation including the random outages of the

generation and transmission units, load forecast errors and fuel price fluctuations are captured by the

proposed stochastic model for the coordinated generation and transmission maintenance scheduling in

power systems in [Wan16a]. In [Ji16] a risk-averse approach for generation maintenance scheduling in

power systems with high penetration of renewable energy resources is proposed. In [Wan16b] the

generation maintenance scheduling is coordinated with transmission maintenance practices considering the

N-1 contingencies in the generation and transmission components. Other factors including renewable

resources and demand response practices were addressed in the generation maintenance scheduling in

[Mol17] and [Wu08a] respectively. While earlier publications were focused on procuring the outage

schedule of the generation and transmission components, the impact of demand-side management and load

control on the generation maintenance scheduling were not addressed. The demand realized by bulk power

systems could be regulated using the controllable generation assets in the distribution networks to provide

improved solutions for maintenance scheduling while maintaining the security and reliability of the bulk

11

power system. In this context, microgrids are among the most viable solutions to regulate the demand in the

power systems. Microgrids are considered as autonomous electric power systems with defined boundaries,

local demand, generation, and/or storage facilities that can operate in grid-connected or island mode [Mic].

Regulating the demand by leveraging local generation assets in microgrids improves the economics and

security measures of the generation maintenance scheduling practices. However, the uncertainty inherently

exists in the available generation capacity in the microgrids, the marginal price of generating electricity and

the demand that should be partly or completely served by bulk power system. Furthermore, the possible

contingencies in transmission network will affect the power flow and the maintenance scheduling of

generation units. Such uncertainties should be captured in the generation maintenance scheduling problem

as they may result in economic inefficiencies and further jeopardize the long-term security of the power

system. The uncertainties in power systems are captured using stochastic programming approach that

leverages the probability distribution of the uncertain variables such as demand, renewable generation, and

availability of the system components [Ozt05],[Wu07],[Wan08],[Rui09],[Tou09][Wan12]. Such approach

provides improved performance over the deterministic solutions, however, its computation burden will

increase with the increase in the number of incorporated scenarios. Furthermore, the probability distribution

functions for the uncertain variables may not be readily available. In order to address these challenges,

robust optimization (RO) is introduced and applied to several engineering problems [Ben09],[ Ber11]. The

solution to RO problems provides risk-averse strategies by capturing the uncertainty boundaries without

requiring the probability distribution function of the uncertain variables. Such solutions are favorable for

long-term operation planning of power systems including maintenance scheduling as the security of the

power system is maintained considering the worst realization of the uncertain variables. Furthermore,

earlier publications [Fu07], [Fu09], [Wan16a], [Wu08b] addressed maintenance as a one-time practice in

the operation horizon, while the minimum and maximum periods among sequential maintenance practices

were ignored. Improving the mathematical model for the maintenance scheduling by addressing such

limitations enables multiple maintenance practices for the generation units in the longer operation horizon.

12

1.5 Coordinated Operation of Electricity and Natural Gas Systems: A Convex Relaxation

Approach

Several research efforts addressed the interdependence among natural gas and electricity networks

[Sha14],[FMO15]. In the electricity network, the natural gas generation units are fast response units that

compensate for the generation scarcity that is triggered by sudden changes in the renewable generation

dispatch or the electricity demand. The essence of capturing natural gas constraints in bulk-power and

distribution network operation is addressed in [Sha05] and [Gei07] respectively. Some research works

ignored the network model for the electricity or natural gas system and addressed the economic and

environmental objectives/constraints by simply balancing the demand and supply using single-node

approach [Mar16]-[Bah16]. Other research works highlighted the challenges associated with the non-

convexity of the network models [Gei07] by presenting a linear representation [Li08], a multi-dimensional

piecewise linear approximation [Sha17], adaptive partitioning [Wu17], or successive linear formulation

[Abd15] to convexify the network constraints and to determine an equilibrium point [Mar12] for the overall

energy flow. These approaches require a large number of piecewise estimates and a considerable number of

iterations for accurate piecewise linear approximation, while there is no guarantee for the exact solution. In

[Liu09], a security-constrained unit commitment problem is solved using Benders decomposition approach

in which the constraints of the natural gas network are incorporated in the sub-problem. The methodology

proposed in [Liu09] to solve the nonlinear and nonconvex natural gas operation sub-problem is an iterative

procedure with a predefined tolerance for error, and Benders decomposition is used to ensure the feasibility

of the natural gas network constraints. Other research efforts leveraged heuristic approaches such as

genetic algorithm [Moe14] to solve for the energy flows in the interdependent electricity and natural gas

networks. Similar convexification, linearization [Uns10], and heuristic approaches [Mar13] were used for

the expansion planning practices in the electricity and natural gas networks. Several scenarios for

coordination among the electricity and natural gas networks are investigated in [Zlo17] to improve the

economic and security measures. An important notion which was ignored in earlier research is that the

electricity and natural gas systems are usually operated by different entities with heterogeneous objectives

[Qad14]. Therefore, the short-term and long-term operational planning for these interdependent

13

infrastructure systems should capture the limitations on the shared information among their respective

system operators [He17]. A consensus operation framework facilitates the coordinated operation of

electricity and natural gas networks. The benefits of the consensus-based distributed approach over the

Lagrange relaxation are presented in [Wen17]. In order to guarantee the convergence, the short-term

operation problem for each infrastructure system is required to be formulated as a convex optimization

problem [He17].

In the natural gas network, the flow in the pipelines is formulated by a nonconvex Weymouth constraint

and in order to convexify this constraint, the direction of natural gas flow in the pipeline is assumed as

known and unchanged [He17]. This assumption may not be valid as the direction of the natural gas flow in

the pipeline may change suddenly to supply the natural gas generation units and compensate for the

variability of renewable generation in electricity networks [Cor14],[Hib12]. The variability in the

generation dispatch of natural gas generation units is referred to the intra-day changes in the dispatch of

these units in response to hour-to-hour changes in system demand that cannot be served by the renewable

generation and other thermal generation technologies. Furthermore, despite knowing the direction of the

natural gas flow, the Weymouth constraint is nonlinear [San16]; therefore, linearization techniques were

used to formulate the electricity network operation problem with natural gas system constraints as a mixed

integer programming (MIP) problem with linear constraints [Li08],[Sha17],[Abd15].

14

Chapter 2

2EXPANSION OF AUTONOMOUS MICROGRIDS IN ACTIVE DISTRIBUTION NETWORKS

The objective of this chapter is to develop a framework for the expansion of the distribution network,

while maximizing the number of autonomous microgrids to maintain the reliability and security of the

energy supply. The procured PCCs determine the boundaries of the microgrids and disjoint the microgrids

from the distribution network in contingencies. The expansion of the distribution lines improves the

reliability of energy supply within microgrids. The main contributions of this chapter are summarized as

follows:

• A bi-level optimization problem is formulated to identify the PCCs by maximizing the number of

microgrids considering the reliability requirements of the customers.

• The expansion of distribution lines ensures the reliability and security of energy supply in the island

operation.

• Eigen decomposition of the Laplacian matrix of the distribution network graph is used to determine the

graph spectra and formulate the connectivity of the distribution network graph.

• The uncertainties in demand and the availability of the components within the microgrids are captured

by developing scenarios using the Monte Carlo simulation.

The rest of this chapter is organized as the follows. In Section 2.1, the application of Fielder theory to

obtain the graph spectra of the distribution network and reconfigure the distribution network to form

microgrids is presented. The problem formulation for the expansion of the distribution network to achieve

the required reliability is discussed in Section 2.2. The effectiveness of the proposed framework is

examined in multiple case studies in Section 2.3. Finally; the summary are presented in Section 2.4.

2.1 Graph spectra and Fiedler theory

In this section, the spectra of the network are utilized to develop a formulation for sectionalizing

the network [Fie75]. The distribution network is represented as a graph G = (V,E) as the buses and lines

15

are presented by vertices 𝐕 ∈ ℕ𝑁𝐵 and edges 𝐄 ∈ ℕ𝑁𝐿 respectively. Graph G has a corresponding

Laplacian matrix 𝐋 ∈ ℤ𝑁𝐵 × ℤ𝑁𝐵in which the diagonal arrays are the total number of edges connecting to

the vertices and the off-diagonal arrays represent the number of edges connecting vertices multiplied by ”

1 ”. The Laplacian matrix of a graph can be decomposed into its eigenvalues 𝜆1 ≤ 𝜆2 ≤ ⋯ ≤ 𝜆𝑁𝐵 as

shown by (1). Here, Q is the eigenvector matrix and Λ is the diagonal matrix of eigenvalues.

1)2.( T

L = QΛQ

A sample graph is represented in Figure 2-1, and the spectra of this graph which represent its

connectivity are given in Figure 2-2. For a connected undirected graph, the smallest eigenvalue of the

Laplacian matrix is always zero and the second smallest eigenvalue, also called Fiedler eigenvalue,

represents the algebraic connectivity of the graph [Fie73]. The number of zero eigenvalues is equal to the

number of isolated subgraphs in a graph [Fie75]. In the graph shown in Figure 2-1b, the first two

eigenvalues are zero as shown in Figure 2-2. One extreme case is the fully connected graph where the first

eigenvalue will be zero and other eigenvalues are equal to the number of vertices in the graph. The other

extreme case is the fully disconnected graph where all the eigenvalues are zero, so the number of sub-

graphs is equal to the number of vertices in the graph.

Figure 2-1 The sample connected graph (a). The sample sectioned graph (b).

4

5

6

1 2

3

4

5

6

1 2

3

(a) (b)

16

Figure 2-2 The spectra of the sample graph.

2.2 Problem Formulation

In this section, a new formulation is proposed to sectionalize the distribution network into

microgrids and the Fielder eigenvalue is used as a measure to determine the number of sub-networks in a

network. The problem is formulated as a bi-level mixed integer nonlinear problem (MINLP) [Bar98] in

which, the binary variables were associated with each edge of the distribution network graph i.e.

distribution lines. If the binary variable is 1 the distribution line remains connected, otherwise, the line will

be disconnected to form microgrid. The objective of the upper-level problem is to procure the maximum

number of microgrids while the lower-level problem ensures the reliability of the formed microgrids.

Therefore, in the upper-level problem, the distribution lines are disconnected to increase the number of

islanded microgrids in the case of a disturbance in distribution network; however, the objective of the

lower-level problem is to expand the distribution network within the formed microgrids to ensure that the

required level of reliability at demand buses is satisfied considering limited budget for such expansions.

Employing the duality theory, the presented bi-level problem is transformed into a single-level problem in

which the lower-level problem is considered as a constraint for the upper-level problem [Arr10]. The

expansion decisions are further validated to avoid over-investing on the distribution assets by minimizing

the cost of expansion plans considering the reliability constraints in the formed microgrids.

2.2.1 Network topology problem (Upper-level problem)

The upper-level problem minimizes the eigenvalues to maximize the number of microgrids in the

distribution network. The eigenvalue minimization problem has a convex feasible set and can be

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

1 2 3 4 5 6

Connected

Sectionned

Number of EigenvaluesE

igen

val

ue

17

represented as a semi-definite programming problem [Boy93]. In the upper-level problem (2.2)-(2.8), the

decision variables are the connection states of the distribution lines, which determine the PCCs in the

distribution network. In order to determine the decision variables, the eigenvalues of the Laplacian matrix

of the distribution network graph are minimized in an ascending order to ensure that 𝜆1 ≤ 𝜆2 ≤ ⋯ ≤ 𝜆𝑁𝐺 .

The sum of the weighted smallest eigenvalues – which are equal in number to the number of DERs – is

minimized in (2.2). In an undirected graph, the sum of eigenvalues of the Laplacian matrix of the graph is

equal to the sum of vertex degrees. In order to procure the maximum number of microgrids, the eigenvalues

of the Laplacian matrix of the distribution network graph should be zero in an ascending order. The first

eigenvalue is always zero. If the first two eigenvalues become zero, the distribution network will be divided

into two microgrids. Similarly, if the first three eigenvalues become zero, the distribution network will be

divided to three microgrids. Thus, the objective function of the upper-level problem is formulated to

procure zero eigenvalues in an ascending order, and the number of formed microgrids cannot exceed the

number of DERs. Here, the priority is to minimize the eigenvalues with lower orders. Minimizing the sum

of the eigenvalues does not guarantee that the eigenvalues will be zero in an ascending order. Therefore,

eigenvalues are weighted in the objective function of the upper-level problem.

Each eigenvalue is determined by multiplying the transposed matrix of eigenvectors, the Laplacian

matrix, and the eigenvector matrix as shown in (2.3). Thus, minimizing the eigenvalues will change the

Laplacian matrix, and consequently the topology of the distribution network. As indicated by (2.4), the

eigenvectors corresponding to each eigenvalue of the Laplacian matrix are orthogonal. The eigenvector

multiplication is a nonlinear but convex function. The eigenvector corresponding to each eigenvalue of the

Laplacian matrix are normalized to unit length as shown in (2.5). In (2.6) and (2.7), the relationship

between the Laplacian matrix of the distribution network and the decision variables associated with the

connectivity of distribution lines (edges of the associated graph) are given. Here, 𝑑𝑖𝑎𝑔(𝐿) represents the

diagonal elements of the Laplacian matrix 𝐿. In (2.8), the investment for adding the new distribution lines

is limited by the total investment budget (𝐵), and 𝑐𝑏,𝑏′ is the cost associated with adding a new

distribution line. Therefore, for the existing distribution lines, this parameter is set to zero. As the upper-

level problem minimizes the weighted eigenvalue spectra of the Laplacian matrix of the distribution

18

network graph – which also represents the connectivity of the distribution network graph – it will limit the

expansion of the distribution network as more distribution lines will increase the connectivity of the

network graph. The expansion of the distribution network is further limited by the available budget for

installing the distribution lines. It worth noting that other economic criteria such as ownership as well as the

PCC and controller costs were not incorporated in the presented formulation. Such criteria could be

addressed in the upper-level problem by adding constraints for the connectivity of certain vertices in the

distribution network graph. Moreover, the costs associated with PCC and controllers could be incorporated

in (2.8).

min ( 1)NG

bb

NB b (2.2)

1,....Tb b b b NB q Lq (2.3)

0 , 1,....Tb b b b b b NB q q (2.4)

1 , 1,....Tb b b b b b NB q q (2.5)

,( ) , 1,....b bb

diag u b b b b NB L (2.6)

, , 1,....bb b b bbl u l b b b b NB L (2.7)

, ,. , 1,....b b b bb

c u B b b b b NB (2.8)

2.2.2 Reliability evaluation problem (Lower-level problem)

The lower-level problem minimizes the expected operation cost of the autonomous microgrids while

maintaining the reliability requirements of the individual demands. The decision variables in the lower-

level problem (2.9)-(2.24) are the real and reactive power dispatch of the DERs and the demand served at

buses in the distribution network considering the determined topology in the upper-level problem. Here, the

decisions made on the connectivity of the distribution lines in the upper-level problem are constrained by

the reliability requirements at demand buses captured in the lower-level problem. This problem is presented

19

as linear programming (LP) problem. The objective function is the expected operation cost of the

autonomous microgrids which includes the operation cost of generating electricity and purchasing

electricity from the main grid, as well as the penalty associated with demand curtailment as shown in (2.9).

The real and reactive power injection at each bus of the distribution network is shown by (2.10) and (2.11),

respectively. At each bus, the available generation is the summation of the power generated by DER and

the imported power from the utility, if the bus is connected to the feeder. The real and reactive power

limitations of DERs are shown in (2.12) and (2.13) respectively. The expected energy not supplied (EENS)

for real and reactive demand at each bus is limited by (2.14) and (2.15) respectively. The EENS for the

demand at each bus is procured by multiplying the loss of energy probability (LOEP) and the total expected

energy consumed. As shown in (2.16), the admittance of distribution line incorporates the binary variable

representing its connectivity, which is determined by the upper-level problem, as well as the binary

parameter representing the line outage in different scenarios. The real and reactive power injection at each

bus is linearized in (2.17) and (2.18). The constraints (2.19), (2.20) and (2.21) present the linearized

formulation for real, reactive and apparent power transmitted through the distribution line [Kho12]. The

power flow in the distribution line is limited by the line capacity as given in (2.22). The voltage magnitude

and phase angle at each bus in the distribution network are limited by (2.23) and (2.24), respectively.

, ,

, , , , ,, , ,

,

( )mint s t s

i g

NS NTs t s t s t s D s d s

t c i g g bi b t b tP P s t i g

F P c P VOLL P P

(2.9)

s.t.

b b

inj ,st ,s t ,s d ,sg i b,t b,t

g GR i GG

t ,sg .P P P PUZ

(2.10)

b b

inj ,st ,s t ,s d ,sg i b,t b,t

g GR i GG

t ,sg .Q Q Q QUZ

(2.11)

, , ,min max .t s t s t s

i ii i iP UX P P UX (2.12)

, , ,min maxt s t s t si ii i iQ UX Q Q UX (2.13)

, ,, ,

0NT NS

s D s d st bb t b t

t s

P P EENS (2.14)

20

, ,, ,

0 d s D sb t b t

Q Q (2.15)

2 2 2 2

t ,s t ,sb,b b,b b,b b,bb,b b,bt ,s t ,s t ,s

b,b b,b b,bb,b b,b b,b b,b

r .UY .u x .UY .uy g Jb J

r x r x

(2.16)

12 1inj ,s

b,t

NBt,s t ,s t ,s t ,s t ,s t ,s t ,s t ,s

b b,b b bb,b b b,b bb b b

P BV G G V V

(2.17)

12 1inj ,sb.t

NBt,s t ,s t ,s t ,s t ,s t ,s t ,s t ,s

b b,b b bb,b b b,b bb b b

GQ V B B V V

(2.18)

, , , , , . ,, , ,

( ) ( )t s t s t s t s t s t s t sbbb b b b b b b b

GPL V V B (2.19)

, , , , , , ,, , ,

( ) ( )t s t s t s t s t s t s t sb bb b b b b b b b

QL B V V G (2.20)

, , ,,, , ,

t s t s t sb bb b b b b b

SL PL QL (2.21)

, max,,

t sb bb b

SL SL (2.22)

min , maxt sb

V V V (2.23)

min , maxt sb

(2.24)

2.2.3 Microgrid topology validation problem

Minimizing the sum of the eigenvalues that are equal to in number to the number of DERs will not

minimize the connectivity inside the formed microgrids. As the only constraint for limiting the number of

installed distribution lines is the limitation on the total budget, unnecessary distribution lines may be

installed. Installing the distribution lines will improve the reliability in the microgrids, however, such

decisions may not be economical once the reliability levels requested by demand entities are satisfied. To

address this issue, a microgrid topology validation problem is proposed to check if the added distribution

lines are necessary to satisfy the reliability requirements. Eliminating the unnecessary distribution lines

within the formed microgrids will not change the number of zero eigenvalues determined in the bi-level

optimization problem and consequently will not change the boundary of the formed microgrids. By

removing the unnecessary distribution lines within the microgrids, the sum of the non-zero eigenvalues is

21

decreased. Therefore, the algebraic connectivity is decreased within microgrids while the reliability

requirements are satisfied. The objective of the microgrid topology validation problem is given in (2.25), in

which the investment cost of adding new distribution lines and the operation cost of the system are

minimized. As this problem checks for the unnecessary installation of the distribution lines, the decision

variables in this problem are limited by the procured decisions (𝑢′𝑏,𝑏′) of the bi-level optimization problem

(2.2)-(2.24) as shown in (2.26). This means that this problem seeks to validate the decisions made in the bi-

level problem. The objective function for this problem is subjected to set of constraints (2.10)-(2.24).

, ,

,

, , , , ,, , ,, ,

, , '

( ) .mint s t s

i g b b

NS NTs t s t s t s D s d s

t c i g g b b b b bi b t b tP P u s t i g b b b

F P c P VOLL P P c u

(2.25)

b,b b,bu u b,b , b b (2.26)

2.2.4 Solution Methodology

The presented bi-level MINLP problem (2.2)-(2.24) is expressed in the general form by (2.27)-(2.32). By

employing the duality theory, the presented problem is transformed into a single-level problem (2.33)-

(2.40). The other approach for such transformation is to use the Karush Kuhn Tucker (KKT) conditions for

the lower-level problem; however, using the duality theory will generate less number of the binary-to-

continuous variable multiplications, which reduces the computation complexity and solution time. The

decision variables in the upper-level problem include the connectivity Ι and the eigenvalue and

eigenvectors η . Here, (2.27) minimizes the weighted eigenvalues in an ascending order as shown in (2.2).

The binary variables are the decision vectors representing the connectivity of distribution network

including the decisions on the location of PCCs and the installation of new distribution lines and the

continuous variables are the eigenvalue and eigenvectors. Set of constraints (2.28) represent the equality

constraints in the upper-level problem including the equality constraint from which the eigenvalues of the

Laplacian matrix are determined (2.3), the properties of the eigenvectors (2.4) and (2.5), and the

relationship between the connectivity of the distribution network graph and its Laplacian matrix (2.6),

(2.7). Constraint (2.8), which represents the limited budget for installing new distribution lines, is presented

by (2.29). The lower-level problem is shown in (2.30)-(2.32) in which the objective function (2.9) captures

22

the binary variables and re-written as (2.30), where vector d represents the parameters in the objective of

the lower-level problem. The set of equality constraints (2.31) represent the equality constraints (2.10),

(2.11), (2.16)-(2.21). Similarly, the set of inequality constraints (2.32) represents the inequality constraints

(2.12)-(2.15), and (2.22)-(2.24). The variables in this problem include the decision variables correspond to

connectivity represented by Ι , the eigenvalues and eigenvectors represented by η , the continuous

variables in the lower-level problem represented by α , and the Lagrange multipliers of the equality and

inequality constraints in the lower-level problem represented by π and τ .

,

minΙ η

η (2.27)

Subjected to:

( , ) 0,1g Ι η Κ Ι (2.28)

( )f Ι (2.29)

min T

αd α (2.30)

Subjected to:

( ) ( ) A B 1Ι α b π (2.31)

( ) ( ) C D 2Ι α b τ (2.32)

By employing the duality theory, the presented problem in (2.27)-(2.32) is transformed into a single-level

problem as shown in (2.33)-(2.40), where the lower-level problem is considered as a set of constraints for

the upper-level problem [Fie73]. The objective is the same as the objective of the upper-level problem,

which minimizes the eigenvalues of the distribution network graph in an ascending order. Here, the

objective and constraints of the upper-level problem are shown in (2.33)-(2.35) and the lower-level

problem (2.30)-(2.32) is shown as a set of constraints (2.36)-(2.40). The constraints (2.36) and (2.37) are

the constraints of the primal lower-level problem, while (2.38) and (2.39) are the constraints correspond to

the dual representation of the lower-level problem. The strong duality condition, which is shown in (2.40),

23

holds once the lower-level problem is linear and therefore the duality gap is zero. Here, the objective of the

primal lower-level problem is equal to the objective of the dual lower-level problem. The binary to

continuous terms in (2.40) are further linearized by (2.41)-(2.44).

min ( )Ι,η,α,π,τ

η (2.33)

Subjected to:

( , ) 0,1g Ι η Κ Ι (2.34)

( )f Ι (2.35)

( )A B 1Ι α b (2.36)

( )C D 2Ι α b (2.37)

T TB D π τ d (2.38)

τ 0 (2.39)

[ ( )] [ ( )]T T TA C 1 2d α π b Ι τ b Ι (2.40)

The equivalent single-level problem has several binary-to-continuous nonlinear terms. Such terms appear

in the AC power flow formulation and in the dual form of the lower problem. In (2.17)-(2.20) the binary

decision variables for the connectivity of the distribution network and installation of the distribution lines

are integrated into the admittance matrix as shown in (2.16).

The nonlinear binary-to-continuous terms illustrated in (2.41) are linearized by presenting the constraints

(2.42)-(2.44). Two auxiliary nonnegative variables, Φ and Ψ are employed and M is an arbitrary large

number.

