ELECTRICITY MARKET OPERATIONS WITH MASSIVE …

Southern Methodist University Southern Methodist University

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Electrical Engineering Theses and Dissertations Electrical Engineering

Summer 2021

ELECTRICITY MARKET OPERATIONS WITH MASSIVE ELECTRICITY MARKET OPERATIONS WITH MASSIVE

RENEWABLE INTEGRATION: NEW DESIGNS RENEWABLE INTEGRATION: NEW DESIGNS

Shengfei Yin Southern Methodist University, [email protected]

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ELECTRICITY MARKET OPERATIONS WITH

MASSIVE RENEWABLE INTEGRATION:

NEW DESIGNS

Approved by:

Dr. Jianhui Wang

Professor

Dr. Jungchih Chiao

Professor

Dr. Mohammad E. Khodayar

Associated Professor

Dr. Harsha Gangammanavar

Assistant Professor

Dr. Xin Fang

External Expert

ELECTRICITY MARKET OPERATIONS WITH

MASSIVE RENEWABLE INTEGRATION:

NEW DESIGNS

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

Shengfei Yin

Master of Science, Electrical Engineering, Illinois Institute of TechnologyBachelor of Engineering, Electrical Engineering, Hunan University

August 4, 2021

Copyright (2021)

Shengfei Yin

All Rights Reserved

iii

ACKNOWLEDGMENTS

I dedicate my sincerest gratitude to my academic advisor, Prof. Jianhui Wang, for

his patience and guidance in my Ph.D. career. This study would not have been complete

without his invaluable wisdom and constructive advice. I always feel privileged and hon-

ored to assist in his works and advance my research. Special thank passes to Prof. Harsha

Gangammanavar and Dr. Xin Fang for their treasured input and help in developing solu-

tion algorithms and practical scenarios. Their fruitful knowledge and experiences helped

me break many bottlenecks and enlightened the synergy between operations research and

electrical engineering. Appreciation further extends to the respected graduation commit-

tee members, including Prof. Jungchih Chiao and Prof. Mohammad Khodayar, who have

provided suggestions of value and set inspiring examples of scientific achievements.

I would like to thank Dr. Yonghong Chen, Dr. Jin Tan, Dr. Xiaohu Zhang, Dr. Feng

Qiu, and Prof. Lei Wu for their consistent support on R&D projects and technical reviews,

bolstering my research from various perspectives. I hope we could keep future collaborations

in related fields and try to improve the industry practice further.

My academic life at SMU would not be so enjoyable without my peer researchers. I would

like to acknowledge Prof. Saeed D. Manshadi for his genuine help and technical support when

I started the journey of Ph.D. I also respect Ying Zhang’s diligence and curiosity, which form

a benign stimulation of my research. Furthermore, my fellow colleagues, Yihao Wang, Xinan

Wang, Mahdi Khodayar, Mohammad Ramin Feizi, Yanling Lin, You Lin, Xinyun Lu, Tao

Wu, Bin Huang, Tu Lan, Abdulraheem Hassan Z Alobaidi, Saeed Fazl Hashemi, take my

heartfelt gratitude for the company and backup at all times.

Last but not least, I keep the deepest affection and thank-you for my family and my

beloved Margaret, who have been my persistent champions as always.

iii

Yin, Shengfei Master of Science, Electrical Engineering, Illinois Institute of TechnologyBachelor of Engineering, Electrical Engineering, Hunan University

Electricity Market Operations with

Massive Renewable Integration:

New Designs

Advisor: Dr. Jianhui Wang

Doctor of Philosophy degree conferred August 4, 2021

Dissertation completed date: July 19, 2021

Electricity market has been transitioning from a conventional and deterministic operation

to a stochastic operation under the increasing penetration of renewable energy. Industry-

level solutions toward the future electricity market operation ask for both accuracy and

efficiency while maintaining model interpretability. Hence, reliable stochastic optimization

techniques come to the first place for such a complex and dynamic problem.

This work starts at proposing a solution strategy for the uncertainty-based power system

planning problem, which acts as a preliminary and instructs the electricity market operation.

Formulated as a multi-year system planning problem from an independent system operator’s

perspective, it minimizes both expansion and operational costs under binary and continuous

uncertainties, i.e., system component contingency and load/generation variation. The origi-

nal hybrid model is highly intractable, but we devise a combined decomposition strategy to

relieve the computational burden. Considering 100% renewable penetration in the future, it

analyzes the cost-effectiveness of renewable energy from a long-term point of view.

After the uncertainty-based system planning problem is well tackled, we turn our di-

rection to the hierarchical market operation considering the transmission and distribution

coordination. As distributed generators are increasingly penetrating the distribution net-

works, the central operator should be aware of their uncertainties. We develop a three-stage

unit commitment model for the market operation of transmission and distribution coordina-

tion under the uncertainties of renewable generation and demand variations. As a multi-stage

iv

stochastic programming problem, the proposed model is mixed-integer nonlinear, and we de-

vise a generalized nested L-shaped algorithm to divide and conquer this problem efficiently.

With the study of new electricity market operations, current market protocols should

be updated according to the growing variable resources in the power system, especially the

ancillary service participation. To better let industry and research communities leverage our

works, we develop an flexible software platform for the full-stack electricity market opera-

tions including both the day-ahead and real-time operations. In this platform, users could

conveniently switch from conventional market models to the novel market framework that

we designed. Various operation novelties, such as ancillary service provision from variable

resources, multi-level energy storage participation, and model updates on the conventional

non-spinning reserve, are integrated in the new model, considering the profitability of the

variable renewable energy.

v

TABLE OF CONTENTS

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

CHAPTER

1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1. Motivations and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2. Research Objectives and Achievements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2. GENERATION AND TRANSMISSION EXPANSION PLANNING

TOWARDS A 100% RENEWABLE FUTURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1. Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3. Mathematical Formulation of the Generation and Transmission Planning . . . 12

2.4. Problem Decomposition and Solution Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.1. Master Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.2. Subproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4.2.1. L-shaped Master Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.2.2. L-shaped Subproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.5. Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5.1. Scenario Generation & Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.5.2. Case Studies: Towards 100% Renewable Penetration . . . . . . . . . . . . . . . . 26

2.5.3. Case Studies: Contingency Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5.4. Case Studies: Long-term Cost-effectiveness . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5.5. Case Studies: Investment Annuitization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.6. Scalability Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

vi

2.7. Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.8. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3. STOCHASTIC MARKET OPERATION FOR COORDINATED

TRANSMISSION AND DISTRIBUTION SYSTEMS . . . . . . . . . . . . . . . . . . . . 39

3.1. Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3. Mathematical Formulation of T-D Coordinated Market Operations . . . . . . . . . 46

3.3.1. The First Stage: ISO’s UC Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.2. The Second Stage: ISO’s ED Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3.3. The Third Stage: DSO’s ED Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.4. Decomposition-based Solution Strategy for the Three-Stage Market Operation 51

3.4.1. ISO’s Master Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4.2. ISO’s Subproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4.3. DSO’s Subproblem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4.4. Generalized Nested Decomposition Algorithm . . . . . . . . . . . . . . . . . . . . . . . 54

3.5. Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.5.1. Tran6+Dist7+Dist9 Test Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.5.2. Tran118+Dist34×5 Test Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4. FUTURE ELECTRICITY MARKET DESIGNS AND GENERIC

SIMULATION PLATFORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.1. Overview for State-of-the-art Market Operations with Renewables . . . . . . . . . . 68

4.1.1. Renewable Integration in the Electricity Market . . . . . . . . . . . . . . . . . . . . . 70

4.1.1.1. Renewable Participation in the Energy Market . . . . . . . . . . . . . 70

4.1.1.2. Renewable Participation in the Reserve Market . . . . . . . . . . . . 74

4.1.1.3. Potential Market Designs for the Solar Integration . . . . . . . . . 75

vii

4.1.1.4. Impact of Future Market Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.1.2. Renewable Uncertainty Modeling in Electricity Markets . . . . . . . . . . . . . 78

4.1.2.1. Sampling-based Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.1.2.2. Boundary-based Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.1.2.3. Learning-based Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.1.3. Future Solution Methodologies in the Renewable-Integrated Elec-tricity Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.1.3.1. Stochastic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.1.3.2. Robust Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.1.3.3. Intelligent Model-free Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.1.4. Outlooks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.2. General Simulation Platform for Current and Future Short-term Elec-tricity Market Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.2.1. Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.2.2. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2.3. Working Mode Design for the Multi-timescale Operation . . . . . . . . . . . . 98

4.2.3.1. Mode 1: DAUC only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.2.3.2. Mode 2: RTED only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.2.3.3. Mode 3: DAUC + RTED. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.2.3.4. Mode 4: DAUC + RTUC + RTED . . . . . . . . . . . . . . . . . . . . . . . . 101

4.2.4. Novelties on Electricity Market Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.2.4.1. Novel AS Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.2.4.2. MIDAS-S ESS Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.2.4.3. Other Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.2.5. Modularized Executables for User Flexibility . . . . . . . . . . . . . . . . . . . . . . . . 107

4.2.5.1. Scheduling Simulation Part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.2.5.2. Dynamic Simulation Part. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

viii

4.2.6. Simulation Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.2.6.1. 18-bus Test System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.2.6.2. Scalability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.2.7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5. CONCLUDING REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

A. Linear Relaxation Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

B. Feasibility Guarantee of the GND Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

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LIST OF FIGURES

Figure Page

1.1 Global wind and solar installation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Future coordinated market hierarchy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Framework of the proposed methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Workflow of the proposed algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 System topology of the modified IEEE 30-bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Regional scenario (10 for each). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.5 Comparison for different sizes of scenarios. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 Power transfer from/to Region 1: one scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.7 Comparison in the contingency criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.8 Investment plans and system cost of the long-term planning. . . . . . . . . . . . . . . . . . . . . 33