, 0,1 (2.41)

(2.42)

0 M (2.43)

24

0 (1 )M (2.44)

The other sources of nonlinearities are the constraints that determine the eigenvalues. These

constraints form convex feasibility sets and an optimal solution of the presented MINLP is procured using

convex optimization approaches. Several approaches may be employed to solve the presented convex

MINLP problem. One of these methods is generating and successively improving the outer approximations

in the neighborhood of a set of optimal solutions for the MINLP. By utilizing the linear outer-

approximation and introducing auxiliary variables, the MINLP is reformulated into MIP problem that could

be solved by branch and bound [Ley01] or cutting plane techniques [Wes02]. Therefore, the presented

problem could be solved utilizing solvers (e.g. Alpha-ECP, BARON, and SCIP) that employ similar

methodologies [Taw04],[Wes03],[Ach09],[Vig13],[Bus14]. Finally, the decisions for adding new

distribution lines will be validated in the microgrid topology validation problem. This problem is

formulated as a MIP problem in which the decision variables are the installation states of the distribution

lines. The objective of this problem is to minimize the installation cost of the distribution lines and the

expected operation cost of the microgrids subjected to generation and distribution network constraints and

the reliability requirements of the demands.

2.3 Case Study

2.3.1 A Sample Meshed Distribution System

In this section, an electricity distribution network, which is composed of 20 buses, is illustrated in Figure

2-3. The presented distribution network is normally connected to the utility feeder while it is divided into

multiple microgrids fed by 5 DERs in the case of any disruption or disturbance in the distribution network.

The distribution network has 23 existing electricity distribution lines, and 20 electricity demands. Nine

electricity distribution lines are considered as candidates in the distribution network. Table 2-1 presents the

characteristics of the DERs and Table 2-2 shows the associated loss of energy probability (LOEP)

[Pra85],[Nii00],[Bas10],[Shaf15] at each bus which determines the priority of demands served in the

distribution network and calculated by (2.45). Table 2-3 describes the characteristics of the electricity

distribution lines including the forced outage rates (FOR), buses that are connected by each line, length and

25

the maximum capacity of the lines. The FOR associated with the feeder is 10.9 f/yr and the price of

electricity at the feeder is 8 ¢/kWh. Here, it is assumed that the available budget is sufficient for 1,000-

meter installation of the distribution lines. The states of the components are calculated based on the

availability of the components using two state Markov Chain process and Monte Carlo simulation. Here,

3000 scenarios are generated to represent the uncertainties in the system; including the generator and

distribution line random outages, and the demand uncertainties. Truncated normal distribution function is

used to represent the errors in load forecast, where the mean value is the forecasted volume and the

standard deviation is percentages of the mean values [Bil96], [Gre03]. Here, the standard deviation is 5%

from the mean value. The load profile is represented by the load duration curve which is divided into four

periods 1 2 3, , , and 4 with 10%, 25%, 60%, and 5% of the total annual hours. The scenario reduction

technique [Dup03],[GAMS] are employed to reduce 3000 scenarios to 12 scenarios. The voltage magnitude

and phase angles of buses are restricted between 0.95-1.05 per unit and (-π) to (π) respectively. The

resistance and inductive reactance of the distribution network cables are r = 0.092W /1000m and

x = 0.121W /1000m respectively. The equivalent single level MINLP is solved by SCIP 3.2.0 that employs

branch and bound algorithm, wherein the relaxed LP is tightened by cutting plane and domain propagation.

In order to handle the computation burden, sets of distribution lines that satisfy primal dual representation

of the reliability evaluation problem are procured and the set with minimum Eigenvalues is selected using

SCIP solver.

,,

s D sb b t b t

t s

LOEP EENS P (2.45)

1b bVOLL LOEP (2.46)

Table 2-1 Power generation unit characteristics

Unit Pmax1

(kW)

Pmax2

(kW)

Fc1

(¢/kWh)

Fc2

(¢/kWh)

Pmax

(kW)

Qmax

(kvar)

Qmin

(kvar)

FOR

(f/yr)

G1 1,200 600 8 15 1,800 900 -900 3.65

G2 900 450 10 28 1,350 625 -625 14.6

G3 1,000 500 10 24 1,500 750 -750 7.3

G4 900 450 9 14 1,350 600 -600 10.9

G5 750 325 12 25 1,075 550 -550 7.3

26

Table 2-2 Required LOEP at each bus

Bus bLOEP Bus bLOEP Bus bLOEP Bus bLOEP

1 0.03 6 0.005 11 0.05 16 0.03

2 0.005 7 0.001 12 0.09 17 0.07

3 0.03 8 0.001 13 0.05 18 0.01

4 0.01 9 0.07 14 0.08 19 0.02

5 0.02 10 0.04 15 0.05 20 0.03

Table 2-4 shows the reliability indices at each bus achieved in this configuration with no

microgrids or DERs. As shown here, the LOEPs of the loads are not within the required range and the

LOEPs exceed the acceptable level at buses 2, 4, 6, 7, 8, 11, 14, and 18. Considering the value of lost load

as defined in (2.46) in $/kWh, the expected annual operation cost of the distribution network is $1.773M.

The total expected demand and the total expected demand curtailment are 17,361 MWh and 2,247 MWh

respectively. The following two cases are considered,

1) Case 1 – heterogeneous LOEP requirement for demands

2) Case 2 – non-heterogeneous LOEP requirement for demands

Table 2-3 Distribution line characteristics

ID FOR

(f/yr)

Length

(m)

(kVA) ID

FOR

(f/yr)

Length

(m)

(kVA)

1 3.65 220 800 17 7.3 90 800

2 7.3 140 1,200 18 3.65 440 1,700

3 3.65 90 800 19 3.65 280 1,200

4 7.3 210 880 20 7.3 260 1,100

5 3.65 120 1,000 21 3.65 200 1,200

6 3.65 80 800 22 3.65 300 1,200

7 3.65 140 1,700 23 7.3 210 900

8 7.3 270 1,200 24 3.65 190 900

9 7.3 350 1,200 25 3.65 460 2,000

10 3.65 200 1,200 26 3.65 260 800

11 3.65 300 1,200 27 7.3 500 1000

12 3.65 120 1,200 28 7.3 450 900

13 7.3 110 1,200 29 3.65 390 900

14 3.65 160 2,000 30 7.3 360 1,000

15 7.3 220 800 31 3.65 500 800

16 7.3 120 1,000 32 3.65 460 1,000

Table 2-4 The LOEP in distribution network with no DER and microgrid

Bus Bus Bus Bus

1 0.015 6 0.015 11 0.342 16 0.015

2 0.015 7 0.015 12 0.022 17 0.019

3 0.022 8 0.015 13 0.019 18 0.015

4 0.015 9 0.019 14 1 19 0.015

5 0.02 10 0.019 15 0.015 20 0.015

maxSL maxSL

bLOEP bLOEP bLOEP bLOEP

27

Figure 2-3 The proposed electricity distribution network

2.3.1..1 Case1: Heterogeneous LOEP requirement for demands

In this case, the values for the required LOEP for demands at each bus are given in Table 2-2. The

spectra of the graph associated with the distribution network are shown in Figure 2-5. The first five

eigenvalues are minimized and the remaining eigenvalues are determined by validating the microgrid

topology. In this case, as the 5 smallest eigenvalues of the corresponding Laplacian matrix are zero, placing

the breakers at PCCs would help to divide the distribution network into five autonomous microgrids as

shown in Figure 2-4. Here, microgrid M1 is composed of buses {1, 2, 3, 8, 9, 10, 13, 16} which is supplied

by the G1. Microgrids M2, M3, M4, and M5 are composed of buses {5, 7}, {6}, {4, 12, 14, 15, 17}, and

{11, 18, 19, 20} which are respectively supplied by G2, G3, G5, and G4. The PCCs for microgrids M1 and

M3 are located on lines L4 and L7. The PCC between microgrids M1 and M2 is located on lines L9 and the

PCC between microgrids M2 and M4 is located on line L10. Moreover, the PCCs between microgrids M1

and M4 are located on lines L8, L17, and L20. The PCCs between microgrids M5 and M1; and between

microgrids M5 and M4; are located on lines L15 and L16, respectively. To address the reliability

requirements, distribution lines L32 and L29 with a total length of 850 meters were installed in the existing

distribution network. Table 2-5 shows the reliability indices achieved in this configuration. As shown in

this table, the LOEPs of the loads are within the required range, while the LOEP reached but not exceeded

G3

G2

7

4

10 2

186

9

3

5

L1

L2

L3 L4

L5 L6

L7

L8 L9

L10

L11

G4

G517

14

20 1211

18

1619

13

15

L12

L13

L14

L15

L16

L17

L18

L19

L20

L21 L22

L23

Feeder

L24

L25

L26

L27L28

L29

L30

L31

L32

Existing Lines

Candidate Lines

G1

28

the maximum acceptable level at bus 4. Considering the value of lost load as defined in (2.46) in $/kWh,

the expected annual operation cost of the distribution network with microgrids is $1.682M which is 2.93%

less than the operation cost of the distribution network in normal condition (2.$1.733M). Although more

expensive DER units serve demands within each microgrid, the reduction in the demand curtailment

reduces the operation cost. Hence, the formation of microgrids with DERs improves the operation cost of

the distribution network while the PCCs will form microgrids in the event of disruptions or failures in the

distribution network.

Table 2-5 The LOEP of demand at each bus in case 1


1 0.0006 6 0.002 11 0.003 16 0.0006

2 0.0006 7 0.00005 12 0.09 17 0.016

3 0.0006 8 0.0006 13 0.0006 18 0.004

4 0.01 9 0.0006 14 0.008 19 0.003

5 0.00005 10 0.0006 15 0.008 20 0.003

The presented framework provides the necessary tool to define boundaries of each microgrid. The

procured microgrids are able to satisfy the reliability requirements of the demands in island mode. The total

expected demand curtailment is 112 MWh while the total expected annual demand is 17,361 MWh.

Forming the microgrids would assign the DERs to serve certain demands by islanding which leads to

heterogeneous LOEPs in the distribution network. Hence the presented approach ensures the reliability

requirements of the demands within the distribution network, while maximizing the number of microgrids

in the distribution network.

29

Figure 2-4 Electricity distribution network Case 1

Figure 2-5 The spectra of the graph associated with distribution network

2.3.1..2 Case 2: Non-heterogeneous LOEP requirement for demands

In this case, the new topology of the distribution network is procured considering a uniform

required LOEP for the demands. The required LOEP is the lowest required LOEP shown in Table 2-2. The

spectra of the distribution network graph are shown in Figure 2-5. Since the 3 smallest eigenvalues of the

related Laplacian matrix are zero, the distribution network is sectionalized into 3 autonomous microgrids as

shown in Figure 2-6. Here, microgrid M1 is composed of buses {1, 2, 3, 4, 6, 8, 9, 10, 13, 15, 16, 17}

which are supplied by the G1 and G3. Microgrids M2 and M3 are composed of {5, 7}, {11, 12, 14, 18, 19,

20} which are respectively supplied by G2; G4 and G5. The PCC between microgrids M1 and M2 are

placed on lines L9 and L10. In addition, PCCs between microgrids M1 and M3 are located on lines L15,

L17, and L21. To address the reliability requirements, the distribution lines L28 and L29 with a total length

G3

G2

7

10 2

18 9

5

L1

L2

L3

L5 L6

L11

G4

G517

14

11

18

16

15

L12

L13

L14L18

L19

L21 L22

L29

L32

19

L4

L7

L16

L17

L20

6

Line

PCC

L9

L10

13

12

20

L15

Boundary

3

G1M1

M2

M3

M4

M5

L23

4

L8

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Distribution Network

Case 1

Case 2

Number of Eigenvalues

Eig

env

alu

e

Eigenvalues corresponding to microgrids network topologies

30

of 860 meters were installed in the existing distribution network. The required reliability for all demand

entities is ensured as shown in Table 2-6. Here the required LOEPs for buses 5 and 7 are the same as those

in Case 1 because the topology of microgrid M2 in this case, is the same as microgrid M2 in Case 1. With

the value of loss load defined in (2.46) in $/kWh, the expected cost of the distribution network is $1.492M

in this case which is 11.24% less than that in Case1. Compared to Case 1, some of the demands require

lower LOEP in Case 2 which leads to larger microgrids (less zero eigenvalues for the graph spectra that

indicates more interconnected distribution network) as shown in Figure 2-6. The total expected demand

curtailment is 709.56 kWh. The LOEP for each demand is calculated in Table 2-6.

Table 2-6 The LOEP of demand at each bus in Case 2


1 0 6 0 11 0 16 0

2 0.0001 7 0.00005 12 0.0006 17 0.00009

3 0 8 0 13 0 18 0

4 0 9 0 14 0 19 0

5 0.00005 10 0 15 0.00009 20 0

Figure 2-6 Electricity distribution network in Case 2

2.3.2 The Modified IEEE 123-bus system

The IEEE 123-bus system [IEEE123] is fed by the utility feeder in normal condition as shown in

Figure 2-7. The network will sectionalize into several microgrids served by 10 DERs in the case of

L17

L15

G3

G2

7

10 2

186

9L1

L2

L3

L5 L6

L7

L11

G4

G5

1714

2012

11

1619

13

15

L12

L13

L14

L16

L18

L19

L20

L22

L23

L28

L29

L9

3

5

Line

PCC

Boundary

L10

L21

18

G1

M1

M2

M3

L4

L8

4

31

disturbances in the network. Here, 8 distribution lines are considered as candidates in the distribution

network to satisfy the required LOEP at the demand buses. The required LOEPs at demand buses are given

in Table 2-7. As the 9 smallest eigenvalues of the corresponding Laplacian matrix of the distribution

network graph are zero, the distribution network is sectionalized into 9 autonomous microgrids. Microgrid

M1 is composed of buses {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 34, 149} which are supplied

by the DER on bus 7 with the FOR of 10.9 f/yr. Microgrid M2 is composed of buses {18, 19, 20, 21, 22,

23, 24, 35, 36, 37, 38, 39, 135} which are supplied by the DER on bus 21 with the FOR of 3.65 f/yr. The tie

switch between buses 18 and 135 is closed and the PCC between microgrids M1 and M2 is placed on line

L13, which connects buses 13 and 18. Microgrid M3 is composed of buses {25, 26, 27, 28, 29, 30, 31, 32,

33, 250} which are supplied by the DER on bus 26 with the FOR of 7.3 f/yr. Here, the PCC between

microgrids M2 and M3 is placed on line L24 that connects buses 23 and 25. Microgrid M4 is composed of

buses {40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 151} which are supplied by the DER on bus 44 with

the FOR of 10.9 f/yr. Here, the tie switch between buses 151 and 300 is open and the PCC between

microgrids M2 and M4 is placed on line L36 that connects the buses 35 and 40. Microgrid M5 is the largest

designated microgrid that is composed of buses {52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 67, 72, 73, 74, 75,

76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 152, 160, 610} which are supplied by the DERs on buses 53 and

76 with the FOR of 10.9 f/yr and 14.55 f/yr respectively. Here, the tie switch between buses 60 and 160 is

closed and the PCC between microgrids M1 and M5 is located at the tie switch that connects buses 152 and

13. Microgrid M6 is the smallest designated microgrid that is composed of buses {62, 63, 64, 65, 66}

which are supplied by the DER on bus 62 with the FOR of 3.65 f/yr. Here, the PCC between microgrids

M6 and M5 is on line L61 that connects buses 60 and 62. Microgrid M7 is composed of buses {87, 88, 89,

90, 91, 92, 93, 94, 95, 96} which are supplied by the DER on bus 89 with FOR of 7.3 f/yr. Here, the PCCs

between microgrids M5 and M6 are on line L86 that connects buses 86 and 87, and on the tie switch that

connects buses 54 and 94. Microgrid M8 is composed of buses {68, 69, 70, 71, 97, 98, 99, 100, 450} which

are supplied by the DER on bus 98 with the FOR of 3.65 f/yr. To satisfy the reliability requirements at the

load point on bus 69, a 200-meter distribution line connects buses 70 and 100. The PCCs between

microgrids M5 and M8 are located on lines L66 and L68 that connect bus 67 to buses 68 and 97

respectively.

32

The FOR of the line L69 that connects buses 68 and 69 is 7.3 f/yr, which leads to multiple outages in the

operation horizon. Such high FOR results in unacceptable LOEP of the demand on bus 69 (0.06) before

forming microgrid M8. The required LOEP for the demand on buses 68 and 69 are 0.06236 and 0.0078

respectively. The network expansion in microgrid M8 by installing the distribution line that connects buses

70 and 100, will further reduce the LOEP of demand on bus 69 from 0.06 to zero, and the LOEP of demand

on bus 68 from 0.349 to 0.059. Thus, network expansion within autonomous microgrids further improves

the reliability of supplied energy.

Finally, microgrid M9 is composed of buses {101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112,

113, 114, 197, 300} which are supplied by the DER on bus 108 with the FOR of 7.3 f/yr. Here, the PCC

between microgrids M8 and M9 is located on the tie switch that connects buses 197 and 97, and the PCC

between microgrids M4 and M9 is located on the tie switch that connects buses 151 and 300.

By considering 3.5% annual increase in demand in the 10-year operation horizon, and the value of

lost load ($/kWh) shown in (2.46), the expected operation cost of the distribution network with microgrids

is $45.6M. To illustrate the merit of forming microgrids in contingencies, the presented approach is

compared with the network reconfiguration technique, which is discussed in [Bor12]. Here, the existing tie-

switches within the IEEE 123 bus test system are utilized to reconfigure the distribution network

considering the outages in the distribution lines. The operation cost of the system with network

reconfiguration capability is $50.9M, which is 11.7% more than the operation cost of the distribution

network with microgrids. Similar to the first case study, forming microgrids supplied by DER units,

enhances the reliability and reduces the operation cost of the distribution network.

33

Table 2-7 Required LOEP of demands in IEEE-123 bus system


1 0.03332 32 0.00516 63 0.01042 94 0.00104

2 0.11646 33 0.00984 64 0.0037 95 0.00516

3 0.2076 34 0.07228 65 0.00262 96 0.12146

4 0.11782 35 0.00056 66 0.01136 97 0.00084

5 0.0329 36 0.00974 67 0.00012 98 0.02432

6 0.24462 37 0.00042 68 0.06236 99 0.0652

7 0.06376 38 0.0002 69 0.0078 100 0.02492

8 0.02628 39 0.00868 70 0.000742 101 0.0023

9 0.00134 40 0.02484 71 0.00624 102 0.0012

10 0.01234 41 0.00364 72 0.00354 103 0.021

11 0.05062 42 0.00952 73 0.00274 104 0.00004

12 0.07316 43 0.00004 74 0.00222 105 0.04598

13 0.01622 44 0.00176 75 0.0611 106 0.00002

14 0.0724 45 0.00242 76 0.00034 107 0.08878

15 0.04926 46 0.08068 77 0.04038 108 0.03392

16 0.00112 47 0.01546 78 0.0006 109 0.00076

17 0.00056 48 0.00892 79 0.00096 110 0.00598

18 0.06462 49 0.06266 80 0.00104 111 0.09222

19 0.00006 50 0.05122 81 0.00006 112 0.00512

20 0.00014 51 0.05926 82 0.00302 113 0.000006

21 0.0024 52 0.00666 83 0.02186 114 0.1334

22 0.00012 53 0.02476 84 0.04486 197 0.02056

23 0.00056 54 0.0151 85 0.05452 160 0.00116

24 0.00182 55 0.02636 86 0.10082 152 0.01408

25 0.00008 56 0.0066 87 0.00338 149 0.00106

26 0.00034 57 0.00056 88 0.01694 135 0.00107

27 0.05124 58 0.003 89 0.00014 300 0.00003

28 0.00034 59 0.0147 90 0.000364 450 0.00001

29 0.00038 60 0.00134 91 0.01478 350 0.00055

30 0.03862 61 0.0372 92 0.11858 250 0.00016

31 0.00454 62 0.00946 93 0.0014

34

Figure 2-7 Modified IEEE 123-bus test system

2.4 Summary

This chapter proposes a framework to sectionalize the distribution network into autonomous

microgrids, considering heterogeneous reliability for the demands. The expansion of distribution network

and formation of microgrids satisfy the reliability requirements in case of disruptions in the distribution

network. The presented problem is formulated as a bi-level optimization problem. In the upper-level

problem, the number of microgrids is maximized by minimizing the weighted sum of the eigenvalues of the

Laplacian matrix of the distribution network graph. In the lower-level problem, the expected operation cost

is minimized considering the reliability constraints for the demands at each bus. The procured microgrid

topologies are further validated by minimizing the expansion and operation cost within the microgrids. The

presented approach is evaluated by two case studies.

32

31

27

33

26

28

25

2930 250

24

22

23

21

20

1918

135

48

44

47

45

46

42

40

49

43

5051

151

35

37

3938

36

11

2

14

910

1149

8

12

713

34

4

3 5 6

16

15 17

95

96

93

94

91

92

89

90

87

88

86

82

83

84

81

80

85

79

7877

76

7574

7372

61061

6716060

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5958

62

63

64

65

66

68

71

70

69

450

98

97

197

110

102

104

103

107

106

105

114

350

300

108

109

111 113110 112

56

152

555453

52

99

10041

35

Chapter 3

3RESILIENT OPERATION OF MULTIPLE ENERGY CARRIER MICROGRIDS

In this chapter, the energy resource interdependence in microgrids is represented by implementing energy

hubs, and preventive reinforcement strategies are offered to the microgrid operators following the

resiliency analysis. The contributions of this chapter are as follows:

• Modeling the disruption as a consequence of deliberate actions in multiple energy carrier

microgrids considering the interdependence of natural gas and electricity infrastructures

• Identify the critical and vulnerable components in multiple energy carrier microgrids

• Quantify the consequences of disruption by calculating the operation cost of multiple energy

carrier microgrid in contingencies

• Apply preventive reinforcement strategies to ensure the resilient operation of microgrids and

determine the reinforcement outcomes

The results can be used by microgrid planners and operators to identify critical components whose

reinforcement will improve the reliability and security measures within the microgrids. The chapter is

organized as follows: Section 3-1 introduces the basic components of the energy hub. Section 3-2 describes

the preventive reinforcement in cyber infrastructure to improve the resilience in the energy supply. Section

3-3 describes the problem formulation. Section 3-4 presents a case study, to show the effectiveness of the

proposed preventive reinforcement procedure, and section 3-5 presents the conclusion.

36

3.1 Energy Hub

Energy infrastructure provides various services to industrial, commercial, and residential consumers. The

optimal expansion planning and operation of various energy carriers are determined as a set of independent

problems without taking the interdependencies and interactions of the energy carrier networks into account.

Here, the nodes in the distribution network with energy conversion facilities are referred to as “energy

hubs”. The energy hub incorporates the interdependencies between natural gas, electricity and heat and

provides operational flexibility within active distribution networks as shown in Figure 3-1.

Here, the heat energy is produced as a bi-product of electricity generation using natural gas in

combined heat and power (CHP) units and curtailing the electricity demand at each hub would result in

curtailment of heat demand, as the electricity drives the pumps to facilitate heat transfer in the energy hubs.

Figure 3-1 Energy hub with electricity, natural gas and heat

The interdependence of electricity and heat demands with electricity and natural gas resources

provided at each energy hub is shown in (3.1), where the electricity and natural gas comprise input vector

E , and the electricity and heat demand comprise output vector L . Here, matrix C is the forward coupling

matrix which represents the input-output energy conversion. The forward coupling matrix is composed of

coupling factors and is determined by characteristics and topology of the energy hub.

e e gas e ee

gash e h gas h

C C EL

EL C C

L EC

(3.1)

37

3.2 Preventive Reinforcement in Cyber Infrastructure

Multiple energy carrier microgrids ensure the continuity of service to critical demands in energy

infrastructure. Hence the resilience of multiple energy carrier microgrids against cyber threats leading to

disruptions in physical components in energy infrastructure is addressed. Multiple energy carrier

microgrids cannot tolerate excessive restoration times, as the recovery and restoration is not considered as a

viable option for the mission critical facilities served by microgrids. Hence the resilience is defined as the

ability of the system to survive attacks without suffering from any service interruptions and the objective is

to provide a solution framework to ensure the continuity of service in multiple energy carrier microgrids.

The set of preventive actions that is required for resilient operation of microgrid, is defined through the

proposed reinforcement procedure.