2.9 Installed capacity for the WECC system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.1 Proposed coordinated market paradigm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 The GND algorithm workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3 Tran6+Dist7+Dist9 system topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.4 Sensitivity analysis on the scenario number. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.5 Algorithmic performance in the Tran6+Dist7+Dist9 case . . . . . . . . . . . . . . . . . . . . . . . 61

3.6 Tran6+Dist7+Dist9 commitment and reconfiguration solutions . . . . . . . . . . . . . . . . . 62

3.7 Boxplot for voltage magnitudes of the two networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.8 Sensitivity analysis on the initial values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.9 Comparative analysis with different stopping criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.1 Operation modes of MIDAS-S. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

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4.2 Two schemes for ESSs providing ASs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.3 Scheduling-dynamic information exchange. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.4 18-bus diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.5 18-bus system’s Mode 4 dispatch results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.6 ESS Schedules in the 18-bus system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

xi

LIST OF TABLES

Table Page

2.1 Data for Candidate Generation Units. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Investment Portfolio Report for the 30-bus System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 Investment Portfolio with the Investment Annuitization. . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1 Comparative Analysis on Isolation and Coordination. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 Maximum SOCP Gaps of the two ADNs (p.u.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1 Comparison of different learning-based approaches and their applications. . . . . . . 82

4.2 Economic Assessment of Different Routines in Day 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.3 Cost Comparison between Four Modes in Day 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.4 Computational Time for Test Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

xii

This dissertation is dedicated to my family and Margaret.

Chapter 1

INTRODUCTION

1.1. Motivations and Preliminaries

The electricity market serves as a system that enables electric power trading by long-term

planning and short-term scheduling through bids to buy and offers to sell. The conventional

deregulated electricity market has been well developed upon multi-timescale co-optimizations

of energy and reserves to achieve the goal mentioned above. In general, a typical electricity

market operation first solves a deterministic ex-ante market optimization that collects the

bids and offers submitted by different stakeholders and clears the market with zonal or

nodal pricing. Then, ex-post dispatch and settlement procedure in real-time are conducted

to realize the cleared power schedules and finalize the payments/charges. This operation

paradigm has been adopted or referred to by many electricity markets around the world,

including the United States [1] and Europe [2].

However, this market framework is now challenged when renewable energy surges in

the power system. Due to the increasing concern of environmental issues, renewable energy

proliferates with generous governmental incentives and community supports. With the global

installations of wind and solar energy depicted in Figure 1.1 (Data source: [3]), the overall

penetration of renewables worldwide has reached 9% in early 2020 [4] and keeps accelerating

phenomenally. From the electricity market’s perspective, however, renewable energy not only

imposes a nature of high capital cost and low operation cost, but also brings uncertainties

that threaten the system reliability. Despite the governmental incentives such as feed-in-

tariffs, renewable energy has been widely questioned for its capital payback and unreliable

power supply [5].

1

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Forecast<latexit sha1_base64="6CIbHyxGKTCaLYPmZx0e9qo/tbQ=">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</latexit><latexit sha1_base64="6CIbHyxGKTCaLYPmZx0e9qo/tbQ=">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</latexit><latexit sha1_base64="6CIbHyxGKTCaLYPmZx0e9qo/tbQ=">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</latexit><latexit sha1_base64="6CIbHyxGKTCaLYPmZx0e9qo/tbQ=">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</latexit>

Figure 1.1. Global wind and solar installation.

On the one hand, from a long-term perspective, before stepping into the daily electricity

market operation, the coordinated system planning should be investigated accounting for the

massive renewable penetration. To what extent that renewable energy should be invested

in the long-term power system planning becomes critical when determining the optimal in-

vestment portfolio of future electricity share. However, the uncertainty of renewable energy

poses a difficult hurdle to the planner since the invested generators might have uncontrollable

variable production. Studying the uncertainty-based system planning in a long-term horizon

contributes to a more reliable and confident short-term or mid-term market operation [6].

On the other hand, the short-term market operation should also be treated very care-

fully since typical renewable energy such as wind and solar possesses short-term variability.

2

One of the most critical concerns is that the downstream markets, i.e., the distribution-level

markets, become increasingly active with the growing penetration of renewable distributed

energy resources (DERs). The uncertainties brought by the DER introduce many unique

challenges, such as the reverse power flow and local net load fluctuation. Hence, an enhanced

market framework that considers the coordination between the transmission and distribution

networks becomes an urgent task [7].

An envisioned overall market hierarchy, as shown in Figure 1.2, provides a road-map

including the key components that should be coordinated in the future market operation.

From the perspective of the central system operator, how to tackle the increasing renewable

uncertainties in both the transmission and distribution networks in the manner of system

operation calls for solutions. Towards the need for a better market operation design, this

dissertation discusses potential solution strategies towards these future challenges while qual-

itatively and quantitatively analyzing the economic values.

1.2. Research Objectives and Achievements

As the proliferating renewable energy introduces more uncertainties to the power sys-

tem, uncertainty-based market operations become a significant research milestone, to which

this dissertation is devoting. As a holistic market operation study, both the long-term and

short-term operation schemes are extensively investigated. From the long-term perspective,

the coordinated system planning problem optimizes the multi-year investment portfolio of

generation mix, including a massive portion of renewable generation, considering practi-

cal system contingencies and uncertainties. From the short-term perspective, transmission

and distribution-coordinated day-ahead market operation considers uncertainties from both

the upstream and downstream markets at different levels, and a new coordinated market

paradigm paves the way for the distribution-aware system operation. We also develop a

general simulation tool considering the renewable impacts in the future electricity market,

which could help both the industry and research communities conduct more comprehensive

studies on future market framework designs.

3

Day-aheadUnit Commitment

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Real-timeDispatch

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Intra-dayUnit Commitment

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Trunk

AncillaryServices

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FinancialTransmission Rights

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OptimalPower Flow<latexit sha1_base64="O8A6QxHZrpfWp9MUjErJx77SBf0=">AAACCHicbVDLSgMxFM3UVx1fVZcuDBahqzLTjS4Lgrizgn1AZyiZ9LYNzSRDkrGU0qUbf8WNC0Xc+gnu/BvTdhbaeuDC4Zx7k3tPlHCmjed9O7m19Y3Nrfy2u7O7t39QODxqaJkqCnUquVStiGjgTEDdMMOhlSggccShGQ2vZn7zAZRmUtybcQJhTPqC9RglxkqdwmlAQRhQTPTxbWJYTHgQuDU5AoWvuRx1CkWv7M2BV4mfkSLKUOsUvoKupGlsX6WcaN32vcSEE6IMoxymbpBqSAgdkj60LRUkBh1O5odM8blVurgnlS1h8Fz9PTEhsdbjOLKdMTEDvezNxP+8dmp6l+GEiSQ1IOjio17KsZF4lgruMgXU8LElhCpmd8V0QBShNhrt2hD85ZNXSaNS9r2yf1cpVktZHHl0gs5QCfnoAlXRDaqhOqLoET2jV/TmPDkvzrvzsWjNOdnMMfoD5/MHPMOZXQ==</latexit><latexit sha1_base64="O8A6QxHZrpfWp9MUjErJx77SBf0=">AAACCHicbVDLSgMxFM3UVx1fVZcuDBahqzLTjS4Lgrizgn1AZyiZ9LYNzSRDkrGU0qUbf8WNC0Xc+gnu/BvTdhbaeuDC4Zx7k3tPlHCmjed9O7m19Y3Nrfy2u7O7t39QODxqaJkqCnUquVStiGjgTEDdMMOhlSggccShGQ2vZn7zAZRmUtybcQJhTPqC9RglxkqdwmlAQRhQTPTxbWJYTHgQuDU5AoWvuRx1CkWv7M2BV4mfkSLKUOsUvoKupGlsX6WcaN32vcSEE6IMoxymbpBqSAgdkj60LRUkBh1O5odM8blVurgnlS1h8Fz9PTEhsdbjOLKdMTEDvezNxP+8dmp6l+GEiSQ1IOjio17KsZF4lgruMgXU8LElhCpmd8V0QBShNhrt2hD85ZNXSaNS9r2yf1cpVktZHHl0gs5QCfnoAlXRDaqhOqLoET2jV/TmPDkvzrvzsWjNOdnMMfoD5/MHPMOZXQ==</latexit><latexit sha1_base64="O8A6QxHZrpfWp9MUjErJx77SBf0=">AAACCHicbVDLSgMxFM3UVx1fVZcuDBahqzLTjS4Lgrizgn1AZyiZ9LYNzSRDkrGU0qUbf8WNC0Xc+gnu/BvTdhbaeuDC4Zx7k3tPlHCmjed9O7m19Y3Nrfy2u7O7t39QODxqaJkqCnUquVStiGjgTEDdMMOhlSggccShGQ2vZn7zAZRmUtybcQJhTPqC9RglxkqdwmlAQRhQTPTxbWJYTHgQuDU5AoWvuRx1CkWv7M2BV4mfkSLKUOsUvoKupGlsX6WcaN32vcSEE6IMoxymbpBqSAgdkj60LRUkBh1O5odM8blVurgnlS1h8Fz9PTEhsdbjOLKdMTEDvezNxP+8dmp6l+GEiSQ1IOjio17KsZF4lgruMgXU8LElhCpmd8V0QBShNhrt2hD85ZNXSaNS9r2yf1cpVktZHHl0gs5QCfnoAlXRDaqhOqLoET2jV/TmPDkvzrvzsWjNOdnMMfoD5/MHPMOZXQ==</latexit><latexit sha1_base64="O8A6QxHZrpfWp9MUjErJx77SBf0=">AAACCHicbVDLSgMxFM3UVx1fVZcuDBahqzLTjS4Lgrizgn1AZyiZ9LYNzSRDkrGU0qUbf8WNC0Xc+gnu/BvTdhbaeuDC4Zx7k3tPlHCmjed9O7m19Y3Nrfy2u7O7t39QODxqaJkqCnUquVStiGjgTEDdMMOhlSggccShGQ2vZn7zAZRmUtybcQJhTPqC9RglxkqdwmlAQRhQTPTxbWJYTHgQuDU5AoWvuRx1CkWv7M2BV4mfkSLKUOsUvoKupGlsX6WcaN32vcSEE6IMoxymbpBqSAgdkj60LRUkBh1O5odM8blVurgnlS1h8Fz9PTEhsdbjOLKdMTEDvezNxP+8dmp6l+GEiSQ1IOjio17KsZF4lgruMgXU8LElhCpmd8V0QBShNhrt2hD85ZNXSaNS9r2yf1cpVktZHHl0gs5QCfnoAlXRDaqhOqLoET2jV/TmPDkvzrvzsWjNOdnMMfoD5/MHPMOZXQ==</latexit>