More than half of the cyber threads are in energy sector [ICS13]. The most frequent cyber threats

are authentication, denial of service, and buffer overflow [ICS13]. Here, it is assumed that the objective of

the attacker is to increase the service interruption by disrupting the physical components through cyber-

attacks. The physical components of the microgrid receive encrypted data packets which contain the control

signals from the microgrid controller. Hence decryption and manipulating the encrypted data packets would

result in the disruption of the physical components. Figure 3-2 illustrates disruption of electricity

distribution line by decrypting the control signals sent from the microgrid controller. Here, the disruption

occurs if the encrypted data is decrypted and manipulated by the attacker. Moreover, it is assumed that the

data packets correspond to each component is encrypted with an existing encryption technique e.g. secured

hash functions (SHA-x) [Mir05], or advanced encryption standard (AES) [Dae02]. However, other

encryption techniques [Den82] could also be applied in the proposed formulation. The higher encryption

strength is achieved by the increase in the encryption cost which eventually leads to higher disruption cost

of the physical component. The cost of providing higher encryption strength is dependent on the hardware

implementations such as processing modules and communication channels. In this chapter, an encryption

cost is assigned to specific encryption strength provided by practical encryption techniques. For instance,

the encryption cost of using AES-256 to encrypt a data packet is more compared to the encryption cost of

using AES-192 considering the same performance measures. While the encryption strategy to improve the

38

encryption strength of the data packets associated with the physical components are discussed in this

chapter, the incorporated encryption technique is out of the scope of this chapter. The presented model

considers an initial encryption strength for the data packets corresponding to each physical component. The

new encryption strategy is to improve the encryption strength for the certain components which are

identified by the solution of the bi-level optimization problem employed. Hence, an improved encryption

strategy decreases the vulnerability of microgrid against disruptions and increase the cost of decryption,

data manipulation, and disruption of the physical components. In other words, each stage of the preventive

reinforcement corresponds to the encryption strategy chosen in the microgrid. The encryption strategy

taken, includes the improvements in the encryption strength of the selected physical components of the

microgrid.

Figure 3-2 Disruption of service in multiple energy carrier microgrids

Here, it is assumed that different types of physical components in the microgrid require different numbers

of data packets. The number of the data packets is dependent on the number of controllable variables in

each component. For instance, it is assumed that generators require larger number of data packets for

monitoring and control compared to the distribution lines.

The preventive reinforcement presented here is defined in several stages, where the microgrid resilience

at each stage is dependent on the encryption strategy incorporated for monitoring and control of the

microgrid. The solution to the bi-level optimization problem at each stage procures the new encryption

39

strategy and the iterative procedure continues until the desired level of resilience is established. The desired

level of resilience is determined by the microgrid planner or operator. To quantify the resilience of the

microgrid against disruptions, a resilience index (r) shown in (3.2) is defined as the exponential of negative

ratio of the increase in operation cost of multiple energy carrier microgrid as a result of disruptions to the

budget of the attacker causing such disruptions. Here, and 0 are operation costs of disrupted and non-

disrupted multiple energy carrier microgrid. Thus low resilience index indicates that the multiple energy

carrier microgrid is highly vulnerable against disruptions.

0( - )( )

e Mr

(3.2)


The operation of multiple energy carrier microgrids exposed to disruptions is presented as a bi-level

optimization problem. The upper level problem maximizes the operation cost of the system subjected to the

limitation of resources to trigger disruptions while the lower level problem minimizes the operation cost of

the multiple energy carrier microgrid under attack. Further, the MIBLP problem is re-formulated into a

single-level mixed integer programming (MIP) problem employing the duality theory.

The objective function of the upper level problem, expressed by (3), maximizes the operation cost while

the decision binary variables are the outage statuses of generation units, natural gas pipelines, and power

distribution lines. The operation cost of system includes the penalty cost for the load curtailments, hence

maximizing the operation cost of the system by the attacker would lead to maximizing the curtailed

demand and the generation cost in the microgrid. The resources for triggering outages in various

components of microgrid are limited as indicated in (3.4). In bi-level optimization problem presented here,

the lower level problem and its constraints are considered as constraints for the upper level problem

[Bar98]. The objective function of the lower level problem is the operation cost of the microgrid which

includes the operation cost of generating electricity and heat energy from natural gas, as well as the penalty

cost for electricity and heat demand curtailment as shown in (3.5). The real and reactive power injection is

shown by (3.6) and (3.7), respectively. For the sake of simplicity the parameter e eC which represents the

40

efficiency of the transformer is considered as unity. The real and reactive power limits for generation units

are formulated in (3.8) and (3.9) respectively. The curtailed real and reactive demand at each energy hub is

less than the total real and reactive demand as presented by (3.10) and (3.11) respectively. The admittance

of each power line shown in (3.12) incorporates the binary variable representing the outage status of the

component. The real/reactive power injection used for ac power flow is linearized in (3.13) and (3.14).

Equations (3.15), (3.16) and (3.17) present the linearized formulation for real, reactive and apparent power

transmitted through the distribution line. Here, ξ is an auxiliary parameter, which is dependent on the load

power factor as calculated in [Kho12]. The limitation on apparent power flow is imposed by distribution

line capacity as given in (3.18). The limitations on voltage magnitude and voltage phase angle in electrical

networks as well as the natural gas pressure within natural gas distribution network are presented in (3.19),

(3.20) and (3.21), respectively. The minimum and maximum capacity for natural gas resources are shown

in (3.22). The interdependence of electricity and heat demands is represented in (3.23), where K is a large

number, which indicates that the heat distribution facility requires electricity to transfer the heat. Thus,

electric demand curtailment at each energy hub would lead to heat demand curtailment. The heat demand is

served by the electricity bi-product or burning natural gas in heater as shown in (3.24). The natural gas

balance at each energy hub is presented by (3.25), in which the injected natural gas at each hub is equal to

the withdrawn volume by the interconnected pipelines as well as the energy hub to serve the electricity and

heat demand. The natural gas is converted into electricity and heat using the respective conversion factors.

Here, any outage in natural gas supply will result in the outage of the supplied generator. The constraint on

natural gas flow in distribution pipelines is given in (3.26). Here, the dependency of the natural gas flow on

temperature which is discussed in [Mar12] is ignored and the linear dependency of natural gas flow

between two interconnected energy hubs and their pressures are expressed in (3.27).The served heat

demand is lower than the total heat demand at each energy hub as shown in (3.28).

,

,, ,

, ,maxl

i j o p

li j o p

UX UY UZ

UX UY UZ (3.3)

Subject to:

41

NG NL NPl

i i l p pi l p

M UX M UY M UZ M (3.4)

, ,

, , , ,, , ,

, ,

,

( , , ) ( ) .( ) ( ) .( )mine h

i k

e d h dj j

l e e e D e d h h h D h di j o p c i i c k kj j j j

P P i j k j

P P

UX UY UZ F P VOLL P P F P VOLL P P

(3.5)

j j

inje e,di j j

i GG j D

P P P

(3.6)

j j

inje e,di j j

i GG j D

Q Q Q

(3.7)

min max(1 ) (1 )ei i i i iP UX P P UX (3.8)

min max(1 ) (1 )ei i i i iQ UX Q Q UX (3.9)

, , ,0 e D e d e Dj j jP P P (3.10)

, , ,0 e D e d e Dj j jQ Q Q (3.11)

2 2 2 2

1 1i ,o j ,o i ,o j ,oj ,o j ,o j ,o

j ,o j ,o j ,o j ,o

r .( UY ) x .( UY )y g Jb J

r x r x

(3.12)

12 1injj

NB

j j, j j ,o j o j,o j oo o j

P BV G G V V

(3.13)

12 1injj

NB

j j, j j ,o j o j,o j oo o j

GQ V B B V V

(3.14)

, , ,( ) ( )j o j o j o j o j oGPL V V B (3.15)

, , ,( ) ( )j o j o j o j o j oQL B V V G (3.16)

, , , ,j o j o j o j oSL PL QL (3.17)

max, ,j o j oSL SL (3.18)

min maxjV V V (3.19)

42

min maxj (3.20)

min maxj (3.21)

min maxs s sv v v (3.22)

,0 h d ejjP K P (3.23)

,0 ( ) / ( )

j j

h d i e i hgas h i gas e kj

i GG k HH

P C P C P

(3.24)

, ,( ) ( )

h eP Pk iA v f fj s p p k is GS p P p P k HH i GGC Cj f j t j j jgas h gas e

(3.25)

,max ,max, , ,.(1 ) .(1 )p p p

p pj o j o j of UZ f f UZ (3.26)

, ,

2 2

( )p j j o op pp pj o j o

j o

Cf K UZ f K UZ

(3.27)

, ,0 h d h Dj jP P (3.28)

The presented MIBLP problem formulation (3.3)-(3.28) is expressed in the general form by (3.29)-(3.34).

By employing duality theory, the presented problem is transformed to single-level MIP problem presented

by (3.35)-(3.42). The proposed bi-level problem can be transformed to a single level MIP problem using

KKT conditions or duality theory. Here, the duality theory is used to decrease the number of the binary-to-

continuous variable multiplications, computation complexity and solving time [Arr10]. The objective

function (3.3) is represented by (3.29) and the disruption resource allocation constraint (3.4) is presented as

(3.30), where * is the optimal solution of lower level problem (3.32). The binary variables in (3.31) are

the decision vectors representing the outage statuses of electricity generation and distribution components

as well as the natural gas pipelines. The lower minimization problem, with determined binary variables,

given in (3.32) represents (3.5) which minimizes the operation cost of the microgrid. Equations (3.33) and

(3.34) represent all the equality and inequality constraints shown in (3.6)-(3.28), respectively. Here and

43

are introduced as the dual variables associated with equality constraints (3.33) and inequality constraints

(3.34), respectively.

*max (U, )U

h (3.29)

Subject to: *( , ) 0g U (3.30)

0,1U (3.31)

min Td

(3.32)

Subject to: 1( )A U B b (3.33)

2( )C U D b (3.34)

In the presented single level MIP formulation (3.35)-(3.42), the objective is to maximize the operation

cost of the system by determining the disruption binary variables U , generation dispatch , equality

constraint dual variable , and inequality constraint variable . The constraints (3.36)-(3.37) are similar to

(3.30)-(3.31) and * is replaced by , which is the decision variable in the equivalent problem (3.35)-

(3.42). In the original formulation (3.29)-(3.34), the binary decision variables which are determined in the

upper level problem (3.29)-(3.31) are fixed for the lower level problem (3.32)-(3.34). However, in (3.38),

the dual form of the lower level problem is shown. In (3.39), it is stated that dual variables of the inequality

constraints of the lower level problem are nonnegative. The constraints (3.40) and (3.41) are the same as

(3.33) and (3.34), respectively. Here, U is the decision variable in the equivalent single level problem. The

equality constraint associated with strong duality is expressed in (3.42). Here (3.38) and (3.42) are

composed of several of binary-to-continuous variables multiplications, which are further linearized.

, , ,max (U, )

Uh

(3.35)

Subject to: ( , ) 0g U (3.36)

44

0,1U (3.37)

T TB D d (3.38)

0 (3.39)

1( )A U B b (3.40)

2( )C U D b (3.41)

1 2[ ( )] [ ( )]T T Td b A U b C U

(3.42)

To solve the current problem, all the nonlinear terms which are composed of binary-to-continuous

variable multiplication are transformed into linear forms. Such nonlinearities exist in three categories: a)

the ac power flow formulation, b) the natural gas flow formulation, and c) the dual form of lower problem

in the equivalent single level MIP problem. In the ac power flow formulation, the binary decision variables

for the distribution lines are embedded in the admittance matrix as shown in (3.12). Equations (3.13)-(3.16)

contain 10 binary-primal continuous variables multiplication terms, e.g. voltage as primal continuous

variable to binary variable associated with distribution line availability status in (3.13). The second

category is the natural gas flow formulations, as shown by equations (3.26) and (3.27). The last category is

the dual form of the lower level problem, which has binary-to-continuous variable multiplications in (3.42).

The linearized form of the term (3.43) is shown in (3.44)-(3.46) in which the lower and upper bounds of the

continuous variable are considered as zero, and a large number, K, respectively. Here, Φ and Ψ are

nonnegative continuous variables.

, 0,1U U (3.43)

(3.44)

0 K U (3.45)

0 (1 )K U (3.46)

45

The constraint (3.27) is the linear approximation of the original Weymouth equation (3.47) which

represents the relationship between the natural gas flow with the pressure at the inlet and the outlet of a

natural gas pipeline. Figure 3-3 shows the hyper plane represented by Weymouth equation, which is

decomposed into several smaller hyper planes. The linearization using Taylor series is valid only if the

difference in natural gas pressure between the inlet and outlet of the pipeline is assumed to be limited, i.e.

there is not significant pressure drop in the pipeline. This is a reasonable assumption for the short pipelines

used in microgrids. The limitation on the node pressure on the gas pipeline network shown in (3.21)

guarantees the accuracy of the approximation. The energy hubs within microgrids operate within specific

range of nodal natural gas pressure, e.g. between 55 and 56 bars. The Weymouth equation is linearized

around the initial point procured by solving optimal energy flow within the microgrid considering no

disruptions.

,

2 2.p

f j o p j oC (3.47)

Figure 3-3 Hyper plane for the Weymouth equation

3.4 Case Study

In this section, a multiple energy carrier microgrid composed of 10 energy hubs is illustrated in

Figure 3-4. The microgrid presented in the case study operates in island mode; however, the presented

preventive reinforcement procedure could be applied for microgrids operating in grid-connected mode by

incorporating the grid interconnection with respective price of electricity and disruption cost into the

formulation. The presented microgrid has 3 distributed generation CHP units with electrical and heat

46

efficiency of respectively 34% and 50% which consume natural gas to produce electricity and heat, 11

electricity distribution lines, 10 electricity demands, 6 heat demands and 5 natural gas pipelines. Table 3-1

presents the characteristics of the natural gas generators. Table 3-2 lists the electricity and heat demands

and their respective value of lost loads which determine the priority of demands. Table 3-3 describes the

characteristics of electricity distribution lines and the natural gas pipelines.

Figure 3-4 Multiple energy carrier microgrid.

Table 3-1 Natural gas power generation unit characteristics

Unit Pmax1

(kW)

Pmax2

(kW)

Fc1

(¢/kWh)

Fc2

(¢/kWh)

1

gas elecC

(SCM/kWh)

2

gas elecC

(SCM/kW

h)

Pmax

(kW)

G1 800 400 8 15 0.00789 0.0092 1,200

G2 1,200 600 10 28 0.00796 0.0102 1,800

G3 1,000 500 10 24 0.00726 0.0083 1,500

Table 3-2 Electricity and heat demand

Hub

ID

,e DP

(kW)

,e DQ

(kVar)

eVOLL

($/kWh)

,h DP

(MBtu)

hVOLL

($/MBtu) m

(bar)

1 80.7 40.4 10 95.23 1 55.34

2 113.0 56.5 8 111.11 1 55.10

3 161.5 80.8 10 142.85 1 55.11

4 242.3 121.2 10 126.98 1 55.00

5 290.7 145.4 10 158.72 1 55.01

6 323.0 161.5 10 158.72 1 55.37

7 80.7 40.4 100 - - -

8 161.5 80.8 10 - - -

9 323.0 161.5 10 - - -

10 323.0 161.5 20 - - -

47

Here, Cp for natural gas pipeline is a constant which is determined by temperature, diameter, gas

composition, length and friction [4]. The natural gas supply could deliver maximum volume of 50 Standard

Cubic Meter (SCM) at the source point. The voltage magnitude and phase angles of energy hubs within the

electrical distribution network are restricted between 0.95-1.05 per unit and -π to π respectively. The

resistance and inductive reactance of the distribution network cables are 0.028 /1000r ft and

0.037 /1000x ft respectively. The natural gas pressure at each hub is limited between 55 and 56 bars.

The equivalent single level MIP is solved by IBM ILOG CPLEX v12.6.0 software package.

Table 3-3 Distribution network characteristics (electricity and natural gas)

ID From

Hub

To

Hub

Length

(m)

maxSL

(KVA) Cp

maxf

(SCM)

L1 1 8 100 1,200 - -

L2 1 10 120 1,200 - -

L3 1 9 160 2,000 - -

L4 6 9 180 800 - -

L5 10 2 120 1,000 - -

L6 3 2 80 800 - -

L7 6 3 140 1,700 - -

L8 2 4 270 1,200 - -

L9 3 5 350 1,200 - -

L10 5 4 200 1,200 - -

L11 5 7 300 1,000 - -

P1 1 6 340 - 3 25

P2 2 3 80 - 2.82 25

P3 6 3 140 - 3 25

P4 3 5 350 - 2.82 25

P5 4 5 200 - 2.82 25

As stated earlier, an encryption cost is associated with a specific encryption strength provided by the

practical encryption techniques. For example, the encryption costs of a single data packet using AES-128,

AES-192, and AES-256 are assumed as $128, $256, and $512 respectively. Similarly, the encryption costs

correspond to the hash function with 128, 256, and 512 bits could be assumed as $128, $256, and $512

respectively. Any Further improvement in encryption strength achieved by cascading encryption using the

standard encryption techniques. The required resources to manipulate an encrypted data packet is assumed

to be 10 times of the encryption cost. Here, the encryption and disruption costs of the initial encryption

strength for each data packet are assumed as $128 and $1,280 respectively. The generator, natural gas

48

pipeline, and electricity distribution line use 7, 6, and 2 data packets. Hence the disruption costs are $8,960,

$7,680, and $2,560 for generator, natural gas pipeline and distribution line, respectively. Disruption

attempts with limited resources ($20,000) were considered in the following cases to evaluate the resilience

of the energy supply in the proposed microgrid:

• Case 1: No preventive reinforcement in the microgrid

• Case 2: First stage of preventive reinforcement, by increasing the disruption costs for the

vulnerable components determined in Case 1.

• Case 3: Second stage of preventive reinforcement by increasing the disruption costs for the


• Case 4: Third stage of preventive reinforcement by increasing the disruption costs for the


3.4.1 Case 1: No preventive reinforcement in the microgrid

In this section, the impact of disruptions on the energy infrastructure within a microgrid is

presented with no preventing reinforcement on the components. Applying the proposed methodology

highlights the vulnerable components of the microgrid against disruption as shown in Figure 3-5. The

attack in this case results in disruption of distribution lines L2, L3, L4, and L7 and the natural gas pipeline

P4. This will lead to the curtailment in electricity demands on energy hubs 2, 3, 4, 5, 7, 9, and 10 as well as

curtailment in heat demands on energy hubs 2, 3, 4, and 5. The operation cost increases from $195 in

normal operation to $26,275 in this case. The resilience index of multiple energy carrier microgrid in this

case is 0.2715. In this case, the electricity and heat demands on energy hubs 1 and 6 are served by

generation unit supplied by the natural gas. Since the electric demand on energy hub 9 is larger than that on

energy hub 8, the impact of disruption on energy hub 9 is higher; hence interruption on L3 and L4 will lead

to disruption of electricity supply on energy hub 9. On the natural gas distribution network, the natural gas

pipeline from energy hub 3 to energy hub 5 is the primary candidate for interruption and this interruption

will disrupt the operation of generator G2. Moreover, disconnecting electricity distribution lines L2 and L7

will result in forming four isolated electrical networks, i.e. {2, 3, 4, 5, 7, 10},{1,8}, {6}, and {9} within the

49

microgrid. The supply and demand balance is reached by curtailing the electricity and heat demand as

required. The total supplied electricity by G1 and G3 in this case is 559.4 kW which is equal to the served

demand in the islands composed of energy hubs {1, 8} and {6}, respectively. In this case, disruption of P4

left G2 without the gas supply, and the generation dispatch of G2 is zero; hence, the island formed by

energy hubs {2, 3, 4, 5, 7, 10} is left without any available electricity generation. As a result, the electricity

demand is curtailed in this island.

Figure 3-5 Microgrid operation condition in Case 1

3.4.2 Case 2: First Stage of Preventive Reinforcement

In this case, the encryption strategy is updated which indicates higher encryption strength for the data

packets utilized to control and monitor the vulnerable components identified by the solution of the bi-level

optimization problem in Case 1. The encryption cost for each vulnerable component is twice as that of the

initial encryption strength. This is represented by increasing the encryption cost of each data packet to $256

for the disrupted components. For instance, the encryption cost of each data packet corresponds to L2 is

increased from the initial cost of $128 to $256 so L2 is reinforced against disruptions in this case, by

spending $512 for the improved encryption strength. In this case, the interruption in distribution lines L1,

G1

Gas Supply

8 1

G3

6

G2

5

9

10 2 3

74ElectricityNatural Gas

Electricity OutageHeat OutageHeat

L1

L2

L3

L4

L5 L6

L7

L8

L9

L10 L11

P4

50

L2, L6, L9, L10 and L11 will lead to the interruption in electricity demand in energy hubs 2, 4, 7, 8, and 10

and the interruption in heat demand in energy hubs 2, 4 and 5. The operation cost will decrease by 20.2%

from $26,275 in Case 1, to $21,860 in this case and the resilience index is 0.3385 in this case.

Here, as the disruption cost of line L7 was doubled, this line is not a good candidate for disruption; instead,

interruption in L6 will result in the electric demand curtailment and consequently heat demand curtailment

on energy hub 2. Moreover, as energy hubs 7 and 8 are fed radially by lines L11 and L1 respectively,

interruption of these lines will lead to the electricity demand curtailment on energy hubs 7 and 8. As shown

in Figure 3-6, the disruption in L11 will lead to the load curtailment on energy hub 7 with very high value

of lost load (VOLL). Higher VOLL on energy hub 10 provokes the interruption of lines L2, L6 and L10

which feeds the demand on energy hubs 2, 4 and 10. As seen in the previous case, disruption in L9 isolates

energy hub 5 from energy hubs {1, 9, 6, 3}. In this case, several isolated energy hubs including {8}, {1, 9,

6, 3}, {10, 2, 4}, {5} and {7} are formed.


G1

Gas Supply

8 1

G3

6

G2

5

9

10 2 3



L1 L2

L3

L4

L5

L6

L7

L8

L9

L10

L11

51

3.4.3 Case 3: Second Stage of Preventive Reinforcement

Similar to Case 2, the encryption strategy is updated based on the solution to the bi-level

optimization problem in Case 2. In this case, L1, L8, L10 and L11 were disrupted which led to the

electricity demand curtailment in energy hubs 4, 7, and 8 and the heat demand curtailment in energy hub 4.

The encryption strategy employed in the second stage of the preventive reinforcement reduced the

operation cost by 71.1% from that of Case 2. The operation cost in this case is $12,777 and the resilience

index is 0.5331. Here, the disruptions in electricity distribution lines L8 and L10 result in the electricity and

consequently the heat load curtailment in energy hub 4. In this case, the energy hub 4 is a good candidate

for the offender as L8 which feeds this energy hub has not been reinforced in the previous reinforcement

stages and still is vulnerable to attacks; hence, it requires lower resources for interdiction compared to other

reinforced distribution lines supplying energy hubs 2 and 3. As shown in Figure 3-7, the heat demand is

curtailed on energy hub 4 as a result of disruption in the electricity supply and the interdependence of heat

and electricity, as shown in (23). Similar to Case 2, interruptions in L1 and L11 will cause electricity

demand curtailment in energy hubs 8 and 7 respectively.

52


3.4.4 Third Stage of Preventive Reinforcement

In this case, the electricity demand on energy hubs 1, 8, 9, and 10 were curtailed as a result of interruption

on L4, L5, and P1. The operation cost in this case is $12,770 which shows a slight decrease compared to

that of Case 3. The resilience index of multiple energy carrier microgrid is 0.5333 in this case. Moreover,

the energy hubs 1, 8, 9, and 10 in the microgrid were isolated as a result of the disruption in L4 and L5 and

G1 is shut down as a result of disruption in the pipeline P1. This will curtail the electricity and heat demand

on energy hub 1 because of disruption in both electricity and natural gas supplies. Although the curtailed

demand in this case is larger compared to that in Case 3, the operation cost is very close to that of Case 3

because of the higher VOLL for electricity demand on energy hub 7 which is not interrupted in this case.

3.4.5 Preventive reinforcement procedure

Increasing the number of iteration to reinforce the disrupted components will result in more costly

disruptions and less outages in the natural gas and electricity infrastructures. The preventive reinforcement

procedure will prioritize the components for further reinforcements to reduce the operation cost of the

G1

Gas Supply

8 1

G3

6

G2

5

9

10 2 3



L1L2

L3

L4

L5 L6

L7

L8

L9

L10 L11

53

microgrids as a result of reduction in curtailed electricity and heat demands. For instance, in the fourth

stage of the reinforcement, the electricity distribution lines L9 and L11 are reinforced. With this preventive

reinforcement, the operation cost is reduced by 51.5% compared to that of Case 4 and reached to $8,424. In

addition, the resilience index is increased to 0.6627. Table 3-4 shows the operation cost, reinforcement

cost, disrupted components and respective resilience index at each stage of reinforcement.

It is assumed that the encryption cost and consequently the disruption cost for the interrupted components

will double at each stage as a result of the applied reinforcements. Figure 4-8 illustrates the reduction in the

operation cost of the disrupted microgrid achieved by improving the encryption strategy at each stage of the

preventive reinforcement. By comparing Case 1 with Case 4, it is observed that with the increase in

disruption cost of the natural gas and electricity distribution network components, fewer islands are formed

within the microgrid and the disruptions would be on the more expensive components which have not been

reinforced earlier such as electricity generators in each island. It is also shown that the electricity and heat

demands with higher VOLL are the primary targets for disruption, and consequently, the electricity

distribution and natural gas pipelines supplying the energy hubs serving high priority demands are the first

options for preventive reinforcements. By adding the operation cost and the reinforcement cost, the total

cost at each stage of reinforcement is procured, which is shown in Figure 4-8. As shown in this figure, the

minimum reinforcement and operation cost is achieved at sixth stage of reinforcement. At this stage, the

total cost is $23,090. Hence, deploying the preventive reinforcement beyond this stage increases the

investment cost on improving the encryption strategy beyond the increase in the operation cost as a result

of disruption.

54

Table 3-4 Outcomes of preventive reinforcement procedure

Stage

Operation

cost

($)

Resilience

Index

(r)

Disrupted

Component

Total

Encryptio

n Cost

($)

0 26,275 0.2715 P4, L2, L3,

L4, L7 9,344

1 21,860 0.3385 L1, L2, L6,

L9, L10,L11 11,136

2 12,777 0.5331 L1,L8

L10,L11 12,298

3 12,770 0.5333 P1,L4,L5 14,720

4 8,424 0.6627 L9, L11 16,512

5 6,753 0.7205 G3, L3, L7 18,048

6 3,122 0.8639 L8, L10 19,968

7 1,924 0.9172 L1, P2 21,504

8 753 0.9725 P3 23,296

9 497 0.9851 P4 24,064

10 322 0.9937 P5 25,600

11 307 0.9945 P1 26,368

12 307 0.9945 P2 27,904

13 307 0.9945 P2 28,672

14 211 0.9993 G1 29,440

15 210 0.9993 G2 31,232

16 200 0.9998 G3 33,024

17 195 1 - 33,920

As shown in Table 3-4, after 17 stages of reinforcement, the microgrid is totally resilient to disruption

attempts with limited resources ($20,000 in this case study). More resources for disruptions will result in

more reinforcement steps within the multiple energy carrier microgrid. Figure 4-8 shows that as the

reinforcement cost increases, the operation cost of the microgrid which reflects the lost electricity and heat

demand, decreases.

55

Figure 3-8 Operation and reinforcement costs of multi-energy carrier microgrid.

3.5 Summary

This chapter introduced a methodology to analyze the resilience of microgrids with multiple energy

carrier networks exposed to interruptions in electricity and natural gas distribution networks. The

vulnerability of multiple energy carrier microgrids is addressed by introducing energy hubs which

emphasizes the interdependence of electricity and natural gas energy infrastructures. It is shown that

disruption in an energy network within the microgrid has adverse effects on other interdependent energy

networks serving electricity and heat demands. It is observed that in order to maximize the adverse effects

of disruptions the attacker inclines to create isolated networks within the microgrid as shown in Cases 1, 2,

and 4. The proposed methodology suggests the reinforcement strategies for microgrid operators and

investors by highlighting the vulnerable components and procuring the disruption outcomes as a result of

component failures considering the interdependence of multiple energy infrastructures within the

microgrid. The recommended preventive reinforcement warrants the resilient operation of multiple energy

carrier microgrids considering limited reinforcement budget and the desired level of resilience. This chapter

is focused on proposing a methodology to identify the vulnerable components and ensure the resilient

operation of coordinated electricity and natural gas infrastructures considering multiple disruptions within

0

5000

10000

15000

20000

25000

30000

35000

40000

10

100

1,000

10,000

100,000

0 2 4 6 8 10 12 14 16

Op

erat

ion

co

st (

$)

Iterations

Operation Cost

Reinforcement Cost

Total Cost

To

tal/

Rei

nfo

rcem

entC

ost

($)

56

the microgrid. Although the diverse load scenarios with different values of lost load in multiple periods

would result in a different preventive reinforcement plans for the operation period, the preventive

reinforcement procedure remains the same. Defining the multi-period problem requires a detailed definition

of the attacks addressing the length of interruption for each attack and the restoration time for the

components if exists. This problem could be addressed as the future work for this chapter.

57

Chapter 4

4A HIERARCHICAL ELECTRICITY MARKET STRUCTURE FOR THE SMART GRID PARADIGM

This chapter is organized as follows: Section 4.1 describes the problem formulation. Section 4.2

introduces the proposed multi-level dynamic game with complete information. Section 4.3 presents a case

study, and section 4.4 presents the summary.


A method for analyzing the competition among GENCOs in the WEM is described in [Li05]. The

competition is modeled as a bi-level optimization problem in which the upper-level problem represents the

profit maximization of the GENCOs and the lower-level problem represents the WEM clearing process. A

framework for calculating multi-participant Nash equilibria in a transmission-constrained electricity market

is presented in [Kho14], where the non-cooperative complete information game is formulated for market

participants with discrete bidding strategies. In this section, the profit maximization problem for each

market participant is presented as a mathematical problem with equilibrium constraints (MPEC) which is

formulated as a bi-level optimization problem. The upper-level problem addresses the profit maximization

while the lower-level problem is the operation cost minimization. The non-cooperative complete

information game for the WEM and the DNEM is represented as an equilibrium problem with equilibrium

constraints (EPEC). An EPEC is formed by several MPECs, in which the lower-level problems are the

same for all MPECs. The EPEC renders a generalized Nash equilibrium problem (GNEP) as the strategy

taken by each market participant is dependent on the decisions made by other participants in the WEM and

the DNEM [Gab12]. Hence the proposed hierarchical market is composed of multiple EPECs in which the

equilibrium at one EPEC is dependent on the equilibrium of the other interdependent EPECs.

58

4.1.1 GENCO’s objective

GENCO j maximizes the payoff function which is shown in (4.1), subjected to the constraints on

the bidding vector as shown in (4.2). The payoff function represents the revenue of GENCO, which is the

first term in (4.1) as well as the generation cost, which is the second term in (4.1). GENCO maximizes the

payoff by proposing the bidding vector, which is formulated in (4.3).

2

,

P ( )maxo

j j j

j o j j j j j jP o B

R LMP P P

ψ (4.1)

j j j ψ ψ ψ (4.2)

2 ;j j j je j je je

P ρ ψ P P (4.3)

4.1.2 Microgrid’s objective

Microgrid m maximizes the payoff function (4.4), subjected to the limitations on the bidding vector

as shown in (4.5). Here, the first term in the payoff function is the revenue of the microgrid and the second

term is the generation cost. The generation dispatch of microgrids is provided by curtailing the IL and

dispatching the DERs. The microgrid may adopt several policies to serve its demand. One policy is to serve

local demand and bid the excess generation in the DNEM while the other is to bid on its total generation

capacity in the DNEM and serve the local demand economically from the distribution network. In both

cases, the bidding vector for generation in microgrid m is shown in (4.6).

20

,maxm m

m m m m m m mP

R P P P ψ

(4.4)

m m m ψ ψ ψ (4.5)

2 ;m m m me m me me

P ρ ψ P P (4.6)

59

Here 0 is the Lagrangian multiplier associated with the generation and load balance constraint

within the distribution network.

4.1.3 LA’s objective

Each LA participates in the WEM and the DNEM. The LA acts as a broker playing two different

roles: a) buy electricity from the WEM and sell the same volume to the DNEM, if the LMP on the LA’s

bus in the WEM is lower than that in the DNEM. b) purchase electricity from the DNEM and sell the same

volume to the WEM, if the MCP in the DNEM is lower than the LMP on the LA’s bus in the WEM.

Considering role “a”, LA l maximizes the payoff function which is shown in (4.7), subjected to the

bidding vector constraint (4.8). Here LMP is the marginal price of electricity on the LA’s bus in the WEM

which is determined as the cost of providing electricity to the distribution network. The LA’s bidding

vector in the DNEM is given in (4.9).

0,

max [( ) ]imp o

l l

impl o l

P o B

R LMP P

lψ

(4.7)

l l l ψ ψ ψ (4.8)

ol

l l oo B

LMP

ρ ψ (4.9)

exp

exp0

,max

ol l l

l o lP o B

R LMP P

ψ

(4.10)

l l l ψ ψ ψ (4.11)

0l l ρ ψ (4.12)

Considering role “b”, the LA l maximizes the payoff function which is shown in (4.10), subjected

to the constraints on the bidding vector (4.11). The market clearing price ( 0 ) in the DNEM is the marginal

cost of providing electricity to the WEM. The bidding vector of LA l in the WEM is shown in (4.12).

60

4.1.4 The WEM clearing process

In the WEM, the ISO minimizes the operational cost subjected to generation and transmission

network constraints. The WEM clearing problem is formulated as (4.13) – (4.17).

expmin j j l l

j l

P P ρ ρ (4.13)

Subjected to: exp

0impD

j ol lj l o l

P P P P (4.14)

exp( . . ) . . , ,

o o oj l l

impo o o D ob b j b b o b b b bl l

o B o B o B

F SF P SF P SF P SF P F b

(4.15)

j j jP P P j (4.16)

exp exp expl l l

P P P l (4.17)

0 ( )T

oo b b b

b

LMP SF (4.18)

The objective function which is shown in (4.13), is the operation cost of the system which

includes the awarded dispatch of market participants multiplied by their bids. The load balance constraint is

shown in (4.14). The transmission line capacity constraint is shown in (4.15). The generation capacity

constraint is shown in (4.16), and the limitation on the electricity exported by each LA is shown in (4.17).

As shown in (4.18), the Lagrangian multipliers associated with the load balance and transmission line

capacity constraints are employed to calculate the LMPs on each bus in the WEM.

4.1.5 The DNEM clearing process

The DNO minimizes the operational cost of distribution network as shown in (4.19)-(4.22).

minimp

m m l lm l

P P ρ ρ (4.19)

Subjected to: exp

0imp D

m l lm l l

P P P P (4.20)

61

m m mP P P m (4.21)

imp imp impl l l

P P P l (4.22)

The load balance in the distribution network is shown in (4.20). The constraints on the awarded

generation for microgrids and the imported power by the LA are shown in (4.21) and (4.22) respectively.

Here, 0 is the Lagrangian multiplier for the generation and load balance constraint in the distribution

network as it is assumed that the electricity delivery is not limited by the line congestions.

4.2 Multi-level complete information game

In case of competition with complete information, each market participant recognizes the

opponents’ payoff functions and bidding strategies. Hence the WEM clearing problem which is shown in

(4.13)-(4.17), and the DNEM clearing problem which is shown in (4.19)-(4.22), are linear programming

problems (LP) that are re-written in general form as (4.23)-(4.25) and (4.26)-(4.28) respectively.

min Tc x (4.23)

Subjected to: A x = b π (4.24)

x x x θ (4.25)

min Tc x (4.26)

Subjected to: Ax = b π (4.27)

x x x θ (4.28)

The bi-level problem for GENCO j is composed of the upper-level problem i.e. the payoff

maximization, which is shown in (4.1)-(4.2); and the lower-level problem, which is shown in (4.23)-(4.25).

Accordingly, the bi-level problem for microgrid m is composed of the upper-level problem, which is shown

(4.4)-(4.5); and the lower-level problem, which is shown in (4.26)-(4.28).

62

Since the LA l participates in multiple markets, if the LA plays role “a”, the upper-level problem

is the payoff maximization problem which is shown in (4.7)-(4.8) and the lower-level problem is shown in

(4.26)-(4.28). If the LA plays role “b”, the upper-level problem is shown in (4.10)-(4.11) and the lower-

level problem is shown in (4.23)-(4.25). In the three types of the bi-level problems presented, the upper-

level problems are the payoff maximization of the GENCOs, microgrids, and LAs while the lower-level

problems represent the operation cost minimization by the ISO or the DNO, subjected to the prevailing

constraints. The proposed bi-level problems are solved for the market participants in the WEM and the

DNEM simultaneously to determine the bidding strategies of market participants in the respective markets.

In this paradigm, microgrids and LAs participate in the DNEMs while GENCOs and LAs bid in the WEM.

Each LA which links the upper level market (WEM) and the lower level market (DNEM), determines its

bidding strategy and role based on the outcomes of each electricity market. The interaction between the

market participants in the WEM and the DNEM continues until none of the market participants in both

markets would update their bidding strategies. The synergy between the market participants in the WEM

and the DNEM is presented as Cournot game while the interaction between the WEM and the DNEMs is

presented as a dynamic game. Here microgrids, GENCOs, and LAs update their bidding strategies, based

on the sensitivity of the payoff function to the bidding strategy chosen. Considering the WEM, the bi-level

linear problems are converted into single level nonlinear problems using the Karush-Kuhn-Tucker (KKT)

optimality conditions for the lower-level problems. Assuming that the lower bound of inequality (4.25) is

zero, the KKT optimality conditions for (4.23)-(4.25) are presented in (4.29)-(4.34). Here ( )Diag X x ,

( )Diag Φ φ , ( )Diag Τ τ and ( )Diag Θ θ .

ΤΑ π + τ -θ = c (4.29)

Α x = b (4.30)

x +φ = x (4.31)

X τ = 0 (4.32)

Φ θ = 0 (4.33)

63

(x ,τ ,θ ,φ ) 0 (4.34)

By differentiating (4.29)-(4.34) with respect to the GENCOs’ and LAs’ bidding vector

components, the equality (4.35) is obtained.

Here the coefficient on the left hand side and the term on the right hand side of the system of

equations (4.35) are Η and d c respectively. In (4.35), H is a low rank sparse matrix, and once the least-

square technique [Woo96] is employed to solve this system of nonlinear equations, the sensitivity function

vector denoted as [ ]Td z π ψ x ψ τ ψ θ ψ is given as (4.36).

T

π

ψc

xA 0 I -Iψ

ψ0 A 0 00

τ0 Τ X 00

ψ0 -Θ 0 Φ0

θ

ψ

(4.35)

1( )T Td d z H H H c (4.36)

In the proposed iterative approach, the payoff of each market participant is maximized by

calculating d z from (4.36) iteratively. The sensitivity functions calculated in (4.36) are employed in (4.37),

(4.38), (4.41), and (4.42) to acquire the sensitivity of the payoff of each market participant with respect to

its bidding strategy at each iteration. The sensitivity of the payoff of GENCO j with respect to the bidding

strategy chosen is shown in (4.37). Other components in (4.37) were calculated in (4.38)-(4.40).

k k k k kj j j j oo

jk k k k kj j j o j

R R P R LMPj B

P LMP

(4.37)

, ,0 .

k kk ko oo b b

b jk k k kbj j j j

LMPSF j B

(4.38)

(2 )

kj k k o

o j j j jkj

RLMP P j B

P

(4.39)

64

kj k o

j jko

RP j B

LMP

(4.40)

Similar formulation is applied for microgrid m in the DNEM, with respective variables and indices.

For LA l with bidding strategy kl , the sensitivity of the payoff with respect to the bidding strategy

chosen, is shown in (4.41). The components which are shown in (4.41), are calculated in (4.42)-(4.44).

exp, exp, exp, exp,

exp,

k k k k kol l l l olk k k k k

l l o ll

R R P R LMPl B

LMPP

(4.41)

, ,0

k kk ko oo b b

b lk k k kbl l l l

LMPSF l B

(4.42)

exp,

0exp,

kk k ol

o lkl

RLMP l B

P

(4.43)

exp,exp,

kk ol

llko

RP l B

LMP

(4.44)

Similar formulation is applied for LA l importing power to the distribution network. Here (4.45)

and (4.46) are employed for updating the bidding strategies of GENCOs and LAs in the WEM,

respectively, and is the constant for each step. Similar formulation is applied for updating the bidding

strategies of microgrids and LAs importing power to the distribution network.

1kjk k

j j kj

R

(4.45)

exp,1

kk k l

l l kl

R

(4.46)

The presented game is considered as a dynamic game as the LAs observe the actions of other

market participants in one market before updating their bidding strategy in the other. Here LAs consider the

decision made by GENCOs once bidding in the DNEM. Similarly, LAs observe the decision made by

65

microgrids once bidding in the WEM. The bidding vector of LAs in each market is determined based on the

awarded dispatch and the price of electricity at the LAs’ bus in the other market. The observability

indicator v is a binary parameter, which indicates that the LAs observed the actions of participants in the

other market at each iteration. The bidding strategy of market participants update iteratively until the

convergence criteria are satisfied and the generalized Nash equilibrium (GNE) is established. The presented

game may have one, multiple or no GNE. However, the market regulations will limit the bidding strategy

of market participants and improve convergence to a state in which the market participants are either

unwilling to unilaterally update their bidding strategy or they cannot do so. As illustrated in Figure 4-1, the

steps (a)-(n) are taken to procure the GNE of the proposed dynamic game. While multiple LAs could

participate in the WEM, for the sake of simplicity, only one LA (LA l ) is considered in this algorithm. The

following approach can be extended to address multiple LAs in multiple DNEMs.

a) Set the initial value of power import/export of the LA to zero with respective arbitrary large initial

marginal costs and initiate

0 0 0 0j m l l ψ ,ψ ,ψ ,ψ for each generator, microgrid, and LA; then go to “b”.

b) Set the iteration indices 1k and LA’s observability indicator 0v ; then go to “c”.

c) If the inequality (4.47) holds LA imports electricity to DNEM; then go to “d”. Otherwise go to “e”.

, 1 exp, 1imp k v k vl l

P P

(4.47)

d) Set the parameters using (4.48)-(4.50) for the DNEM while the LA’s role is to import electricity to this

market; then go to “f”.

1k k vo oLMP LMP (4.48)

, , 1 exp, 1imp k imp k v k vl l l

P P P

(4.49)

exp,0

kl

P (4.50)

e) Set the parameters using (4.48), (4.51), and (4.52) for DNEM while LA’s role is to export electricity

from this market; then go to “f”.

66

, , 1imp k imp k vl l

P P

(4.51)

exp, exp, 1k k vl l

P P

(4.52)

f) DNO clears DNEM; then go to “g”.

g) The bidding strategy of microgrids and LA l within the distribution network are updated using the

sensitivity functions of their payoff with respect to their bidding strategy; then go to “h”.

h) If the inequality (4.53) holds, LA exports electricity to the WEM; then go to “i”. Otherwise go to “j”.

, exp, 1imp k k vl l

P P

(4.53)

Figure 4-1 Competition in the hierarchical electricity market

LA export electricity

from distribution

market (e)

LA export electricity(h)?


Market of LA l

Choose initial bidding strategies for all GENCOs

and LAs and microgrids (a)

Initial iteraction indices and indicators k=1, v=0 (b)

DNO clears the distribution network market (f)

ISO clears the wholesale electricity market(k)

Microgrids/LA l update bidding strategies (g)

Y

LA export electricity

to wholesale market

(j)

LA import electricity

from wholesale

market (i)

N

GENCOs/LA l update bidding strategies (l)

LA imported electricity(c)?N

Y LA import electricity

to distribution market

(d)

Toggle vObserved(m)?

Y

k=k+1

N

Nash Equilibiruim

(Optimal bidding Strategy for each market participant)

Converged(n)?

Y

N

67

i) Set the parameters using (4.54)-(4.56) for the WEM while LA’s role is to export electricity to the WEM;

then go to “k”.

10 0k k v (4.54)

exp, exp, 1 ,k k v imp kl l l

P P P

(4.55)

,0

imp kl

P (4.56)

j) Set the parameters using (4.54) and (4.57) for the WEM while LA’s role is to import electricity from the

WEM.

exp, exp, 1k k vl l

P P

(4.57)

k) ISO clears the WEM; then go to “l”.

l) The bidding strategy of the GENCOs and LA l within the WEM are updated using the sensitivity

functions of their payoff with respect to their bidding strategy.

m) Check if the observability indicator is equal to 1.0, go to “n”. Otherwise, toggle the observability

indicator go to “c”.

n) If the conditions (4.58),(4.59) hold for all market participants in the WEM and DNEM, then GNE is

established; otherwise, set 1k k , set the observability indicator to zero, and go to “c”.

1(.) (.)k k (4.58)

1(.) (.)k k (4.59)

Assuming the existence of an equilibrium, convergence to different local equilibriums depends on

the initial bidding strategies. As the feasible region for each market participant is non-convex, the

equilibrium among MPECs represent a local optimum solution. In [Rui12], the MPECs are rewritten as a

mathematical problem with primal dual constraints (MPPDC). Considering fixed dual variable associated

68

with the strong duality constraint of the MPPDC, the EPEC is further formulated as a parameterized mixed-

integer linear problem (MILP). The solution of the MILP is a local Nash equilibrium or a saddle point

depending on the choice of the fixed dual variable. In [She84], the existence of a unique equilibrium under

certain assumptions is proved, while the existence and uniqueness of an equilibrium is proved in [DeM09]

by relaxing some constraints. The alternative method to the sensitivity function employed in this chapter, is

to reformulate the problem in to nonlinear complementarity problem (NCP) [Ley10], [Fle04].

4.3 Case study

In order to evaluate the effectiveness of the proposed approach, a WEM with two generators, and

one DNEM including two microgrids and one LA are considered. Figure 4-2 shows the proposed network

configuration. Here, the DNEM and the WEM have the total demand of 75 MW and 260 MW respectively.

Table 4-1 shows the network data for both electricity markets. The coefficients of the generation

cost curve for GENCOs and microgrids are shown in Table 4-2. As shown in this table, the marginal cost of

electricity generation for the microgrids are higher than that for the GENCOs. The lower and upper limits

for the bidding strategy vector for market participants are 0.5 and 4.0 respectively. The lower limit for the

bidding strategy of LA is 1.0. The initial bidding strategy for LA is set to 1.0. The multiplier and the

threshold for convergence are set to be 1E-5 and 1E-4, respectively. The initial marginal cost of LA in

the WEM and DNEM is 1000$/MWh.

Table 4-1 Transmission and distribution network data

Line From

bus

To

bus

Reactance

(p.u)

Capacity limit

(MW)

L1 3 4 0.0098 160

L2 4 5 0.0105 250

L3 5 6 0.02 190

L4 6 3 0.01 130

L5 1 3 0.005 100

L6 2 3 0.008 90

Table 4-2 Generation cost curve of microgrids and GENCOs

Market

Participant

Min.

capacity

(MW)

Max.

capacity

(MW)

($/MW2)

($/MW)

($)

MG1 0 60 0.125 25 0

MG2 0 60 0.250 20 0

G1 0 210 0.073 6.926 0

G2 0 210 0.066 7.352 0

69

Figure 4-2 Power network topology

The following three cases are presented:

Case 1) higher marginal cost of electricity in microgrids with no congestion in the wholesale electricity

market.

Case 2) higher marginal cost of electricity in microgrids with congestion in the wholesale electricity

market;

Case 3) lower marginal cost of electricity in microgrids with no congestion in the wholesale electricity

market.

4.3.1 Case 1) higher marginal cost of electricity in microgrids with no congestion in the wholesale

electricity market

In this case, the microgrids provide higher marginal cost for electricity compared to the GENCOs as shown

in Table 4-2. The bidding strategies and the awarded dispatches for each market participant are listed in

Table 4-3. Since there is no congestion in the WEM, the MCP is 80 $/MWh, while the MCP in the DNEM

is 140 $/MWh. In this case, the lower price of electricity in the WEM provides incentives for the LA to

import electricity from this market to the DNEM. The payoff of MG1, MG2, LA, G1 and G2 are $4,400,

MG1 MG2

2

G2

5

3

6

50MW 25 MW

100 MW

4

G1

90 MW

70 MW

1


Electricity Market

Wholesale

Electricity Market

70

$2,300, $900, $8,800, and $8,604 respectively. Hence the LA spent $1,200 to purchase 15MW electricity

and earned $2,100 by selling the same volume in the DNEM. The exchanged power, the LMP, and the

bidding strategies of the market participants in the WEM and the DNEM are illustrated in Figure 4-3.