TransmissionPlanning

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MaintenancePlanning

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ContingencyAnalysis

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StateEstimation<latexit sha1_base64="tKqbFA2LttwTEfE9fcq7P3rJtOI=">AAACBnicbVA9SwNBEJ2LXzF+RS1FWAxCqnCXRsuACJYRzQfkjrC3mSRL9vaO3T0hhFQ2/hUbC0Vs/Q12/hs3yRWa+GDh8d7M7MwLE8G1cd1vJ7e2vrG5ld8u7Ozu7R8UD4+aOk4VwwaLRazaIdUouMSG4UZgO1FIo1BgKxxdzfzWAyrNY3lvxgkGER1I3ueMGit1i6c+Q2lQcTkgd4Ya9P3CtTY8yvySW3HnIKvEy0gJMtS7xS+/F7M0sjOZoFp3PDcxwYQqw5nAacFPNSaUjegAO5ZKGqEOJvMzpuTcKj3Sj5V90pC5+rtjQiOtx1FoK+16Q73szcT/vE5q+pfBhMskNSjZ4qN+KoiJySwT0uMKmRFjSyhT3O5K2JAqymwwumBD8JZPXiXNasVzK95ttVQrZ3Hk4QTOoAweXEANbqAODWDwCM/wCm/Ok/PivDsfi9Kck/Ucwx84nz8ePpjM</latexit><latexit sha1_base64="tKqbFA2LttwTEfE9fcq7P3rJtOI=">AAACBnicbVA9SwNBEJ2LXzF+RS1FWAxCqnCXRsuACJYRzQfkjrC3mSRL9vaO3T0hhFQ2/hUbC0Vs/Q12/hs3yRWa+GDh8d7M7MwLE8G1cd1vJ7e2vrG5ld8u7Ozu7R8UD4+aOk4VwwaLRazaIdUouMSG4UZgO1FIo1BgKxxdzfzWAyrNY3lvxgkGER1I3ueMGit1i6c+Q2lQcTkgd4Ya9P3CtTY8yvySW3HnIKvEy0gJMtS7xS+/F7M0sjOZoFp3PDcxwYQqw5nAacFPNSaUjegAO5ZKGqEOJvMzpuTcKj3Sj5V90pC5+rtjQiOtx1FoK+16Q73szcT/vE5q+pfBhMskNSjZ4qN+KoiJySwT0uMKmRFjSyhT3O5K2JAqymwwumBD8JZPXiXNasVzK95ttVQrZ3Hk4QTOoAweXEANbqAODWDwCM/wCm/Ok/PivDsfi9Kck/Ucwx84nz8ePpjM</latexit><latexit sha1_base64="tKqbFA2LttwTEfE9fcq7P3rJtOI=">AAACBnicbVA9SwNBEJ2LXzF+RS1FWAxCqnCXRsuACJYRzQfkjrC3mSRL9vaO3T0hhFQ2/hUbC0Vs/Q12/hs3yRWa+GDh8d7M7MwLE8G1cd1vJ7e2vrG5ld8u7Ozu7R8UD4+aOk4VwwaLRazaIdUouMSG4UZgO1FIo1BgKxxdzfzWAyrNY3lvxgkGER1I3ueMGit1i6c+Q2lQcTkgd4Ya9P3CtTY8yvySW3HnIKvEy0gJMtS7xS+/F7M0sjOZoFp3PDcxwYQqw5nAacFPNSaUjegAO5ZKGqEOJvMzpuTcKj3Sj5V90pC5+rtjQiOtx1FoK+16Q73szcT/vE5q+pfBhMskNSjZ4qN+KoiJySwT0uMKmRFjSyhT3O5K2JAqymwwumBD8JZPXiXNasVzK95ttVQrZ3Hk4QTOoAweXEANbqAODWDwCM/wCm/Ok/PivDsfi9Kck/Ucwx84nz8ePpjM</latexit><latexit sha1_base64="tKqbFA2LttwTEfE9fcq7P3rJtOI=">AAACBnicbVA9SwNBEJ2LXzF+RS1FWAxCqnCXRsuACJYRzQfkjrC3mSRL9vaO3T0hhFQ2/hUbC0Vs/Q12/hs3yRWa+GDh8d7M7MwLE8G1cd1vJ7e2vrG5ld8u7Ozu7R8UD4+aOk4VwwaLRazaIdUouMSG4UZgO1FIo1BgKxxdzfzWAyrNY3lvxgkGER1I3ueMGit1i6c+Q2lQcTkgd4Ya9P3CtTY8yvySW3HnIKvEy0gJMtS7xS+/F7M0sjOZoFp3PDcxwYQqw5nAacFPNSaUjegAO5ZKGqEOJvMzpuTcKj3Sj5V90pC5+rtjQiOtx1FoK+16Q73szcT/vE5q+pfBhMskNSjZ4qN+KoiJySwT0uMKmRFjSyhT3O5K2JAqymwwumBD8JZPXiXNasVzK95ttVQrZ3Hk4QTOoAweXEANbqAODWDwCM/wCm/Ok/PivDsfi9Kck/Ucwx84nz8ePpjM</latexit>