In the first iteration an arbitrary large number is assigned as the LMP at the LA’s bus in the WEM

(1000$/MWh) and LA bids into the DNEM based on the assigned LMP at the WEM. The awarded dispatch

imported to DNEM and the MCP at the DNEM is determined. As shown in Figure 4-3a, in iterations 2 to

36, the bidding strategy of G1 is greater than G2, which in turn is greater than the bidding strategy of the

LA in the WEM. Similarly, as shown in Figure 4-3b, the bidding strategy of MG1 is greater than that of

MG2, which in turn is greater than the bidding strategy of the LA in the DNEM. As shown in Figures 4c

and 4d, the LA is awarded 20MW in the DNEM and the MCP in the DNEM is significantly larger than the

LMP at LA’s bus in the WEM. At iteration 95, the LMP at the LA’s bus in the WEM becomes closer to the

MCP of DNEM which would lead to zero power import by the LA in the DNEM. As the demand in the

DNEM is inelastic to the MCP, MG1 bids lower than MG2 to increase the generation in the DNEM as

shown in Figure 4-3b. At iteration 171, the LMP at the LA’s bus in the WEM is larger than that in the

DNEM which inspires the LA to export electricity from the DNEM as illustrated in Figure 4-3a. However,

the response of the GENCOs to such decision will alter the exported dispatch of the LA in the WEM. At

iteration 189, G2 reaches the upper limit of the bidding strategy regulated by the market rules. In response,

the bidding strategy of G1 reaches 3.64, and the LMP of LA’s bus in the WEM will be 80 $/MWh as

shown in Figures 4a and 4d. Ultimately as shown in Figure 4-4, the LA is awarded 15 MW in the DNEM at

iteration 204, as the MCP at the DNEM is higher than the LMP of LA’s bus at the WEM. This situation

continues until MG2 reaches its highest possible bidding strategy (i.e. 4.0) limited by the market regulation.

As illustrated in Figure 4-3b, the bidding strategy of MG1 reaches 3.74, and the MCP of DNEM becomes

140 $/MWh.

71

(a) Bidding strategy of market participants in the WEM

(b) Bidding strategy of market participants in the DNEM

(c) Exchanged power for the LA (negative/positive values represent imported power

to/exported power from the DNEM)

(d) LMP of LA’s bus in the WEM and MCP of DNEM

Figure 4-3 Bidding strategies of market participants, exchanged power, and energy price in Case

0

1

2

3

4

1 51 101 151 201 251 301B

idd

ing S

trat

egy

Iteration

LAG1G2

0

1

2

3

4

1 51 101 151 201 251 301 351 401 451 501

Bid

din

g S

trat

egy

Iteration

MG1

MG2

LA

-40

-20

0

20

40

0 50 100 150 200 250

Exch

nag

ed

Po

wer

[M

W]

Iteration

-10

40

90

140

1 51 101 151 201 251 301 351 401 451 501

LM

P [

$/M

Wh]

Iteration

MCP of DNEM LMP of LA in WEM

72

Table 4-3 Bidding strategy and awarded dispatch in Case 1 MG1 MG2 LA G1 G2

Bidding Strategy 3.74 4 1.05

(imports power) 3.64 4

Awarded Dispatch

(MW) 40 20 15 140 135

Figure 4-4 Market clearing results in Case 1

4.3.2 Case 2) higher marginal cost of electricity in microgrids with congestion in the wholesale

electricity market

In this case, the impact of transmission line congestion is considered in two different cases.

4.3.2..1 Congestion in Line 1

In this case, the capacity limit of line L1 is set to 70 MW. Because of the congestion on L1, the

LMP on bus 3 in the WEM is increased to 841.20 $/MWh while the MCP in the DNEM is 180 $/MWh as

shown in Figure 4-5. This would provide incentives for the LA to export 27.25MW from the DNEM to the

[40 MW]

MG1

[20 MW]

MG2

2

[135 MW]

G2

5

3

6

L4

L1

50MW 25 MW

100 MW

4

L2

L3

[140 MW]

G1

90 MW

70 MW

1


Electricity Market

Wholesale Electricity

Market

[MCP: 80 $/MWh]

[MCP: 140 $/MWh]

[15 M

W]

1 3.74 2 4

1.05l

1 3.64

2 4

73

WEM. As listed in Table 4-4, G2 is only awarded 22.75 MW with the LMP equal to 48 $/MWh, while G1,

is awarded its maximum generation capacity of 210 MW with the LMP equal to 253.78 $/MWh on bus 4.

In this case MG1’s strategy is to sell more electricity with lower price while MG2’s strategy is to sell less

electricity with higher price. The payoff of MG1, MG2, LA, G1 and G2 are $8,850, $6,313, $18,802,

$48,620, and $886 respectively. Here, the payoff of G2 decreased dramatically compared to that in Case 1,

as a result of congestion while the payoffs of other market participants especially G1 and LA were

increased.

Table 4-4 Bidding strategy and awarded dispatch in case 2 (L1 is congested) MG1 MG2 LA G1 G2

Bidding Strategy 1 4 4

(exports power) 4 4

Awarded Dispatch (MW) 60 42.25 27.25 210 22.75

Figure 4-5 Market clearing result in Case 2.1

4.3.2..2 Congestion in Line 4

In this case, the capacity limit of line L4 is set to 30MW. As a result of the congestion on line L4,

the LMP on bus 3 in the WEM is 72.53 $/MWh which is lower than that in Case 1. The awarded dispatch

[60 MW]

MG1

[42.25 MW]

MG2

2

[22.75 MW]

G2

5

3

6

L4

L1< 70 MW

50MW 25 MW

100 MW

4

L2

L3

[140 MW]

G1

90 MW

70 MW

1


Electricity Market


Market

[MCP: 180 $/MWh]

[27

.25

MW

]

1 1 2 4

4l

1 3.64

2 4

[LMP: 841.2 $/MWh]

[LMP: 48 $/MWh]

[LMP: 253.8 $/MWh]

74

for the LA in the DNEM is 15 MW. The imported electricity to the DNEM mitigates the congestion on line

L4 in the transmission network, and the LA has the incentive to increase the electricity import to the

DNEM to reduce the LMP at bus 3 in the WEM.

As shown in Figure 4-6, the MCP in the DNEM is 130 $/MWh, which is lower than that in Case 1. As

shown in Table 4-5, G2 is awarded 117.52 MW with the LMP equal to 80 $/MWh while G1 is awarded

157.48 MW with the LMP equal to 88 $/MWh as a result of congestion on line L4. MG1 and MG2 were

awarded 20 MW and 40 MW in the DNEM respectively. In this case, MG2 proposes lower bid resulting in

the increase in its awarded generation and the decrease in the imported power from the WEM. The payoff

of MG1, MG2, LA, G1 and G2 are $2,050, $4,000, $862, $10,956, and $7,626 respectively. In this case,

the payoffs of LA, MG1, and G2 decreased compared to those in Case 1, while MG2 and G1 had much

higher payoffs as a result of congestion on line 4.

Table 4-5 Bidding strategy and awarded dispatch in case 2 (L4 is congested)

MG1 MG2 LA G1 G2

Bidding Strategy 4 2.89 1.02

(imports power) 4 4

Awarded

Dispatch (MW) 20 40 15 157.48 117.52

Figure 4-6 Market clearing results in Case 2.2

[60 MW]

MG1

[42.25 MW]

MG2

2

[22.75 MW]

G2

5

3

6

L4

L1< 70 MW

50MW 25 MW

100 MW

4

L2

L3

[140 MW]

G1

90 MW

70 MW

1


Electricity Market


Market

[MCP: 180 $/MWh]

[27.2

5 M

W]

[LMP: 841.2 $/MWh]

[LMP: 48 $/MWh]

[LMP: 253.8 $/MWh]

75

4.3.3 Case 3) lower marginal cost of electricity in microgrids with no congestion in the wholesale

electricity market

In this case, the generation costs for microgrids are reduced to 20% of those shown in Table 4-2,

which is lower than the marginal costs of GENCOs in the WEM. The bidding strategy and the awarded

dispatch for each market participant are listed in Table 4-6. Here, the LMP at all buses in the WEM is 80

$/MWh, while the MCP in the DNEM is 36 $/MWh. The payoffs of MG1, MG2, LA, G1 and G2 are

$1,770, $1,740, $1,980, $8,800, and $5,077 respectively. Here the LMP at LA’s bus in the WEM is higher

than the MCP in the DNEM, hence, the LA exports power from the DNEM. Compared to Case 1, G1 has

the same profit, the profit of G2 is decreased, and the profit of LA is increased. Since the LMP at LA’s bus

in the WEM is higher than that in the DNEM, the LA exports electricity to the WEM.

Table 4-6 Bidding strategy and awarded dispatch in case 3 MG1 MG2 LA G1 G2

Bidding Strategy 1.75 4 1

(exports power) 3.64 4

Awarded Dispatch

(MW) 60 60 45 140 75

Once the market participants bid on their marginal prices of electricity, the payoff of MG1, MG2,

LA, G1 and G2 are $697, $667, $84, $400, and $577 respectively and the LMP at all buses in the WEM is

20 $/MWh and the MCP in the DNEM is 18.12$/MWh. In this case, the LA export 45 MW to the WEM.

The total payoff of all market participants is decreased by $15,046 in this case compared to the case in

which the market participants bid strategically.

Figure 4-8 shows the bidding strategies of the market participants in the WEM and the DNEM, the

LMP of LA’s bus in the WEM, as well as the MCP of DNEM. Similar to Case 1, in the first iteration an

arbitrary large number is assigned as the LMP at the LA’s bus in the WEM (1000$/MWh) and LA bids in

the DNEM based on the assigned LMP at WEM. In this case, the awarded dispatch imported to the DNEM

is zero and the MCP at DNEM is determined. As shown in Figure 4-8a, in all of the iterations, the bidding

strategy of G2 is greater than that of G1, which in turn is greater than the bidding strategy of LA in the

WEM. As shown in Figure 4-8b, in iterations 2 to 353, the bidding strategy of MG1 is close to that of MG2

and the bidding strategies of MG1 and MG2 are greater than the bidding strategy of the LA in the DNEM.

76

As shown in Figures 4-8c, the LMP at LA’s bus in the WEM is significantly larger than the MCP in the

DNEM, hence the LA is awarded 45MW in the WEM. At iteration 283, G2 reaches the upper limit for the

bidding strategy regulated by the market. In response, the bidding strategy of G1 reaches 3.64, and the

LMP of LA’s bus in the WEM will be 80 $/MWh as shown in Figures 4-8a and 4-8c. The bidding strategy

of MG1 reaches 1.75 in iteration 363. This situation continues until MG2 reaches its highest possible

bidding strategy (i.e. 4.0) limited by the market regulation, and the MCP of DNEM becomes 36 $/MWh, as

illustrated in Figure 4-8b.

Figure 4-7 Market clearing results in Case 3

[60 MW]

MG1

[60 MW]

MG2

2

[75 MW]

G2

5

3

6

L4

L1

50MW 25 MW

100 MW

4

L2

L3

[140 MW]

G1

90 MW

70 MW

1


Electricity Market


Market

[MCP: 36 $/MWh]

[45 M

W]

[MCP: 80 $/MWh]

77

(a)

(b)

Figure 4-8 Bidding strategies of market participants and energy prices in Case 3 (a) Bidding

strategy of market participants in WEM. (b) Bidding strategy of market participants in DNEM.

(c) LMP of LA bus in WEM and MCP of DNEM

0

1

2

3

4

1 51 101 151 201 251 301

Bid

din

g S

trat

egy

Iteration

LAG1G2

0

1

2

3

4

1 201 401 601 801

Bid

din

g S

trat

egy

Iteration

MG1

MG2

LA

0

20

40

60

80

100

1 201 401 601 801

LM

P [

$/M

Wh]

Iteration

MCP of DNEM LMP of LA in WEM

4.4 Summary

Microgrids are the building blocks of active distribution networks in the smart grid paradigm

which provide energy and ancillary services in distribution and wholesale electricity market. In this

chapter, the hierarchical market structure for the smart grid paradigm which is composed of the WEM and

the DNEM is proposed. Here the LA represents as the middle agent which participates in the distribution

network and the WEMs competing with microgrids and GENCOs. The competition among market

participants in both electricity markets with complete information is presented as a dynamic game. The

bidding strategy chosen by each market participant is procured using bi-level linear programming problem

with the upper-level problem representing the market participant’s payoff maximization and the lower-level

problem minimizing the operation cost of the network. The bi-level problems are solved by developing

78

sensitivity functions for the market participants’ payoff with respect to their bidding strategies. The result

shows the impact of LA’s bidding strategy on the bidding strategy and payoffs of other market participants

in the WEM and the DNEMs. While the non-cooperative dynamic game with complete information is

proposed in this chapter, the proposed approach can be extended to address the non-cooperative dynamic

game with incomplete information.

79

Chapter 5

5RISK AVERSE GENERATION MAINTENANCE SCHEDULING WITH MICROGRID

AGGREGATORS

This chapter presents a risk-averse formulation for generation maintenance scheduling using microgrid

aggregators in the bulk power system. Microgrid aggregators capture the characteristics of aggregated

distributed generation units and demand served by a large number of microgrids. The system operators

leverage the proposed framework in order to determine the most effective long-term generation

maintenance schedule while ensuring the secure and economic operation of power systems through

balancing the demand and supply. The system operator incorporates the characteristics of the generation

units for maintenance practices including the minimum and maximum periods in which the unit should be

on scheduled outage, the duration of the maintenance, and respective maintenance costs. Moreover, the

system operator determines the marginal cost of the generation units and local generation assets for

microgrid aggregators based on the submitted marginal costs (bids) for the short-term operation practices.

The individual characteristic of the local generation in the microgrids is inherently captured by the

aggregated generation bid, generation capacity, and respective uncertainties in such values. The presented

formulation is a two-stage model in which the first stage captures multiple maintenance practices in long-

term operation horizon by introducing the minimum and maximum time for the maintenance practices, and

in the second stage, the worst realization of uncertainties in the long-term maintenance scheduling is

revealed. The uncertainty in available generation resources of microgrids, the demand of the power system,

the marginal cost of the microgrids as well as contingencies in the transmission network are addressed and

further limited by the budget of uncertainty. The contributions of this chapter are as follows:

• The long-term generation maintenance scheduling of power system is formulated as a two-stage

robust optimization problem in which the first stage addresses the maintenance scheduling of

the generation units, while the second stage captures the operation decisions once the worst

realization of the uncertainties is revealed.

• A new formulation for long-term maintenance scheduling is presented that addresses multiple

80

maintenance practices for generation units considering the minimum and maximum periods for

the availability of the units prior to maintenance, as well as the required number of periods for

the maintenance practices.

• The presented formulation determines the worst realization of uncertainties in the marginal cost

of the microgrid generation, the capacity of the local generation assets in microgrids, electricity

demand, and the availability of the transmission network components considering a certain

budget for the uncertainties in the power system. Furthermore, the impact of the budget of

uncertainty on the operation and maintenance costs is evaluated.

• The effect of the aggregated microgrids on the long-term generation maintenance scheduling is

addressed by performing sensitivity analysis on the capacity of local generation and the

marginal cost of electricity in microgrids.

The rest of the chapter is organized as follows, the problem formulation and solution methodology are

described in Sections 5.1 and 5.2 respectively. A case study to show the effectiveness of the problem is

shown in Section 5.3. The summary is presented in Section 5.4.


The mathematical formulation for the generation maintenance scheduling is presented in (5.1)-(5.24). The

objective is to minimize the maintenance and the operation cost of the power system as shown in (5.1). The

first term in (5.1) represents the maintenance cost for all generation units in the power system and the

second term represents the operation cost of the system. The operation cost of the system is affected by

several uncertain variables in the operation horizon. These uncertain variables are the demand in power

system, the available local generation assets in microgrids that reduces the net demand realized by the

power system, the marginal cost of the generation units in microgrids, and the contingencies in the

transmission networks that changes the power flow pattern in the system and affects the operation cost. The

second term in (5.1) represents the operation cost minimization that is further maximized over the variables

in the uncertainty set. The operation cost of each generation unit is a quadratic cost function of the

generation dispatch, which is further linearized using a piece-wise linearization technique as shown in

(5.2)-(5.4) [Sha02]. The dispatch of the generation unit is limited by the maximum capacity of the unit as

81

shown in (5.5). The relationship between the binary variables representing the transition of the unit to

outage state for maintenance, the binary variables representing the transition from outage to available state,

and the state of the unit at each period is shown in (5.6). As shown in this constraint, once the unit goes on

maintenance from the available state, ,i tX will be 1 and the unit will change its state from available to

unavailable. Similarly, once the unit becomes available from an outage state in the maintenance period, ,i tY

will be 1 and the state of the unit ( ,i tI ) will change from 0 to 1. The number of periods in which the

generation unit was available at the beginning of each period is determined by (5.7) and (5.8). Here, if the

state of the generation unit is not transitioning from outage to available mode, the number of periods that a

generation unit was available at the beginning of each period is increased by one. The number of periods

the generation unit should be available is less than a maximum limit as shown in (5.9) and more than a

minimum limit as enforced by (5.10). Therefore, the period in which the unit should go on maintenance is

defined by constraints (5.9) and (5.10). Once the unit is recovered from the scheduled maintenance, the unit

will be available for the minimum number of periods and can go on maintenance for the second time

between the minimum and maximum periods assigned for the maintenance practices. The transition states

are mutually exclusive as shown in (5.11). The number of periods in which the unit is on maintenance is

determined by (5.12) and (5.13). The number of periods the unit should be on maintenance is enforced by

(5.14)-(5.15). Here, once the unit is recovered from the outage state to the available state, ,i tX is 0, and

,i tY is 1. Therefore, the number of periods the unit is on maintenance is enforced to be equal to iRHF . The

generated power at each bus is determined by (5.16) and the nodal power balance in the power system is

determined by (5.17). The power transmitted through the transmission line is dependent on the difference

between the voltage angles of the interconnected buses as shown in (5.18)-(5.19) if the transmission line is

available. Here, once the transmission line is disconnected because of contingency, the voltage angles of

the buses connected by the transmission line are relaxed. The power transmitted through the transmission

line is limited by the capacity of the transmission line as shown in (5.20). The capacity of the local

generation asset in microgrids connected to each node of the power system, is limited to a certain portion of

the demand at the corresponding node as shown in (5.21). The uncertainty in the marginal cost of the

generation units in microgrids, the uncertainty in the capacity of local generation assets in microgrids, and

82

the uncertainty in the demand of the power system are limited by (5.22), (5.23), and (5.24) respectively. As

shown in (5.22)-(5.24), the uncertain variables are identified by the nominal values and the uncertainty

interval. The available generation capacity of microgrid aggregators is dependent on the available energy

resources (e.g. renewable resources and fuel). Furthermore, the marginal cost of generation within

microgrids is dependent on several factors including the generation technology, the fuel cost, and the

inflation and interest rates. In order to capture the operation and maintenance costs, the investment costs,

the inflation and interest rates, and the labor cost associated with the electricity generation within the

microgrids, the Levelized Cost of Energy (LCOE) for the generated electricity within microgrids can

replace the marginal cost. As the number of microgrids is large and the generation technology utilized in

each microgrid could be different, capturing the marginal cost and generation capacity with respective

uncertainties for each microgrid is challenging and practically infeasible in this problem. Therefore, the

marginal cost and capacity of the local generation resources in microgrids with respective uncertainties are

aggregated and associated with the microgrid aggregators.

,, , , ,

, ,, , , ,

, , ,1 1 1 1 1

min max ming g d

i tl t b t b t b t

NT NG NT NG NB NMg m g m

i i t i t b t b tPu C P Pt i t i b m

MC X F C NW P

(5.1)

, ,1

NSs s

i t i i ts

F NW C P

1,: i t

(5.2)

, ,1

NSs

i t i ts

P P

2,: i t

(5.3)

,max,s s

i t iP P

1, ,: i t s

(5.4)

max, ,i t i i tP P I

2,: i t

(5.5)

, , , , 1i t i t i t i tY X I I (5.6)

, , 1 , 1 ,i t i t i t i i tHO HO I MHO Y (5.7)

, , 1 , 1 ,i t i t i t i i tHO HO I MHO Y (5.8)

83

, ,(1 )i t i i tHO MHO Y (5.9)

, ,i t i i tHO NHO X (5.10)

, , 1i t i tX Y (5.11)

, , 1 , 1 ,(1 )i t i t i t i i tHF HF I RHF X (5.12)

, , 1 , 1 ,(1 )i t i t i t i i tHF HF I RHF X (5.13)

, ,(1 )i t i i tHF RHF X (5.14)

, ,i t i i tHF RHF Y (5.15)

, , ,1

NG

i b i t b ti

A P P

3,: b t

(5.16)

,, , , ,,

NLg md

b t b t b l l tb tm l

P P P S f

4,: b t

(5.17)

, , ,(1 )NB

tl t l t b l b l

b

f M u S X

6,: l t

(5.18)

, , ,(1 )NB

tl t l t b l b l

b

f M u S X

7,: l t

(5.19)

max, ,l t l l tf f u

3 4, ,: ,l t l t

(5.20)

,, ,,

g m db t b tb t

m

P P

5,: b t

(5.21)

, , ,0 , , ,0 ,, , , , ,

,g m g m g m g m g mb t b t b t b t b t

C C C C C (5.22)

0 0, , , , ,,b t b t b t b t b t

(5.23)

84

,0 ,0, , ,, ,

,d d d d db t b t b tb t b t

P P P P P (5.24)

The presented problem is formulated as a two-stage robust optimization problem [Ber13] in which, the

first stage problem determines the first stage i.e. “here-and-now” variables including the scheduled

maintenance of the generation units in the power system, and the second stage problem will determine the

second stage i.e. “wait-and-see” variables once the uncertainties are revealed. These variables include the

generation dispatch of the available units, the flow of the transmission lines, the dispatch of local

generation assets, as well as the realization of the uncertainties including the marginal cost and capacity of

the microgrids’ generation, availability of the transmission lines, and the power system demand. The

uncertain variables determined at this stage belong to a polyhedral uncertainty set and the objective is to

minimize the sum of first-stage and second-stage costs considering a budget for the uncertainty that

represents the conservativeness of the decision maker and the size of uncertainty interval.

5.2 Solution Methodology

The column-and-constraint generation approach as a cutting plane procedure is proposed to solve the

presented two-stage robust optimization problem [Zen13]. This approach has superior computation

efficiency compared to the Benders decomposition [Ji16]. The problem presented in (5.1)-(5.24) is

presented in general form as (5.25)-(5.28) in which x is the first stage decision variable while y and u are

the second stage (i.e. recourse) decision variables. The objective function (5.1) is represented as (5.25) and

the set of constraints (5.26) represents all constraints with binary decision variables including the states and

the transition states of the units that were formulated in (5.5)-(5.15). The set of constraints shown by (5.27)

captures the second stage decision variables y such as the dispatch of generation units as well as the

realization of the uncertain variables u such as the demand of power system and the capacity of local

generation assets in microgrids. The feasibility set for first stage and second stage decision variables are

shown in (5.28). As enumerating all scenarios with different realization of u is practically challenging, the

solution approach leverages partial enumeration over a subset u considering the budget of uncertainty to

determine a valid relaxation to the original problem. The presented column-and-constraint generation

85

procedure further determines significant scenarios that contribute to the worst realization of the system

operation cost.

min maxminT T

x yu

c x b y

(5.25)

Subjected to: Ax d (5.26)

Fx Gy h Hu (5.27)

, x yx Ω y Ω (5.28)

The column-and-constraint generation procedure is implemented as a master problem and a sub-problem.

The algorithm for this procedure is shown as below:

a) Set iteration 0, ,LB UB with the determined realization of uncertainty *(0)u ,

b) Solve

,min

T ex y

c x (5.29)

Subjected to: ( )Te b y (5.30)

Ax d (5.31)

( ) *( ) Fx Gy h Hu (5.32)

, x yx Ω y Ω (5.33)

c) Set * *TLB e c x where *x and *e are the solution of (5.29)-(5.33); and go to (d),

d) Solve sub-problem (5.34)-(5.36) to determine the new realization of the uncertainties.

maxminT

yu

b y (5.34)

Subjected to: * ( 1) Fx Gy h Hu (5.35)

( 1), yy Ω u U (5.36)

e) Set * *T TUB c x b y where

*y is the solution of (5.34)-(5.36); and check if

UB LB

UB

. If the

86

condition is satisfied terminate the process, otherwise, add (5.37) and (5.38) to (5.29)-(5.33), then set

1 and go to (b).

( 1) ( 1)* Fx Gy h Hu (5.37)

( 1)Te b y (5.38)

Assuming that the worst realization of uncertainties occur at extreme points within the polyhedral

uncertainty set, the sub-problem (5.34)-(5.36) is re-formulated as a mixed-integer programming problem

shown in (5.39)-(5.52) by capturing the dual representation of inner minimization problem.

In order to determine the worst realization of the uncertain variables, binary variables ,db tu , ,

db tv , ,b tu ,

,b tv , ,l tu , ,g

b tu and

,g

b tv are introduced. Each pair of variables is mutually exclusive as shown in (5.47)-

(5.49). Moreover, an N-k contingency analysis is presented in (5.50), in which up to k components could be

unavailable in each period. The budget of uncertainty limits the combination of the binary variables that

yields the worst realization of the uncertain variables as shown in (5.51) and (5.52). Here, E and L are the

budget of uncertainty chosen by the decision maker.

4 ,0 4, , , , ,,,max 1 max 2

, , , ,,5 0 ,0 5 ,01 1 1, , , , , ,, ,

max 3,

ˆ

max

d d d db t b t b t b t b tNG NS NB b ts

i t s i i t i ti td di s b

b t b t b t b t b t b tb t b t

l l t

P u v P

P P I

P u v P

f

u,v,u ,vu ,u ,v

1

4 6 7, , , , ,

1

. 1

NT

tNL

l t l t l t l t l tl

u M u

(5.39)

1, 1i t

,: i tF (5.40)

1 1 2, , , , 0s

i i t i t s i tC NW ,: s

i tP (5.41)

2 2 3, , , ,

1

0NB

i t i t i b b ti

A

,: i tP

(5.42)

3 4, , 0b t b t

,: b tP (5.43)

, ,0 , , ,4 5, , , , , ,

( ).g m g m g m g m

b t b t b t b t b t b tNW C u v C

,,

:g m

b tP

(5.44)

87

4 6 7 3 4, , , , , ,

1

. 0NB

l b b t l t l t l t l tb

S

,: l tf (5.45)

6 7, , , ,

1

0NL

l b l t l l b l t ll

S X S X

: t

b (5.46)

, , 1d db t b tu v

(5.47)

, , 1b t b tu v (5.48)

, ,1

g gb t b t

v u (5.49)

,1

NL

l tl

u k

(5.50)

, , , , , ,1 1

NB NTg gd d

b t b t b t b t b t b tb t

u v u v v u E

(5.51)

,1 1

NL NT

l tl t

u L

(5.52)

The binary-to-continuous variable multiplication in the objective function (5.39) requires further

linearization as shown in (5.53)-(5.56) by introducing new auxiliary continuous variable . Here, is the

continuous variable and u is the binary variable of the binary-to-continuous multiplication, while M is a

large number which is considered as the upper bound of the continuous variable .

(1 )M u (5.53)

(1 )M u (5.54)

M u (5.55)

M u (5.56)

The solution to this problem is used to add more constraints (5.37), (5.38) to the master problem once the

condition at step (e) is not satisfied. Utilizing the solution of (5.39)-(5.52) and (5.57)-(5.59), a new

88

realization of uncertainties ( *u ) is captured by adding the constraints (5.37) and (5.38) to the master

problem.

,0, , , , ,,

ˆ ˆd d d d d db t b t b t b t b tb t

P P P u P v (5.57)

0, , , , , ,ˆ ˆb t b t b t b t b t b tu v

(5.58)

,0, , , , , ,

ˆ ˆg g g g g gb t b t b t b t b t b t

C C C u C v (5.59)

5.3 Case Study

In this section, two case studies were presented to show the effectiveness of the presented

approach as well as the impact of considerable penetration of microgrids on the maintenance scheduling

practices in the power system. The first case study uses a sample 6-bus power system while the second case

study utilize the modified IEEE-118 bus system.

5.3.1 6-Bus Power System

In this section, a 6-bus power system, which is composed of 3 thermal generation units and 7

transmission lines is utilized. The characteristics of the transmission line and the thermal generation units

are shown in Tables 5-1 and 5-2, respectively. The generation cost curve for the generation units is

piecewise linearized with four equal segments. The generation units G1, G2, and G3, are connected to

buses 1, 2, and 5 respectively. The minimum and maximum available periods as well as the required

maintenance periods for units G1-G3 were shown in Table 5-2. The maintenance cost for G1, G2 and G3

are $40,000, $30,000 and $10,000 respectively. The time step for this case study is one week and the

operation horizon is one year (52 weeks). The weekly system demand profile is shown in Figure 5-1.

In this case study, the contingency in transmission lines is ignored and the following cases are

considered:

Case 1 – Maintenance scheduling without aggregated microgrids

Case 2 – Maintenance scheduling with aggregated microgrids

Case 3 – Maintenance scheduling without aggregated microgrids considering the uncertainties

89

Case 4 – Maintenance scheduling with aggregated microgrids considering the uncertainties

Case 5 – Maintenance scheduling with aggregated microgrids knowing the probability distribution of

uncertainties

Table 5-1 Transmission line characteristics

Line

ID

From

Bus

To

Bus

Impedance

(p.u.)

Maximum

Power Flow

(MW)

L1 1 2 0.170 100

L2 1 4 0.258 100

L3 2 4 0.197 70

L4 5 6 0.140 60

L5 3 6 0.018 120

L6 2 3 0.037 150

L7 4 5 0.037 70

Table 5-2 Thermal units characteristics

Unit C1

($/MWh)

C2

($/MWh)

C3

($/MWh)

C4

($/MWh)

Pmax

(MW)

NHO

(Week)

MHO

(Week)

RHF

(Week)

G1 15 30 40 62.5 520 10 48 4

G2 20 32 46.25 73 360 10 48 5

G3 30 38 75 98 200 5 25 2

Figure 5-1 Demand profile for one year

5.3.1..1 Case 1 – Maintenance scheduling without aggregated microgrids

In this case, no microgrid is considered in the power system and the maintenance schedules of the

generation units are determined considering the nominal values for the demand in the power system. The

total maintenance and operation cost is $63.494M. In this case, unit G1 goes on maintenance in weeks 14-

17 when the demand is low; unit G2 goes on maintenance in weeks 42-46, and unit G3 goes on

maintenance in weeks 12-13 and 39-40. The total capacity on outage, in this case, is shown in Figure 5-2.

0

50

100

150

200

250

300

350

400

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

Lo

ad (

MW

)

Period (Week)

90

Figure 5-2 Scheduled capacity on outage in Case 1

As shown in this figure, the minimum and maximum maintenance period for unit G3 are 5 and 25 weeks

respectively. Therefore, once G3 goes on maintenance in weeks 12 and 13, it needs to go on maintenance in

weeks 39 and 40.

In this case, the transmission lines L1, L2, L6 and L7 will be congested. By increasing the

capacity of these lines to twice of the capacity shown in Table 5-1, the maintenance scheduling pattern will

change as shown in Figure5-3. In this case, the total maintenance and operation cost will decrease to

$62.473M. As shown in this case, the congestion in the transmission line will impact the maintenance

schedule of the generation units as well as the total maintenance and operation cost.

Figure 5-3 Scheduled capacity on maintenance in Case 1 with the increase in the transmission line capacity

5.3.1..2 Maintenance scheduling with aggregated microgrids

Integrating microgrids in the power system provides higher flexibility at the demand side. Local

generation assets in microgrids can serve the demand once the locational marginal price (LMP) of

electricity falls beyond their marginal cost. In this case, the generation capacity of the microgrids is

91

assumed as 30% of the demand on each bus. The marginal cost of the local generation assets in microgrids

is 30 $/MWh. Here, the total operation and maintenance cost is $62.217M and units G1 and G2 are

scheduled for maintenance in weeks 14-17, and 40-44 respectively. Unit G3 goes on maintenance in weeks

9-10 and 34-35. The effect of microgrid on the net demand profile is shown in Figure 5-4. The net demand

is defined as the realized demand by the thermal generation units. In other words, the net demand is the

demand of the power system which is not served by the generation assets within the microgrids. As shown

in this figure, unit G1, which is the largest and cheapest available unit, will be on outage for maintenance in

periods that the net demand is low. Similarly, unit G2 goes on maintenance in the periods that the net

demand is reduced by utilizing the local generation of microgrids.

Figure 5-4 Demand profile with and without microgrids

The microgrids will change the net demand profile and affect the scheduled maintenance for unit

G3. As shown in this case, the scheduled maintenance is shifted from weeks 12-13 and 39-40 in Case 1, to

week 9-10 and 34-35 in this Case. Moreover, as a result of deviation in the net demand the maintenance

and operation cost is decreased. As the generation capacity in the microgrids increases, the maintenance

and operation cost will further decreases. Once the microgrids are completely self-sufficient (i.e. the total

demand can be served by local generation units in microgrids), the total maintenance and operation cost is

decreased to $62.155M and units G1 and G2 will be on planned outage for maintenance in weeks 13-16

and 21-25 respectively. Moreover, unit G3 is maintained in weeks 23-24 and 48-49.

5.3.1..3 Maintenance scheduling without aggregated microgrids considering the uncertainties

In this Case, the uncertainty in demand is considered as ±10% of the nominal values that are shown in

Figure 5-1, and the budget of uncertainty is 100 uncertain variables in the operation horizon. The total

0

50

100

150

200

250

300

350

400

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

Lo

ad (

MW

)

Period (Week)

Demand (MW)

Net Demand with Microgrids

92

maintenance and operation cost, in this case, is $72.08M. The generation units G1 and G2 will be on outage

for maintenance in weeks 11-14, and 42-46 respectively. Unit G3 will be on outage for maintenance in

weeks 15-16, and 40-41.

By comparing Case 3 to Case 1, it is shown that the total operation and maintenance cost is

increased from $63.494M in Case 1 to $72.08M in Case 3 as a result of the introduced uncertainties in the

long-term operation horizon. Introducing the uncertainties will also affect the pattern of the generation

maintenance practices in the long-term operation horizon. As the budget of uncertainty decreases, the

operation cost will decrease because the decrease in the flexibility to determine the worst realization of the

uncertain variables. Once the budget of uncertainty is reduced to 30, the total maintenance and operation

cost is reduced to $66.736M. Figure 5-4 shows the total maintenance and operation cost with respect to the

determined budget of uncertainty. As shown in Figure 5-5, once the budget of uncertainty reached 240, the

total maintenance and operation cost is not changed. Moreover, the increase in the uncertainty interval will

lead to the increase in the total maintenance and operation cost. As shown in Figure 5-5, once the

uncertainty interval increases to ±15% of the nominal values, the total operation and maintenance cost will

increase. Similarly, reducing the uncertainty intervals will reduce the total operation and maintenance cost.

Figure 5-5 Increase in the total operation and maintenance cost because of the increase in the budget of

uncertainty

5.3.1..4 Maintenance scheduling with aggregated microgrids considering the uncertainties

In this Case, microgrids will serve the power system demand and the uncertainties associated with

the marginal cost of the local generation assets in microgrids as well as the uncertainties in the generation

capacity of microgrids are captured. The uncertainty bound for the marginal cost of generation assets in

microgrids is ±20% of the nominal marginal cost. The uncertainty bound for the local generation capacity

is ±10% of the nominal generation capacity within the microgrids. Once the budget of uncertainty is

60

65

70

75

80

85

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400

Op

erat

ion a

nd

Mai

nte

nan

ce C

ost

($

)M

illi

ons

Budget of Uncertainty

Deviation=10%Deviation=15%Deviation=5%

93

considered as 100 uncertain variables, the total maintenance and operation cost of the system is $67.886M,

which is lower than that in Case 3. The scheduled outage for maintenance for units G1, G2 are in weeks 14-

17, and 40-44 respectively. Unit G3 will be on outage for maintenance in weeks 26-27 and 51-52. Similar

to Case 3, as the budget of uncertainty increases, the total maintenance and operation cost will increase. In

this case, as the budget of uncertainty increases to 420, the maintenance and operation cost increases to

$72.183M. Moreover, the maintenance schedule pattern for G1 and G2 are shifted to weeks 44-47 and 48-

52 and the maintenance of G3 is in weeks 15-16 and 42-43. By comparing Case 4 to Case 2, it is shown

that the total maintenance and operation cost increased from $62.217M to $67.886M as a result of

introducing the uncertainty sets in the risk-averse formulation. Furthermore, the generation maintenance

pattern was changed.

5.3.1..5 Maintenance scheduling with aggregated microgrids knowing the probability

distribution of uncertainties

In this case, the probability distribution functions of the uncertain variables are known and a risk-neutral

solution using stochastic programming is proposed for the generation maintenance scheduling problem.

The probability distribution function for electricity demand, the marginal cost of generation in the

aggregated microgrids, and the generation capacity of the aggregated microgrids are considered as normal

distribution functions with the mean values equal to the forecasted values and the standard deviation equal

to 3.33%, 6.66% and 3.33% of the mean values. In the proposed probability distribution functions, the

uncertain variables are between ±10%, ±20%, and ±10% of the mean values with 99.7% confidence

interval. In this case, the uncertainties are captured by generating 3,000 scenarios using Monte-Carlo

simulation and fast backward/forward method is used to reduce the number of effective scenarios to 12 by

eliminating the low probability scenarios and bundle the comparable scenarios [Dup03], [GAM].

The total maintenance and operation cost is $64.168M which is higher than that in Case 2

($62.217M) and lower than that in Case 4 ($67.886M). Moreover, in this case, the generation units G1 and

G2 were on maintenance in weeks 23-26, and 31-35 respectively. The generation unit G3 is on

maintenance in weeks 15-16, and 42-43.

94

5.3.2 IEEE 118-bus system

The modified IEEE 118-bus system with the total demand profile shown in Figure 5-6 is considered. The

system is composed of 54 generation units and 186 transmission lines. Generation units G4, G10, G11,

G27-G29, G36, G39, G40, G43-G45 with generation capacity over 300 MW are considered as candidates

for the maintenance. The marginal cost of the aggregated microgrids is shown in Figure 5-7. The capacity

of the local generation in microgrids is 30% of the demand on the network bus. Unlike previous case study,

the uncertainties in this case study include the contingencies in the transmission network. The candidate

lines for N-1 outage are L1, L10, L15, L38, L51, L90, L94, L103, and L126. The deviation of demand, the

marginal cost of the microgrid, and the local generation assets of the microgrids is ±10% of their respective

nominal values. Cases 1-4 that were introduced for the previous case study are considered here.

In Case 1, where there is no microgrid in the system and there is no contingency considered, the

total maintenance and operation cost is $649.422M and the planned outages for generation units were

shown in Table 5-3. Integrating microgrids in the power system in Case 2 decreases the total maintenance

and operation cost to $529.38M, which is lower than that for Case 1.

Figure 5-6 Demand profile for IEEE 118-bus power system

0

1000

2000

3000

4000

5000

6000

1 3 5 7 9 111315171921232527293133353739414345474951

Dem

and (

MW

)

Period (week)

95

Figure 5-7 Marginal cost of the local generation for the aggregated microgrids

The total maintenance and operation cost is further affected by the marginal cost (MC) of the generation

units in microgrids as shown in Figure 5-8. In this figure, the marginal cost of the microgrid aggregators is

changed by applying marginal cost multiplier (MCM). Here, with the increase in the marginal cost of the

local generation assets in microgrids, the generation units are used less frequent and the operation cost will

increase. Moreover, increasing the capacity of the local generation assets in microgrids can further affect

the net demand profile and reduces the total maintenance and operation cost of the power system. As

shown in Figure 5-8 with the increase in the capacity of the local generation assets with lower marginal

costs, the maintenance and operation cost decreases. As the marginal cost of the microgrids increases, the

effect of microgrids on the total maintenance and operation cost of the power system decreases. Similarly,

reduction in the installed capacity of low-cost local generation assets in microgrids will increase the total

operation and maintenance cost.

In Cases 1 and 2, no contingency in transmission line was considered and the demand is set to the

nominal values as shown in Figure 5-6. In Cases 3 and 4, the demand and the marginal cost of the

microgrids, as well as the capacity of the generation resources are considered as uncertain variables.

Moreover, the worst realization of the N-1 contingencies is also captured considering the budget of

uncertainty for the contingencies in the transmission network. The budget of uncertainty for the

contingency and other uncertainties in the operation horizon are 5 and 30 respectively.

0% 33% 66% 100%

Percentage of the full generation capacity

Marg

inal

Co

st (

$/M

Wh

)

15

30

45

96

Figure 5-8 Sensitivity of the total operation and maintenance cost to the installed generation capacity and

marginal cost multiplier (MCM) of microgrids

Table 5-3 Scheduled maintenance in weeks for cases 1-4

Component Case 1 Case 2 Case 3 Case 4

G4 14-18 14-18 13-17 42-46

G10 40-42 40-42 14-16 12-14

G11 44-46 11-13 40-42 14-16

G27 14-16 50-52 16-18 49-51

G28 41-44 17-20 43-46 41-44

G29 48-50 14-16 41-43 18-20

G36 50-52 42-44 50-52 48-50

G39 43-45 14-16 41-43 34-36

G40 17-20 39-42 14-17 2-5

G43 40-42 42-44 46-48 40-42

G44 13-16 45-48 44-47 45-48

G45 46-48 45-47 18-20 15-17

Table 5-4 The periods of transmission network contingencies for cases 1-4

Component Case 1 Case 2 Case 3 Case 4

L1 - - - -

L10 - - - -

L15 - - - -

L38 - - 2, 7, 8, 9, 10, 24, 25

29, 30, 31, 33 17

L51 - - 14, 46 12,30,14,16

L90 - - 5 21,22,26,12,2,9,10

L94 - - - 21, 19, 50, 43, 5, 14, 32,

48

L103 - - - 24,31

L126 - - - 29

0

100

200

300

400

500

600

700

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Oper

atio

n a

nd M

ainte

nan

ce C

ost

($)

Mil

lions

Capacity of Local Generation in Microgrids (Percentage of Demand)

MCM = 0 MCM = 0.5MCM = 1 MCM = 1.5MCM = 2

97

Table 5-3 and 5-4 show the determined outages for maintenance and the outages considered to

yield the worst realization of the transmission network contingencies respectively. As shown in Table 5-3,

dispatching the local generation resources in microgrids will impact the maintenance schedule of the

generation assets in the power system. Moreover, microgrids can reduce the total maintenance and

operation cost of the power system by adjusting the net demand profile during the operation horizon. The

total maintenance and operation cost in Case 3 is $657.84M which is larger than that in Case 1 as a result of

uncertainties captured. The total maintenance and operation cost in Case 4 is $533.45M. Table 5-4 shows

the periods in which the contingencies in transmission network occurred for Cases 1-4. As shown in this

Table, most contingencies applied to lines L38, and L94 in Cases 3 and 4 respectively.

5.4 Summary

This chapter presents an approach for long-term maintenance scheduling in power systems

considering with large penetration of microgrids and the uncertainties in the operation horizon. The

uncertainty set captures the variation in the power system demand, the marginal cost of the microgrids, the

installed generation capacity within the microgrids, as well as the contingencies in the transmission

network. A two-stage robust optimization problem is formulated and column-and-constraint generation

procedure is proposed to solve the presented problem. A budget for uncertainty is considered to address a

trade-off between the conservativeness of the solution and the performance of the solution methodology.

The presented approach is applied to two case studies. The sensitivity of the total maintenance and

operation cost to the installed capacity of generation units in microgrids, the marginal cost of the

microgrids, and the considered budget of uncertainty was shown in a case study. It is shown that if the

marginal cost of the local generation in microgrids is small, leveraging the generation capacity of

microgrids to regulate the demand will decrease the total operation and maintenance cost of the power

system. Moreover, the total operation and maintenance cost is further decreased with the increase in the

capacity of the low-cost local generation assets in microgrids. It is also shown that the risk-averse solution

that captures the uncertainties in the long-term operation horizon will lead to higher maintenance and

operation cost compared to the deterministic solution. The presented risk-averse solution further compared

with risk-neutral solution knowing the probability distribution of the uncertain variables. It is shown that

98

the risk-averse solution will lead to higher operation and maintenance cost as the worst realization of the

uncertainties was captured.

99

CHAPTER 6

6COORDINATED OPERATION OF ELECTRICITY AND NATURAL GAS SYSTEMS: A CONVEX

RELAXATION APPROACH

In the electricity network, the unit commitment problem, which is solved by the independent system

operator (ISO), involves binary decision variables that represent the commitment of the generation units.

This problem has a discrete nonconvex feasibility region as a result of incorporated binary decision

variables. The contributions of this chapter are as follows: 1) develop an exact, tight, and computationally

inexpensive convex relaxation for the short-term operation problem in electricity and natural gas networks;

2) propose a consensus optimization framework for the coordinated operation of the electricity and natural

gas networks. The presented algorithm can be used by system operators to determine the tight convex

relaxation of the operation problems that lead to global solutions for the short-term operation of electricity

and natural gas networks. Furthermore, compared to earlier research [Sha05],[Gei07],[Mar12],[Cor14] that

assumed the electricity network operator has access to the natural gas network information, the presented

consensus optimization framework requires limited information being exchanged among the system

operators. The rest of the chapter is organized as follows, the problem formulation is presented in Section

6.1. The convex relaxation of the proposed problems and the solution methodology is presented in Section

6.2. A case study is presented in Section 6.3 to show the effectiveness of the proposed relaxations. The

summary is presented in Section 6.4.


In this section, the short-term operation problems for the natural gas and electricity networks are

formulated and the challenges for solving the coordinated operation of these systems using the consensus

optimization framework are discussed.

6.1.1 Short-term Operation of Natural Gas Network

The mathematical formulation for the natural gas network operation is presented in (6.1)-(6.8). The

100

objective is to minimize the natural gas consumption cost and the objective function is formulated as shown

in (6.1). The natural gas pipeline capacity constraint is given in (6.2). The natural gas supply volume at

each resource is further limited by (6.3). The operation limits on the natural gas pressure at each junction

are given in (6.4). The Weymouth equation presents the dependence of natural gas flow in the pipeline to

the pressure at the junctions on both sides of the pipeline as given in (6.5) and (6.6) [Men05],[Cfg]. The

nodal natural gas flow balance at each junction is illustrated in (6.7) where ( )H Pi i can be replaced by its

piecewise linear form. The natural gas compressor is modeled as (6.8) [Tom07].

,min .s GS j

s s tt

v

(6.1)

,max , ,max, , ,p p t pj o j o j of f f (6.2)

,s t ssv v v (6.3)

,j t (6.4)

( ).,

, , , ,, ,sgn ,p t p

j t o t j t o tj o j of C

(6.5)

1 , ,sgn ( )

1 , ,, ,,

j t o t

j t o tj t o t

(6.6)

, ,

, ,( ), ,, , ,s GS p P p P k HH i GGj f j t j j j

p t p t h H Ps t i i tj o j o k tsjA v f f P

(6.7)

, ,, ,, 1p p

j t o tj o j o

(6.8)

The presented problem is formulated as a mixed integer nonlinear optimization problem (MINLP). In

order to determine the decision on the natural gas flow direction in the pipeline, (6.5)-(6.6) are further

formulated as (6.9), where ,,t

j ou is a binary variable that is 1 if , ,j t o t and is 0 if , ,j t o t . Therefore,

the Weymouth equation represents a nonlinear non-convex constraint with discrete decision variables.

, ,, , , ,, , , ,2

p t p ptj t o t j t o tj o j o j o j of C u C

(6.9)

101

6.1.2 Short-term Operation of Electricity Network

The short-term operation problem for the electricity network is formulated in (6.10)-(6.17) as a unit

commitment problem. As shown in (6.10), the objective is to minimize the operation cost of the system.