Multi-regionalCoordination

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Trunk

ISOMarket

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TransmissionSystem

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BulkSystem

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TransmissionLMP

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DistributionLMP

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DSO<latexit sha1_base64="N0uqafgEXxfVo5mP2pwLxdjboXw=">AAACAHicbVC7TsMwFHXKq5RXgIGBxaJCYqqSLjBWgoGNIuhDaqLKcW9aq44T2Q6iirLwKywMIMTKZ7DxN7htBmg50pWOzrnXvvcECWdKO863VVpZXVvfKG9WtrZ3dvfs/YO2ilNJoUVjHstuQBRwJqClmebQTSSQKODQCcaXU7/zAFKxWNzrSQJ+RIaChYwSbaS+feRREBokE0PsaXjUQZhd3d3kfbvq1JwZ8DJxC1JFBZp9+8sbxDSNzGuUE6V6rpNoPyNSM8ohr3ipgoTQMRlCz1BBIlB+Njsgx6dGGeAwlqaExjP190RGIqUmUWA6I6JHatGbiv95vVSHF37GRJJqEHT+UZhyrGM8TQMPmASq+cQQQiUzu2I6IpJQE4mqmBDcxZOXSbtec52ae1uvNmpFHGV0jE7QGXLROWqga9RELURRjp7RK3qznqwX6936mLeWrGLmEP2B9fkD27OWfQ==</latexit><latexit sha1_base64="N0uqafgEXxfVo5mP2pwLxdjboXw=">AAACAHicbVC7TsMwFHXKq5RXgIGBxaJCYqqSLjBWgoGNIuhDaqLKcW9aq44T2Q6iirLwKywMIMTKZ7DxN7htBmg50pWOzrnXvvcECWdKO863VVpZXVvfKG9WtrZ3dvfs/YO2ilNJoUVjHstuQBRwJqClmebQTSSQKODQCcaXU7/zAFKxWNzrSQJ+RIaChYwSbaS+feRREBokE0PsaXjUQZhd3d3kfbvq1JwZ8DJxC1JFBZp9+8sbxDSNzGuUE6V6rpNoPyNSM8ohr3ipgoTQMRlCz1BBIlB+Njsgx6dGGeAwlqaExjP190RGIqUmUWA6I6JHatGbiv95vVSHF37GRJJqEHT+UZhyrGM8TQMPmASq+cQQQiUzu2I6IpJQE4mqmBDcxZOXSbtec52ae1uvNmpFHGV0jE7QGXLROWqga9RELURRjp7RK3qznqwX6936mLeWrGLmEP2B9fkD27OWfQ==</latexit><latexit sha1_base64="N0uqafgEXxfVo5mP2pwLxdjboXw=">AAACAHicbVC7TsMwFHXKq5RXgIGBxaJCYqqSLjBWgoGNIuhDaqLKcW9aq44T2Q6iirLwKywMIMTKZ7DxN7htBmg50pWOzrnXvvcECWdKO863VVpZXVvfKG9WtrZ3dvfs/YO2ilNJoUVjHstuQBRwJqClmebQTSSQKODQCcaXU7/zAFKxWNzrSQJ+RIaChYwSbaS+feRREBokE0PsaXjUQZhd3d3kfbvq1JwZ8DJxC1JFBZp9+8sbxDSNzGuUE6V6rpNoPyNSM8ohr3ipgoTQMRlCz1BBIlB+Njsgx6dGGeAwlqaExjP190RGIqUmUWA6I6JHatGbiv95vVSHF37GRJJqEHT+UZhyrGM8TQMPmASq+cQQQiUzu2I6IpJQE4mqmBDcxZOXSbtec52ae1uvNmpFHGV0jE7QGXLROWqga9RELURRjp7RK3qznqwX6936mLeWrGLmEP2B9fkD27OWfQ==</latexit><latexit sha1_base64="N0uqafgEXxfVo5mP2pwLxdjboXw=">AAACAHicbVC7TsMwFHXKq5RXgIGBxaJCYqqSLjBWgoGNIuhDaqLKcW9aq44T2Q6iirLwKywMIMTKZ7DxN7htBmg50pWOzrnXvvcECWdKO863VVpZXVvfKG9WtrZ3dvfs/YO2ilNJoUVjHstuQBRwJqClmebQTSSQKODQCcaXU7/zAFKxWNzrSQJ+RIaChYwSbaS+feRREBokE0PsaXjUQZhd3d3kfbvq1JwZ8DJxC1JFBZp9+8sbxDSNzGuUE6V6rpNoPyNSM8ohr3ipgoTQMRlCz1BBIlB+Njsgx6dGGeAwlqaExjP190RGIqUmUWA6I6JHatGbiv95vVSHF37GRJJqEHT+UZhyrGM8TQMPmASq+cQQQiUzu2I6IpJQE4mqmBDcxZOXSbtec52ae1uvNmpFHGV0jE7QGXLROWqga9RELURRjp7RK3qznqwX6936mLeWrGLmEP2B9fkD27OWfQ==</latexit>

TrunkP

hysica

lPow

erFlow

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al

Coord

inatio

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DistributionMarket

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T-DCoordination

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DistributionLMP Analysis

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Distribution-levelDER Aggregation

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UtilityBidding Auction

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Real-time Coordinationwith ISO and

Transmission System<latexit sha1_base64="tou/F1/Cz5K8nlX1Kyunz8tATlA=">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</latexit><latexit sha1_base64="tou/F1/Cz5K8nlX1Kyunz8tATlA=">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</latexit><latexit sha1_base64="tou/F1/Cz5K8nlX1Kyunz8tATlA=">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</latexit><latexit sha1_base64="tou/F1/Cz5K8nlX1Kyunz8tATlA=">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</latexit>

UnitCommitment

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EconomicDispatch

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Lower-levelMarket Clearing

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DistributionSystem

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Trunk DERCoordination

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Distribution-levelState Estimation

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Market Scheduleand Dispatch

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(Optimal)Power Flow<latexit sha1_base64="d7Dp3s/aBSDyvOfBjMAOFCnpdrE=">AAACCnicbVDLSgMxFM34rPU16tJNtAh1U2a60WVBEHdWsA/olJJJb9vQTDIkGUsZunbjr7hxoYhbv8Cdf2PazkJbD1w4nHNvcu8JY8608bxvZ2V1bX1jM7eV397Z3dt3Dw7rWiaKQo1KLlUzJBo4E1AzzHBoxgpIFHJohMOrqd94AKWZFPdmHEM7In3BeowSY6WOexJQEAYUE31cvI0Niwg/D4J8VY5A4WsuRx234JW8GfAy8TNSQBmqHfcr6EqaRPZdyonWLd+LTTslyjDKYZIPEg0xoUPSh5algkSg2+nslAk+s0oX96SyJQyeqb8nUhJpPY5C2xkRM9CL3lT8z2slpnfZTpmIEwOCzj/qJRwbiae54C5TQA0fW0KoYnZXTAdEEWrD0Xkbgr948jKpl0u+V/LvyoVKMYsjh47RKSoiH12gCrpBVVRDFD2iZ/SK3pwn58V5dz7mrStONnOE/sD5/AESv5nC</latexit><latexit sha1_base64="d7Dp3s/aBSDyvOfBjMAOFCnpdrE=">AAACCnicbVDLSgMxFM34rPU16tJNtAh1U2a60WVBEHdWsA/olJJJb9vQTDIkGUsZunbjr7hxoYhbv8Cdf2PazkJbD1w4nHNvcu8JY8608bxvZ2V1bX1jM7eV397Z3dt3Dw7rWiaKQo1KLlUzJBo4E1AzzHBoxgpIFHJohMOrqd94AKWZFPdmHEM7In3BeowSY6WOexJQEAYUE31cvI0Niwg/D4J8VY5A4WsuRx234JW8GfAy8TNSQBmqHfcr6EqaRPZdyonWLd+LTTslyjDKYZIPEg0xoUPSh5algkSg2+nslAk+s0oX96SyJQyeqb8nUhJpPY5C2xkRM9CL3lT8z2slpnfZTpmIEwOCzj/qJRwbiae54C5TQA0fW0KoYnZXTAdEEWrD0Xkbgr948jKpl0u+V/LvyoVKMYsjh47RKSoiH12gCrpBVVRDFD2iZ/SK3pwn58V5dz7mrStONnOE/sD5/AESv5nC</latexit><latexit sha1_base64="d7Dp3s/aBSDyvOfBjMAOFCnpdrE=">AAACCnicbVDLSgMxFM34rPU16tJNtAh1U2a60WVBEHdWsA/olJJJb9vQTDIkGUsZunbjr7hxoYhbv8Cdf2PazkJbD1w4nHNvcu8JY8608bxvZ2V1bX1jM7eV397Z3dt3Dw7rWiaKQo1KLlUzJBo4E1AzzHBoxgpIFHJohMOrqd94AKWZFPdmHEM7In3BeowSY6WOexJQEAYUE31cvI0Niwg/D4J8VY5A4WsuRx234JW8GfAy8TNSQBmqHfcr6EqaRPZdyonWLd+LTTslyjDKYZIPEg0xoUPSh5algkSg2+nslAk+s0oX96SyJQyeqb8nUhJpPY5C2xkRM9CL3lT8z2slpnfZTpmIEwOCzj/qJRwbiae54C5TQA0fW0KoYnZXTAdEEWrD0Xkbgr948jKpl0u+V/LvyoVKMYsjh47RKSoiH12gCrpBVVRDFD2iZ/SK3pwn58V5dz7mrStONnOE/sD5/AESv5nC</latexit><latexit sha1_base64="d7Dp3s/aBSDyvOfBjMAOFCnpdrE=">AAACCnicbVDLSgMxFM34rPU16tJNtAh1U2a60WVBEHdWsA/olJJJb9vQTDIkGUsZunbjr7hxoYhbv8Cdf2PazkJbD1w4nHNvcu8JY8608bxvZ2V1bX1jM7eV397Z3dt3Dw7rWiaKQo1KLlUzJBo4E1AzzHBoxgpIFHJohMOrqd94AKWZFPdmHEM7In3BeowSY6WOexJQEAYUE31cvI0Niwg/D4J8VY5A4WsuRx234JW8GfAy8TNSQBmqHfcr6EqaRPZdyonWLd+LTTslyjDKYZIPEg0xoUPSh5algkSg2+nslAk+s0oX96SyJQyeqb8nUhJpPY5C2xkRM9CL3lT8z2slpnfZTpmIEwOCzj/qJRwbiae54C5TQA0fW0KoYnZXTAdEEWrD0Xkbgr948jKpl0u+V/LvyoVKMYsjh47RKSoiH12gCrpBVVRDFD2iZ/SK3pwn58V5dz7mrStONnOE/sD5/AESv5nC</latexit>