The objective function is the summation of the operation costs of all generation units. The generation

dispatch of each generation unit is limited by the minimum and maximum capacity of the unit as shown in

(6.11). Minimum up/downtime constraints are given in (6.12)-(6.13). The ramp up/down limits are

enforced by (6.14)-(6.15). It is assumed that unit i sets at minimum generation iP prior to shutting down and

after starting up. The system-wide generation and demand balance is shown in (6.16). The power flow on

the transmission line l is limited by (6.17), considering the respective shift factors that relate the power flow

of the transmission line to the nodal injected power.

2, , ,

,min

i i

i i t i i t i i tP I t i

P P I (6.10)

Subjected to: , , ,. .i i t i t i i tP I P P I (6.11)

,( 1) ,( 1) , 0on oni t i i t i tX T I I

(6.12)

,( 1) ,,( 1)0

off offi t i tii t

X T I I

(6.13)

max, ,( 1) , ,( 1) , ,( 1)0 (1 ( )). ( )i t i t i t i t i i t i t iP P I I RU I I P (6.14)

max,( 1) , ,( 1) , ,( 1) ,0 (1 ( )). ( )i t i t i t i t i i t i t iP P I I RD I I P (6.15)

, ,D

b t i tb i

P P (6.16)

, ,. .

b

b b Dl l i t l b t l

b i B

F SF P SF P F

(6.17)

102

6.2 Proposed Convex Relaxation

6.2.1 Background

A theoretically strong convex relaxation that leverages the moment relaxation is proposed in [Las10].

The first order of moment relaxation represents the semi-definite relaxation of the original problem and as

the order of the moment relaxation increases, the asymptotic convergence to the global optimal solution for

the relaxed problem is guaranteed in polynomial time [La10]. However, the increase in the number of

variables in the moment relaxation matrices will lead to a considerable computational burden that impedes

the utilization of such relaxations in practice.

By employing the chordal property of the electricity network graph, a sparse moment relaxation is

formulated in [Mol15]. Despite the fact that the sparse formulation is a step forward to practically utilize

the theoretically perfect moment relaxation approach, the computational burden is still an obstacle even for

the sparse formulation presented for the medium-sized networks [Mol15]. The solution procured by

formulating lower order moment relaxations is computationally less expensive compared to those for

higher order moment relaxations; however, such solutions are not feasible for the original problem unless

the rank of the moment matrix is one. If the rank of a moment matrix is larger than 1, a higher order of

moment relaxation is required [Las10]. In such cases, the procured solution with a higher rank of the

moment matrix is a lower (upper) bound – yet infeasible solution – for the expected global optimal solution

of the minimization (maximization) problem. Since utilizing the higher order of moment relaxation is

computationally expensive, the alternative is to employ perturbation to procure a rank-1 solution for the

first order moment relaxation problem. However, the perturbation will lead to a rank-1 solution which

renders a feasible yet not necessarily optimal solution for the original problem. Another approach to

reducing the rank of the first order moment relaxation is to add valid constraints that include the elements

of the first order moment relaxation matrix [Koc16]. An example of such valid constraints is the

McCormick and disjunctive constraints. Although valid constraints will help to reduce the rank of the first

order moment relaxation matrix, the rank reduction may not lead to a rank-1 solution (i.e. the optimal

solution that is feasible for the original non-convex problem.) Therefore, it is vital to develop a

comprehensive approach to procuring a computationally inexpensive tight convex relaxation that renders a

103

rank-1 solution by incorporating the lessons learned from the earlier attempts.

As shown in the natural gas and electricity network operation problems, the binary decision variables

introduce non-convexity in the feasibility region. In order to handle these variables, the binary decision

variables are characterized by introducing constraints (6.18)-(6.19). For the sake of simplicity, the index of

hour t is eliminated from the presented formulation.

0 1rI (6.18)

2

r rI I (6.19)

By using (6.18)-(6.19), the constraints (6.9) and (6.11) with discrete decision variables are

transformed into nonlinear and nonconvex constraints with continuous decision variables ( rI ).

Incorporating (6.18)-(6.19) in (6.10)-(6.17) and (6.1)-(6.8) will lead to an NLP problem that is NP-hard

with no guarantee for global or local optimal solution in polynomial time. In the next section, tight convex

relaxations of the presented operation problems in electricity and natural gas networks will be formulated.

The solution to the formulated relaxed problems is the solution to the original problem.

6.2.2 Convex Relaxation of Natural Gas Network Operation Problem

The non-convexity associated with the operation of the natural gas network is because of introducing

constraint (6.9). Figure 6-1 shows the developed algorithm to solve the short-term operation problem for

the natural gas network. The steps of this algorithm are further described as follows:

Step (a) Define a reference vector of the first order monomials for the pressure of all junctions as given in

(6.20).

1 2[ .... ]NJ

π (6.20)

Step (b) Define monomials associated with the pipelines connected to each junction as given in (6.21).

Such monomials could be further expanded in the solution procedure.

j o j oj o j oj u u

, ,

( )[1 ]x y y y (6.21)

104

Figure 6-1 The algorithm to procure an optimal solution for the natural gas network operation problem.

Define a reference monomial vector associated with

natural gas pressures of all junctions (a)

Reformulate the natural gas transportation network

in SDP relaxed form (d), then solve the problem

Feasible

optimal

solution

Tightening the feasible region by adding RLT

constraints (e), then solve the problem

Tightening the feasible region by adding

McCormick and disjunctive constraints for element

of minors with higher ranks (g)

Add the elements of those 2by2 minors with higher

rank to the monomial vector associated with natural

gas pipelines connected to each junction (h)

Define the monomial vector associated with natural

gas pipelines connected to each junction (b)

Construct the sparse first order moment matrices

based on the monomial vector associated with

natural gas pipelines connected to each junction (c)

The ranks of all

moment matrices are 1?

The ranks of all 2by2

minors of the moment

relaxation matrix are 1? (f)

The ranks of all 2by2

minors of the moment

relaxation matrix are 1? (f)

No Yes

No

Yes

Yes

No

105

Here, (.)y represents the lifting variable corresponds to the nonlinear terms in the non-convex

optimization problem. For example, j o

y is the lifting variable that represents the nonlinear term

j o . Therefore, the nonlinear terms are substituted in the relaxed formulation using the corresponding

lifting variables (.)y . The general representation of the vector of monomials is given in (6.21). The first

element in this vector is 1 which represents the normalization of the monomial. The second vector is the

monomials associated with the direction of each pipeline that connects junctions j and o . It should be

noted that if the direction is a priori, the binary decision variable j o

u,

in monomials will be fixed. The third

vector of monomials represents the nonlinear relationship among nodal gas pressure at both sides of the

natural gas pipeline. This vector of monomials is formed for all pipelines that are connected to each

junction regardless of the decision on the direction of natural gas flow in them. The fourth monomial vector

is introduced to represent the binary to continuous terms associated with all the pipelines that are connected

to the junction j , where The elements of the monomial vector only capture the pipelines for which the

decision on the natural gas flow (j o

u,

) is a variable. Go to step (c).

Step (c) The moment matrix of the presented monomials for the junction j in (6.21), introduces a set of

lifting variables as presented in (6.22).

( ), ,

( () ), , , ,

) )( ( (), , ,

) )( ( ( () ), , , ,

1

2

u uj o j oj o j o

u u u uj o j o j o j oj o j o

u u uj o j oj o j o j oj o j o

u u u uj o j oj o j o j o j oj o j o

jj o

y y y

y y y yX

y y y y

y y y y

(6.22)

There are six types of independent variables in the moment matrix (6.22). Three of them (i.e. j o

u

,

y

,j o

y , and u j o j o

( ),y ) are the lifting variables given in (6.21). Two of them (i.e.

j and

o ) are given in

step (a) that are utilized in various moment matrices of each junction as represented by (6.20). The last

variable (i.e. u j oj o )( ,

y ) is a lifting variable that is introduced in (6.22) to complete the moment matrix.

Go to step (d).

106

Step (d) By utilizing the lifting variable that was introduced in step (c), the SDP relaxation for the short-

term operation of the natural gas network given in (6.1)-(6.8) is reformulated as (6.23)-(6.28). Here the

decision variables are the elements of the moment matrix (6.22) as well as the vector of pressure given in

(6.20).

,min . ( )s GS j j o

pss j j o

o

A C

y (6.23)

,

,max ,max, , ,( )

( 2 )j oj o j o

p p pj o j o j ou

f C f

y y (6.24)

, ( )j o

psj sj os

o

v A C v

y

(6.25)

, ,, 1p p

j j oj o j o (6.26)

,

,

( ) ( 2 )( ), , ,

( 2 ) ( )( ), ,

s GS p Pj f j

p P k HH i GGt j j j

p psA C C uj j o j oj o j o j o j oo

p hC H Pu i ij o j o j o j o kP

y y y

y y

(6.27)

0jX (6.28)

The presented SDP problem is solved to determine the optimal solution for the original problem. If the

rank of all sparse moment matrices associated with each junction is one, the procured optimal solution is

feasible for the original non-convex problem (6.1)-(6.8). Otherwise, the algorithm proceeds to step (e) to

further tighten the presented relaxation and to render rank-1 moment matrices.

Step (e) In this step, a tighter SDP relaxation is formulated by leveraging reformulation-linearization

technique (RLT). The convex-relaxed problem (6.23)-(6.28) is presented in a compact form as shown in

(6.29)-(6.34). The nodal balance constraint given in (6.27) is converted into two inequalities as shown in

(6.30), (6.31). The inequalities given in (6.24)-(6.26) are also presented in the general form in (6.30)-(6.31).

RLT is used to form valid constraints (6.32) and (6.33) to tighten the relaxation, while the moment matrices

defined in step (c) (i.e. j

X ) and the monomials are given in step (b) (i.e. j

x ) are utilized to formulate these

107

constraints.

,

minj j

Tj j

j

x Xξ x (6.29)

1 1 0j j

j g x h (6.30)

2 2 0j j

j g x h (6.31)

1 1 1 11 1 1 10

T T T Tj j j jTj j j jj j j g X h x g x hg g h h (6.32)

2 2 2 22 2 2 20

T T T Tj j j jTj j j jj j j g X h x g x hg g h h (6.33)

0jX (6.34)

Step (f) Solving the SDP relaxation problem with additional valid constraints will reduce the rank of the

moment matrices corresponding to the procured solution (i.e. the lower bound to the optimal solution.)

However, the rank of some of the sparse moment matrices may remain higher than one. To add a new set of

valid constraints, it is necessary to exploit those elements of the moment matrix that result in the solution

with a rank higher than 1. Theorem 1 is employed to identify such elements in moment matrices.

Theorem 1. The rank of a moment relaxation matrix is one, if and only if the rank of all 2by2 minors (e.g.,

( , ),( , )i k n mjX where i, k, n, m are the indices for the rows and columns of jX ) of that matrix are one,

( , ),( , )( ) 1 ( ) 1 , , ,

i k n mj jrank rank i k n m X X . The proof of this theorem is discussed in [Koc16].

In this step, the ranks of all 2by2 minors are checked. If the ranks of all 2by2 minors are one, the convex

relaxation is tight and the optimal solution for the original problem is procured. Otherwise, if the previous

step was step (e) then go to step (g); if the previous step was step (g) then go to step (h).

Step (g) Once the minors with a rank higher than 1 are identified in step (f), McCormick constraints and

disjunctive cuts are employed to reduce the rank of each minor within any moment matrix. The McCormick

constraints for a bi-linear term xy are given in (6.35)-(6.38), where the upper and lower bounds of variable

x and y are given as x y, and x y, respectively. This technique is applied to each term associated with the

higher than rank-1 minors of each moment matrix.

xy x y x y x y (6.35)

108

xy x y x y x y . (6.36)

xy x y x y x y (6.37)

xy x y x y x y (6.38)

A list of such valid constraints are given in (6.40)-(6.43) and in the Appendix, where the scalar

lifting variables that are used as the elements of the moment matrix (6.22) are captured. As an example

here, the upper and lower bounds of ( ),u j o j o

y

in the 2by2 minor given in (6.39) are presented in (6.40)-

(6.43).

)( ,

( () ), ,

1u j o

u uj o j oj o j o

y

y y

(6.39)

Here, the lower and upper bounds of the off-diagonal elements in the 2by2 minor are ( ),

0 1u j o

y and

j o

y

( )

0 . Using (6.35) and considering the lower bounds as zero for the off-diagonal elements,

(6.40) is formed. All elements of the moment matrix are nonnegative. Similar to (6.36), the upper bounds

of the off-diagonal elements of the 2by2 minor (6.39) are used in (6.41) to form a lower bound for the

diagonal element (( ),u j o j o

y

) in the 2by2 minor shown in (6.39). The upper bounds of the element

( ),u j o j oy

are enforced in (6.42)-(6.43) using the general form (6.37)-(6.38). As

u j oy

( ),0 1 , the upper

bound of( ),u j o j o

y

is j o

y ( )

as given in (6.42). Using (6.37), j o

y ( )

and its upper bound ( ) are

multiplied by zero which is the lower bound of u j o

y )( ,. Similarly,

j oy

( ) is multiplied by 1 which is the

upper bound of u j o

y )( ,. Using (6.38),

u j oy )( ,

and its upper bound (i.e. 1) are multiplied by zero which is the

lower bound ofj o

y ( )

. Therefore, the only non-zero term in (6.43) is u j o

y )( , multiplied by which is

the upper bound ofj o

y ( )

. The rest of the valid constraints are presented in the Appendix.

u j o j oy

( ),0 (6.40)

109

u uj o j oj o j o

y y y

( ), ,

( )(1 ) (6.41)

( ),u j o j o j oy y

(6.42)

u uj o j oj o

y y

( ), ,

( ) (6.43)

In this step, only the valid constraints associated with the 2by2 minors with a rank higher than 1 – that

were determined in step (f) – will add more cutting planes to the feasible region of the problem to further

tighten the relaxation. Go to step (f) to further check the rank of the 2by2 minors.

Step (h) The elements of the 2by2 minors with a rank higher than 1 will be added as new

monomials to the vector of monomials in step (b). Traditionally, it is necessary to employ a higher order

moment matrix to procure a rank-1 solution [Las10]. However, in this step, by applying Theorem 1, only

certain elements that are present in the 2by2 minors with a rank higher than 1, will be added to the vector of

monomials to tighten the feasible region of the procured moment relaxation. Therefore, instead of adding

all elements of the higher order moment matrix for each junction, adding a limited number of monomial

will not increase the size of the moment matrix dramatically. Go to step (b).

6.2.3 Convex Relaxation of the Electricity Network Operation Problem

The algorithm developed for the convex relaxation of this problem is similar to the algorithm

developed for the natural gas network operation problem. The objective is to determine a tight convex

relaxation for the electricity network operation problem considering the binary decision variables that

represent the commitment of the generation units. The following steps are taken to achieve this objective.

Step (a) A vector of monomials is introduced as shown in (6.44).

NG NG

I I P Py y y y

1 1

[1 ... ... ]z (6.44)

Step (b) The associated first-order moment matrix of the presented monomials is given in (6.45) in which

the binary-to-continuous terms are incorporated.

110

1 1

1 1 1 1 1 1

1 1

21 1 1 1 11

21 1

1 ... ...

... ...

... ... ... ... ... ... ...

... ...

... ...

... ... ... ... ... ... ...

... ...

NG NG

NG NG

NG NG NG NG NG NG

NG NG

NG NG NG NG NG NG

I I P P

I I I I I P I I

I I I I I P I P

P I P I P P PP

P I P I P P P P

y y y y

y y y y y

y y y y y

y y y y y

y y y y y

Z

(6.45)

Step (c) The lifting variables introduced in (6.45) are utilized to reformulate the electricity network

operation problem (6.10)-(6.17) to (6.46)-(6.51). Here, the inequality constraints presented in (6.48)-(6.49)

represent nodal power balance in the electricity network. If the rank of the moment matrix (6.45) is one, the

procured lower bound solution is an optimal solution for the original problem and the algorithm will

terminate, otherwise, proceed to step (d).

21

minii

i i P i IPZ i

y y y (6.46)

Subjected to: . .i i ii I P i IP y y P y (6.47)

i i

Db I P

b i

P y (6.48)

'

2

'i i

Db PP

b i i

P y

(6.49)

. .i

b

b b Dl l P l b l

b i B

F SF y SF P F

(6.50)

0Z (6.51)

Step (d) Add the RLT constraints to tighten the procured relaxation. If the tightened relaxed problem

renders a rank-1 solution, the procured lower bound is the optimal solution for the original problem.

Otherwise, employ Theorem 1 to identify the minors with a rank higher than 1 and proceed to step (e).

Step (e) Once the 2by2 minors with a rank higher than 1, were identified, employ McCormick and

disjunctive constraints to reduce their rank. This would also reduce the rank of the moment matrix

introduced in step (b). By using these valid constraints, the size of the problem will not increase, however,

the procured solution would be optimal for the original problem. If the relaxed problem that is tightened by

111

adding the valid constraints procures a rank-1 solution, the algorithm is terminated. Otherwise, identify the

2by2 minors with higher than rank-1 and proceed to step (f).

Step (f) The elements of the minors with a rank higher than 1, are added to the vector of monomials

introduced in step (a).

Using the proposed approach, the feasibility of the optimal solution to the convex-relaxed electricity

network operation problem is guaranteed. With the increase in the order of the moment relaxation, the

solution for the convex-relaxed problem will converge to the optimal solution of the original problem

[Men05]. The proposed algorithm is computationally less expensive compared to employing higher order

moment relaxation as it will only add the necessary elements to the vector of monomials to construct the

moment matrix.

6.2.4 Coordination among Electricity and Natural Gas Networks Operation

Here, a consensus optimization framework is developed to capture the interactions among the natural gas

and electricity networks. The presented consensus optimization is solved using the alternating direction

method of multipliers (ADMM) [Boy11]. The vector of shared variables among the natural gas and

electricity networks is the volume of the natural gas demand for the natural gas generation units. The

outline of the proposed problem is shown in (6.52)-(6.54). Here, ( )f x represents the objective function of

the natural gas operation problem given in (6.23) and ( )g z represents the objective function of the electricity

network operation problem given in (6.46). The convex-relaxed problems procured in sections III.B and

III.C are represented by (6.52) and (6.53) respectively. The feasible regions are shown as x and z

respectively, and λ is the Lagrange multiplier of the consensus constraint.

( ) . . , :xMin f x s t x x z (6.52)

( ) . . , :zMin g z s t z z x (6.53)

The consensuses equality constraint is relaxed with respective Lagrange multiplier in the

augmented Lagrangian function shown in (6.54) and is a scalar.

112

2

2( , , ) ( ) ( ) ,

2L x z f x g z x z x z

(6.54)

In order to solve this problem, (6.52)-(6.53) are decomposed into two separate problems. The first

problem (6.55) is the x-update problem and the second problem (6.56) is the z-update problem. As shown

in (6.55), the x-update problem is solved in each iteration, while the variables associated with the z-update

problem are fixed. Similarly, in the z-update problem (6.56) is solved while the variables associated with

the x-update problem are fixed. Here, the x-update problem is the convex-relaxed natural gas network

operation problem and the z-update is the convex-relaxed electricity network operation problem. The

vector of Lagrange multipliers for the consensus constraints is updated in each iteration as shown in (6.57).

1 arg min ( , , )xx

x L x z

(6.55)

1 1arg min ( , , )zz

z L x z

(6.56)

1 1 1( )x z (6.57)

6.3 Case Study

A sample 6-bus electricity network and 7-junction natural gas network are considered as shown in Figure

6-2. The 6-bus electricity network is composed of 2 natural gas generation units (G1 and G2) and 1 fossil

fuel generation unit (T1). The characteristics of the generation units and transmission lines are shown in

Tables 6-1 and 6-2, respectively. The fuel price of the thermal unit is 3.5 $/MMBtu. The characteristics of

the Natural gas resources and pipelines are shown in Tables 6-3 and 6-4, respectively. The minimum and

maximum pressures at junctions of the natural gas network are 105 and 170 Psig respectively. The natural

gas demands at junctions J1, J2, and J3 are 15%, 50% and 35% of the total natural gas demand,

respectively. The electricity demand on buses 3, 4, and 5 are 20%, 40%, and 40% of the total electricity

demand, respectively. The normalized demand profiles for the electricity and natural gas are given in

Figure 6-3. The peak demands for electricity and natural gas network are 256 MW at hour 17 and 6780 kcf

113

at hour 20, respectively. Here, the number of binary variables in the natural gas and electricity network

operation problems is 24 and 72 respectively. The number of continuous variables for the natural gas and

electricity network operation problems is 504 and 216 respectively. The following four scenarios are

presented:

Scenario 1) Uncoordinated operation of the electricity and natural gas networks

Scenario 2) Coordinated operation of the electricity and natural gas networks

Scenario 3) Scenario 2 with congestion in the electricity network

Scenario 4) Scenario 3 with congestion in the natural gas

Figure 6-2 The 6 bus-system interconnected with 7-junction natural gas

Figure 6-3 The normalized demand profile for electricity and natural gas

PL6J4

C1

PL2J2

J6PL5

J5

PL4J3

PL3

PL1J7

123

456

L1

L2

L3

L4L5

L6L7

G1

G2

T1

v1

v2

GT Gas resourceCompressorJunctionGas-fired unitsThermal units

J1

0.5

0.6

0.7

0.8

0.9

1

1.1

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

No

rmal

ized

Dem

and

Pro

file

Time (Hours)

Electricity Demand Profile Natual Gas Demand Profile

114

Table 6-1 Generation unit charcteristics

Unit

a

(MMBtu/MW

h2)

b

(MMBtu/MW

h)

c

(MMBtu/h

)

Pmin

(MW)

Pmax

(MW)

RU/RD

(MW/h)

,offon

i iT T

(h)

G1 0.028 12.196 170.48 10 320 100,100 2,2

T1 0.0676 18.543 289.9 50 160 50,50 4,3

G2 0.058 10.69 136.29 10 60 50,50 1,1

Table 6-2 Transmission line characteristics

Line

ID

From

Bus

To

Bus

Impedance

(p.u.)

Ling Rating

(MW)

L1 1 2 0.170 80

L2 1 4 0.258 80

L3 2 4 0.197 70

L4 5 6 0.140 60

L5 3 6 0.018 80

L6 2 3 0.037 150

L7 4 5 0.037 70

Table 6-3 Natural gas resources characteristics

Resource Minimum Capacity

(kcf/h)

Maximum Capacity

(kcf/h)

Cost

($/kcf)

v1 1500 5000 2.6

v2 1000 6000 3.2

Table 6-4 Natural gas pipeline characteristics

Line

ID

From

Junction

To

Junction

Pipeline

Constant (kcf

/Psig)

Compressor

factor

PL1 J1 J2 51 0

PL2 J2 J4 0 1.21

PL3 J2 J5 83 0

PL4 J3 J5 64 0

PL5 J5 J6 44 0

PL6 J4 J7 63 0

6.3.1 Scenario 1 – Uncoordinated operation of the electricity and natural gas networks

The hourly unit commitment solution is presented in Table 6-5. At peak demand in hour 17, the

generation dispatches of G1 and G2 are 196 MW and 60 MW, respectively. The total operation cost of the

electricity network is $298,722. The volumes of natural gas withdrawn from resources v1 and v2 at hour 17

are 5000 kcf and 5827.7 kcf respectively. The total operation cost of the natural gas network is $675,887

and junction pressures of this network are shown in Table 6-6.

115

Table 6-5 Hourly unit commitment in Scenario 1

Unit Hours (0-24)

G1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

G2 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Table 6-6 Natural gas pressures in scenario 1 at hour 17

Junction ID J1 J2 J3 J4 J5 J6 J7

Pressure

(Psig) 131.44 137.80 105.00 137.80 137.55 178.42 165.96

The increase in the natural gas supply to serve the natural gas generation units leads to the violation of the

operation limits of the natural gas network. For example, to serve the 3565.58 kcf natural gas demand of

G1, the pressure at junction J6 is increased to 178.42 psig, which is higher than the maximum limit. Thus,

the natural gas network is unable to supply the natural gas generation units.

6.3.2 Scenario 2 – Coordinated operation of the electricity and natural gas networks

In this scenario, the operational planning of the electricity network is coordinated with that for the

natural gas network. However, the capacity limits of the transmission lines and pipelines are relaxed to

avoid any congestion. In this scenario, in addition to units G1 and G2, T1 which is a more expensive unit is

also committed as shown in Table 6-7. Compared to scenario 1, unit T1 is committed at hours 16-20, and

G2 is committed at hour 7.

At hour 17, all units are committed to serve the peak electricity demand. The generation dispatch of G1,

T1 and G2 is 153.5 MW, 50 MW, and 52.5 MW respectively. The generation dispatches of natural gas

units G1 and G2 are decreased compared to those in scenario 1. Furthermore, the volumes of natural gas

withdrawn from resource v1 and v2 are 4883.82 kcf and 4871.62 kcf respectively. The junction pressures in

the natural gas network are given in Table 6-8. In this scenario, the shared variables among the electricity

and natural gas networks are the consumption volumes of the natural gas generation units. At hour 17, the

natural gas consumption volumes for G1 and G2 are 2701.97 MMBtu and 1067.38 MMBtu, respectively.

The flow of natural gas in pipeline PL3 is 113.83 kcf/h from junction J5 to J2 and the junction pressures are

within their operation limits.

116

The total operation cost of the electricity network is increased to $303,956 which is 1.75% higher

than that in scenario 1. Furthermore, the total operation cost of the natural gas network is decreased to

$661,132.


Unit Hours (0-24)

G1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0

G2 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1



Pressure

(Psig) 122.91 128.76 105.00 128.76 128.76 170.00 150.19

In order to show the performance of the presented algorithm the characteristics of the sparse moment

matrices associated with each junction of the natural gas network were considered. The procured convex

relaxation is exact if the rank of the all moment relaxation matrices is one. If the rank of a matrix is one, all

of its eigenvalues are zero but one. Alternatively, if the ratio of the largest eigenvalue to the second largest

eigenvalue of the moment matrix is very large (close to infinity), the rank of the matrix is one. Therefore,

this ratio is used as a measure of the tightness for the proposed relaxation. The ratio of the two largest

eigenvalues of the moment matrix associated with each junction of the natural gas network is evaluated for

all hours in the operation horizon. As an example, this ratio is given for scenario 2 at peak hour (hour 17) in

Table 6-9. Here, the rank-1 moment relaxation matrix corresponds to junction J4 is procure in step (d) of

the algorithm before adding any RLT constraints, or new variables to the vector of monomials. However,

the rank-1 moment relaxation matrices correspond to junctions J6 and J7 are procured after adding the RLT

constraints in step (e). Once the valid constraints added in step (g), the rank-1 moment relaxation matrix is

procured for junctions J1 and J3. Finally, the rank-1 moment relaxation matrix is procured at step (h) for

junctions J2 and J5 for which the variables associated with the rank higher than 1 2by2 minors are added to

the vector of monomials in addition to the constraints added in the previous steps. YALMIP [Lof04] is used

to solve the problem and Mosek 7 is used as the SDP solver on a PC with 3.2 GHz Intel i5 processor and 8

GB of memory. The computation time at the last iteration of the ADMM approach to reach the optimal

117

solution for 7-junction natural gas and 6-bus electricity network operation problems is 14.65 sec and 100.62

sec respectively. The total computation time to converge to a solution using the ADMM approach is 829.94

sec.

Table 6-9 The ratio of the two largest eigenvalues in scenario 2 at hour 17

Junction

ID J1 J2 J3 J4 J5 J6 J7

Step (d) 5.7E1 2.1E0 4.3E1 1.7E9 1.8E0 6.4E2 3.8E2

Step (e) 7.8E2 1.6E1 1.5E2 - 2.3E1 9.8E9 1.5E10

Step (g) 7.9E12 7.3E3 9.2E10 - 6.5E2 - -

Step (h) - 9.4E11 - - 8.3E9 - -

6.3.3 Scenario 3 – Scenario 2 with Congestion in the Electricity Network

In this scenario, the capacity limits of the pipelines are relaxed to avoid congestions in the natural

gas network, and the power flow limits given in Table 6-2 are enforced for the electricity network. As

shown in Table 6-10, the generation units are committed for more number of hours compared to scenario 2

to avoid congestion in the electricity network. Compared to scenario 2, the more expensive generation unit

T1 is committed at hours 10-15 and 21-22 while the natural gas unit G2 is committed at hours 2-6. The

total operation cost of electricity network is increased to $313,817, which is 3.24% higher than that in

scenario 2.


Unit Hours (0-24)

G1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T1 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0

G2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

The decrease in the total dispatch of the natural gas generation units leads to the decrease in

consumption in the natural gas network. The total operation cost of the natural gas network is decreased to

$631,587 which is 4.46% less than that in scenario 2. At this hour, the volumes of natural gas withdrawn

from the resources v1 and v2 are 5000 kcf and 4587.1 kcf, respectively. The total volume of natural gas

withdrawn is decreased compared to scenario 2 as the overall natural gas consumption of the natural gas

generation units is decreased. The junction pressures in the natural gas network at hour 17 are given in

Table 6-11. The natural gas consumptions for G1 and G2 at hour 17 are 1792.3 MMBtu and 272.81

118

MMBtu, respectively. Compared to scenario 2, it is shown that congestion in the electricity network leads

to the changes in the demand of the natural gas generation units. At hour 17, the flow of natural gas in

pipeline PL3 is 554.15 kcf/h from junction J5 to J2 and the junction pressures are within the operation

limits.



Pressure

(Psig) 118.71 125.72 105.0 125.72 125.9 169.6 145.28

The ratio of the two largest eigenvalues of the moment relaxation matrix associated with each junction of

the natural gas network at hour 17 in scenario 3 is considered as a tightness measure for the proposed

convex relaxation as given in Table 6-12. Here, the rank-1 moment relaxation matrix corresponds to

junction J4 is procured in step (d) of the algorithm. Adding RLT constraints in step (e) would lead to the

rank-1 moment relaxation matrices correspond to junctions J6 and J7. The rank-1 moment relaxation

matrices correspond to junctions J1, J2 and J3 are procured after adding the related valid constraints in step

(g). Finally, the rank-1 moment relaxation is procured in step (h) for junction J5, where the variables

associated with higher than 1 rank 2by2 minors are added to the vector of monomials in addition to the

constraints added in previous steps. Compared this scenario with scenario 2, it is shown that the rank-1

moment relaxation matrix corresponding to junction J2 is procured in step (g) in scenario 3 while that was

procured in step (h) in scenario 2.



Step (d) 9.2E2 2.1E1 6.5E1 9.8E9 2.1E1 4.1E3 2.5E3

Step (e) 1.3E3 2.2E2 5.1E3 - 9.7E1 6.9E8 2.6E9

Step (g) 1.1E14 9.4E9 8.3E11 - 4.7E4 - -

Step (h) - - - - 7.4E8 - -

6.3.4 Scenario 4 – Scenario 3 with Congestion in the Natural Gas Network

In this scenario, the capacity of pipeline PL3 is limited to 500 kcf/h while the capacity limits of the

transmission lines are also considered. The congestion in the natural gas network will change the

generation dispatch for some hours but the commitment status of the generation units remains the same as

119

scenario 4. The congestion in the natural gas network affects the volume of natural gas withdrawn from the

resources. As the natural gas demand of G1 alleviates the congestion in the natural gas network, the natural

gas price for G1 is lower than that for G2 in this scenario. Therefore, the total operation cost of electricity

network is decreased to $278,239 which is 11.33% less than that in scenario 3. While the generation

dispatch and natural gas demand of natural gas generation units remain the same as those in scenario 3, the

total operation cost of the natural gas network is increased to $634,025 which is 0.39% more than that in

scenario 3. At hour 17, the volume of the natural gas withdrawn from resource v1 is decreased to 4945.85

kcf and the volume of the natural gas withdrawn from resource v2 is increased to 4641.24 kcf. Although

the resource v1 is cheaper compared to v2, the limited capacity of pipeline PL3 (500 kcf/h) will reduce the

withdrawn volume of natural gas from this resource. The pressures at the junctions at hour 17 are given in

Table 6-13. The efficiency of the presented algorithm for tightening the convex relaxation in this scenario

is shown in Table 6-14.



Pressure

(Psig) 118.75 125.75 105.0 125.75 125.90 168.78 145.74


Junction

ID J1 J2 J3 J4 J5 J6 J7

Step (d) 7.6E2 3.4E1 1.5E0 8.2E9 9.4E1 3.9E3 3.5E3

Step (e) 9.7E2 3.8E2 6.1E3 - 2.2E2 7.4E8 7.2E9

Step (g) 2.7E13 2.3E10 9.8E10 - 6.9E8 - -

Step (h) - - - - - - -

6.3.5 Discussion

The generation profile of the generation units G1, G2, and T1 in scenarios is presented in Figure 6-4,

Figure 6-5, and Figure 6-6, respectively. By comparing scenarios 1 and 2, it is shown that the coordination

among the electricity and natural gas networks will decrease the generation dispatch of the natural gas

generation units at hours where the natural gas demand is increased in the network. During hours 16-20 the

generation dispatch of G1 is decreased in scenario 2 compared to scenario 1 as shown in Figure 6-4; and

the generation dispatch of T1 is increased as shown in Figure 6-6. A comparison of scenarios 2 and 3

120

highlights the impact of congestion within the electricity network on the dispatch profile of the generation

units. As shown in Figure 6-4, the generation dispatch of G1 is decreased during hours 2-6 while the

generation dispatch of G2 is increased in the same period as shown in Figure 6-5. Additionally, the

generation dispatch of T1 is increased during hours 10-18. Comparing scenarios 3 and 4 shows the impact

of congestion in the natural gas network on the generation dispatch of the generation units in the electricity

network. As congestion in the natural gas network will decrease the marginal cost of the generation unit G1

during hours 2-6, the generation dispatch of G1 will increase compared to that in scenario 3. The generation

dispatch of G2 decreases accordingly. By comparing the generation dispatch of G1 for scenarios 2, 3, and

4, it is shown that although congestion in the natural gas network will increase the dispatch of G1, the

limited capacity of the transmission lines in the electricity network will limit the increase in the generation

dispatch to avoid congestion in the electricity network.

Furthermore, the presented formulation captures the variability in the electricity and natural gas network

by addressing the bi-directional flow of natural gas in the pipelines. Here, the flows of natural gas in

pipeline PL3 in scenario 3 and scenario 4 are given Figure 6-7. As the flow of natural gas in this pipeline is

limited to 500 kcf/h in scenario 4, the variation in the generation dispatch of the natural gas generation units

in Figure 6-4, 6-5, and 6-6 will result in multiple changes in the direction of the natural gas flow in the

pipeline PL3 in the operation horizon. The presented framework procures the flow direction in the pipelines

as a result of the variation in the generation dispatch of the natural gas generation units.

Moreover, the congestion in the natural gas and electricity networks will impact the volume of

natural gas withdrawn from the natural gas resources. The hourly volumes of natural gas withdrawn from

the sources v1 and v2 in scenarios 3 and 4 are shown in Figure 6-8. In congestion period for natural gas

network in scenario 4, the natural gas withdrawn from source v1 is decreased compared to that in scenario 3

and the natural gas withdrawn from source v2 increased to compensate for the shortage in the natural gas

supply. As a result of congestion in the natural gas network, the total operation cost of this network is

increased from $631,587 in scenario 3 to $634,025 in scenario 4. As the increase in the natural gas demand

of G1 mitigates the congestion in the natural gas network, the dispatch of this generation unit is increased

as shown in Figure 6-4.

121

Figure 6-4 The generation dispatch of G1 in scenarios

Figure 6-5 The generation dispatch of G2 in scenarios

Figure 6-6 The generation dispatch of T1 in scenarios

100

120

140

160

180

200

220

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Gen

erat

ion D

isp

atch

[M

W]

Time [Hours]

Scenario 1 Scenario 2 Scenario 3 Scenario 4

0

10

20

30

40

50

60

70

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Gen

erat

ion D

isp

atch

[M

W]

Time [Hours]


0

10

20

30

40

50

60

70

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Gen

erat

ion D

isp

atch

[M

W]

Time [Hours]


122

Figure 6-7 The flow of natural gas in pipeline PL3 in scenarios 3-4.

Figure 6-8 The profile for the natural gas volume withdrawn from sources in scenarios 3-4.

6.3.6 IEEE 118-Bus System with 12-junction Natural Gas network

In this case, a modified IEEE-118 bus system is supplied by a 12-junction natural gas network. The

electricity network has 46 fossil fuel generation units, 8 natural gas generation units, 186 branches, and 91

demand entities. The total capacity of the natural gas generation is 575 MW. The peak load is 3700 MW

which occurs at hour 21. The natural gas network that supplies the electricity network is composed of 12

junctions, 12 pipelines, two compressors, and 12 natural gas demand entities including the natural gas

generation units as shown in Figure 6-9. The natural gas peak demand of 14,500 kcf that occurs at hour 19.

Here, the number of binary variables for the natural gas and electricity network operation problem is 168

-1500

-1000

-500

0

500

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Nat

ura

l G

as F

low

[kcf

/h]

Time [Hour]

Scenario 3 Scenario 4

2200

2700

3200

3700

4200

4700

5200

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Nat

ura

l G

as F

low

[kcf

/h]

Time [Hour]

v1 in Scenario 4

v2 in Scenario 4

v1 in Scenario 3

v2 in Scenario 3

123

and 3888 respectively. The number of continuous variables for the natural gas network operation problem

is 648 and the number of continuous variables for the electricity network operation problem is 3888. In the

last iteration of the ADMM approach, the computation time for the 12-junction natural gas network

operation problem is 20.94 sec. Here, the proposed algorithm for the natural gas operation problem is used

to determine the sparse formulation for the IEEE 118-bus network operation problem. The computation

time for the large-scale electricity network is reduced by leveraging the sparse formulation of the unit

commitment problem and the solution time is 160.1 sec.

Figure 6-9 12-junction natural gas network

In this case study, the uncoordinated operation scheme (scenario 1) is compared to the

coordinated operation scheme (scenario 2). As the natural gas network is constrained by the operation

limits on the junction pressures, the natural gas generation units may not operate at their full capacity. As a

result of imposing the pressure limits at junction J3, the dispatches of units G4 and G5 are decreased from

100 MW and 61.9 MW in scenario 1 to 76.3 MW and 0 MW at hour 21 in scenario 2, respectively. Here, as

the natural gas demand increases the pressure at junctions will violate the operation limits. The natural gas

pressure at junction J3 will increase to 172.9 psig at hour 21 which violates the upper limit for natural gas

pressure at this junction. The commitment states of the generation units in scenario 1 are given in Table 6-

15. Here the total operation cost is $1,603,326.

To ensure that the unit commitment solution in scenario 1 provides a feasible solution for the

natural gas operation problem, the demands of the natural gas generation units are imposed to the natural

J3

v2

v1

J12

L4

J1

J8

J7

J9

J2J5

J4

L1

G1

J6

L3

G4

G5

L2

G2

G3

J11

v3G6

J10

G7

G8

Gas resource

Compressor

Junction

124

gas network. In this case, the natural gas network failed to serve the demand of the natural gas generation

units. In scenario 2, the operation of the electricity network is coordinated with the operation of the natural

gas network. The changes in the commitment of the generation units compared to those in scenario 1 are

given in Table 6-16. Here, the total operation cost is increased to $1,629,829 which is 1.65% higher than

that in scenario 1. However, unlike scenario 1, the presented solution for unit commitment is feasible for

the natural gas network as there is no supply shortage for the natural gas generation units and there is no

violation of the operation constraints of the natural gas network. Compared to scenario 1, the natural gas

units G3 and G5 are not committed at hours 6-24 and 3-24 in scenario 2, respectively. Therefore, the more

expensive fossil fuel units T13 and T34 are committed during hours 1-22 and 8-13, respectively. Moreover,

G4 is committed at hours 5-13, G2 is committed at hour 4, T14 is decommitted in hour 8 and T46 is

decommitted during hours 1-14 and 21-24.


Unit Hours (1-24)

T1-T3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

T4-T5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

T7 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T8-T9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

T10-T11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T12 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0

T13 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

T14 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T15 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

T16 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T17-T18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

T19 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T20-T21 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T22-T23 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T24-T25 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T26 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T27-T29 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T30-T31 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

T32 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T33 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

T34 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1

T35-T36 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T37 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T38 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

T39-T40 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T41-T42 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

T43-T45 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T46 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0

G1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

G2 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

G3 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

G4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1

G5-G8 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

125

Table 6-16 Changes in the hourly commitment in scenario 2 from scenario 1

Unit Hours (1-24)

T13 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0

T14 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T34 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

T46 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0

G2 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

G3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

G4 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

G5 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6.4 Summary

This chapter presents a consensus optimization framework for the coordinated operation of electricity and

natural gas networks. Tight convex relaxations for the electricity and natural gas operation problems with

nonconvex feasible regions were proposed. The nonconvex feasible region is characterized by the

Weymouth equation in the natural gas network, and the introduced unit commitment binary decision

variables in the electricity network. A computationally tractable convex representation with several valid

constraints is formulated that procures a rank-1 optimal solution for the natural gas and electricity network

operation problems. The alternating direction method of multipliers is used to solve the convex-relaxed

consensus optimization problem. Two case studies showed the effectiveness of the proposed approach to

capture the interdependence among electricity and natural gas networks.

The presented approach exploits the sparsity of the natural gas network. If the natural gas network

is dense, procuring the rank-1 moment relaxation matrix will be more challenging. Additionally, in order to

reach the global solution, the number of monomials within the vector of monomials could increase

dramatically. Consequently, a very large moment relaxation for each junction of the natural gas network is

formed. Although the presented algorithm would avoid forming the unnecessary large moment relaxation

matrices, very large moment relaxation matrices may be required to procure a rank-1 moment relaxation

matrix. This work could be further extended to capture a more detailed model of the natural gas network

considering the nonlinearities in power consumption of the compressors.

126

Chapter 7

7SUMMARY

The aging electricity grid is the largest machine in the world which is so vulnerable that may fail

under extreme conditions. Although alternating the inexpensive bulk power generation from coal and oil

with the clean energy resources (e.g. natural gas and volatile renewable energy) will reduce the air

pollution and carbon footprint of human activities, a comprehensive economic justification is required to

encourage the investment in grid modernization and clean energy generation. The research presented in this

dissertation employs the cutting-edge mathematical techniques to resolve these key challenges to making

the electricity grid smart.

With the rise in the reliability and resiliency concerns in electricity networks with high penetration

of distributed renewable energy resources, the expansion of microgrids in active distribution network has

become a crucial interest. This problem is analytically addressed by utilizing an advanced graph theory

practice. The expansion of microgrids will facilitate the controllability of distributed energy resources

(DERs) within the distribution network. The proposed framework includes several decision-making

components to identify the most controllable and reliable plan according to attributes and uncertain

parameters including the renewable generation, demand, and component outages. Thus, the planners can

determine the points of common couplings (e.g. boundaries) for microgrids. This mathematical

methodology is a powerful tool that to utilize in undertaking other similar problems in smart grid that

certain level of autonomy in control and privacy is desired while the quality and reliability of the service in

the network are maintained. Another goal that is carried out in the presented dissertation is the

enhancement in the resiliency of energy supply in distribution network through multiple energy carrier

microgrids. Abundance, expanded transportation network, availability at a low price, and minimum CO2

emissions of natural gas prompted the growth in the penetration level of DERs supplied by natural gas. On

the other hand, more than half of the cyber threats occur in the energy sector, so it is pivotal to safeguard

the cyber layer of the cyber-physical system, e.g. the control of multiple energy carrier microgrids. A

mathematical model is developed to identify and then reinforce the vital components within microgrids

127

throughout a preventive reinforcement procedure. In addition, a quantitative resiliency measure is

introduced to analyze the operation of microgrids under threat. There is a tradeoff between minimizing the

investment on the reinforcing the components and maximizing the resiliency of energy supply within the

microgrid. The presented research provides a decision support system and resiliency measure that assists

the microgrid planners to allocate the resources and reach the best strategy based on their budget and their

desired level of resiliency.

Investment on microgrids is unobtainable without a financial model that justifies a decent return

on the investment. Due to the technical limitations and ineligibility of microgrids to directly participate in

the wholesale electricity market, a hierarchical electricity market structure is proposed to facilitate

acquiring profit from microgrids in the grid-connected mode. In the hierarchical market, microgrids

participate in the wholesale electricity market through load aggregators (LAs). A dynamic game theory

approach is employed to model the role of LAs as intermediate agent coupling the wholesale electricity

market with several distribution networks. The proposed hierarchical structure utilized dynamic game

theory is a promising tool to address the interdependence of multiple competitive environments.

Exploring additional aspects of multiple challenges that is addressed in the presented dissertation

requires solving the nonconvex optimization problems. Thus, the convex relaxation techniques is employed

to develop a solid problem-solving tool for overcoming this challenge particularly for smart grid

applications. An algorithm based on sparse semidefinite programming (SDP) relaxation approach is

developed to procuring an at least near-global optimal solution of these problems. By leveraging the

sparseness of the networks, the presented algorithm construct several small, tight, and traceable SDP

relaxation matrices. This technique opened a new door to solve several challenging problems including the

non-convexity issue within the natural gas network.

128

APPENDIX

The research conducted to address the problems discussed in this dissertation lead solution methodology to

overcome a wide-range of challenges in grid modernization. It should be noted that only a selected number

of works are discussed in this dissertation. The relevant publications are listed as followings.

- S. D. Manshadi and M. E. Khodayar. Coordinated Operation of Electricity and Natural Gas

Systems: A Convex Relaxation Approach. IEEE Transactions on Smart Grid, accepted for

publication, 2018.

- S. D. Manshadi and M. E. Khodayar. A Tight Convex Relaxation for the Natural Gas Operation

Problem. IEEE Transactions on Smart Grid, accepted for publication, 2018.

- S. D. Manshadi and M. E. Khodayar. Risk-Averse Generation Maintenance Scheduling with

Microgrid Aggregators. IEEE Transactions on Smart Grid, in press, 2017.

- S. D. Manshadi and M. E. Khodayar. Wireless Charging of Electric Vehicles in Electricity and

Transportation Networks. IEEE Transactions on Smart Grid, in press, 2017.

- A. Vafamehr, M. E. Khodayar, S. D. Manshadi, I. Ahmed, adn J. Lin. A Framework for

Expansion Planning of Data Centers in Electricity and Data Networks under Uncertainty. IEEE

Trans. on Smart Grid, in press, 2017.

- S. D. Manshadi and M. E. Khodyar. Expansion of Autonomous Microgrids in Active Distribution

Networks. IEEE Transactions on Smart Grid, in press, 2016.

- M. E. Khodayar, S. D. Manshadi, and A. Vafamehr. Short-term Operation of Microgrids in

Transactive Energy Architecture. The Electricity Journal, 29(10): 41–48, Dec. 2016.

- M. E. Khodayar, S. D. Manshadi, H. W, and J. Lin. Multiple Period Ramping Processes in Day-

Ahead Electricity Markets. IEEE Transactions on Sustainable Energy, 7(4):1634–1645, Oct. 2016.

- S. D. Manshadi and M. E. Khodyar. A Hierarchical Electricity Market Structure for the Smart

Grid Paradigm. IEEE Transactions on Smart Grid, 7(4):1866–1875, Jul. 2016.

129

- S. D. Manshadi and M. E. Khodyar. Resilient Operation of Multiple Energy Carrier Microgrids.

IEEE Transactions on Smart Grid, 6(2):2283–2392, Sep. 2015. UNDER REVIEW

- S. D. Manshadi, G. Liu, M. E. Khodayar, J. Wang, and R. Dai. A Convex Relaxation Approach to

Find a Feasible Solution for the Power Flow Problem. IEEE Transactions on Power Systems,

Under Review.

- S. D. Manshadi and M. E. Khodayar. Strategic Behavior of In-motion Wireless Charging

Aggregators in the Electricity and Transportation Networks. IEEE Transactions on Smart Grid,

Under Review.

- S. D. Manshadi, M. E. Khodayar, and C. Chen. Topology Estimation in Mesh Distribution

Network with Limited Measurements. IEEE Transactions on Smart Grid, Under Review.

- S. D. Manshadi and M. E. Khodayar. Decentralized Operation Framework for Hybrid AC/DC

Microgrids. Proceedings of the North America Power Symposium (NAPS 2016), Sep. 2016.

130

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