Distribution States<latexit sha1_base64="T/e0qcGUrBi3lLwzZOHUdejKg5g=">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</latexit><latexit sha1_base64="T/e0qcGUrBi3lLwzZOHUdejKg5g=">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</latexit><latexit sha1_base64="T/e0qcGUrBi3lLwzZOHUdejKg5g=">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</latexit><latexit sha1_base64="T/e0qcGUrBi3lLwzZOHUdejKg5g=">AAACLnicdVBNSwMxEM3Wr1q/qh69BIvgQUrSg21vBRU8VrQqtEvJptMams0uSVYoS3+RF/+KHgQV8erPMFtbUNGBwOO9eTOTF8RSGEvIs5ebm19YXMovF1ZW19Y3iptblyZKNIcWj2SkrwNmQAoFLSushOtYAwsDCVfB8CjTr25BGxGpCzuKwQ/ZQIm+4Mw6qls86UxmtPUg8FNSJoRQSg8yQKuHxIF6vVahtXGHg7KghRrgY3eVFkGSDcDnllkw3WJp5sUzL555Mc0kVyU0rWa3+NjpRTwJ3VAumTFtSmLrp0xbwSWMC53EQMz4kA2g7aBiIRg/nZw6xnuO6eF+pN1TFk/Y746UhcaMwsB1hszemN9aRv6ltRPbr/mpUHFiQfGvRf1EYhvhLDvcExq4lSMHGNfC3Yr5DdOMu2RMwYUw+yn+H1xWypSU6Vml1NifxpFHO2gX7SOKqqiBTlETtRBHd+gBvaBX79578t6896/WnDf1bKMf5X18Ak4hpbQ=</latexit>

UnitStates

<latexit sha1_base64="XGZeeKEF/iozwIQKAWABevqI6n8=">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</latexit><latexit sha1_base64="XGZeeKEF/iozwIQKAWABevqI6n8=">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</latexit><latexit sha1_base64="XGZeeKEF/iozwIQKAWABevqI6n8=">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</latexit><latexit sha1_base64="XGZeeKEF/iozwIQKAWABevqI6n8=">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</latexit>

EnergyManagement

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MaintenanceScheduling

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DistributionPlanning

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NetworkReconfiguration

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Coordination<latexit sha1_base64="4dbmDpo4Rrv/It/eTZcBRw5hc78=">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</latexit><latexit sha1_base64="4dbmDpo4Rrv/It/eTZcBRw5hc78=">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</latexit><latexit sha1_base64="4dbmDpo4Rrv/It/eTZcBRw5hc78=">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</latexit><latexit sha1_base64="4dbmDpo4Rrv/It/eTZcBRw5hc78=">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</latexit>

DistributionOperation

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PowerTransaction

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Customer

s

Third-

party

DERs

Pro

sum

ers

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Utility-levelDER Aggregation

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Transactive Market<latexit sha1_base64="t6I76gCURcsujpwdRSE57VSQEHM=">AAACBXicbVA9SwNBEN3zM8avU0stFoNgFe7SaBmwsREi5AuSEPY2c8mSvb1jdy4YjjQ2/hUbC0Vs/Q92/hs3H4UmPhh4vDfDzLwgkcKg5307a+sbm1vbuZ387t7+waF7dFw3cao51HgsY90MmAEpFNRQoIRmooFFgYRGMLyZ+o0RaCNiVcVxAp2I9ZUIBWdopa571kZ4wCDMqpopwziKEdA7poeAk3zXLXhFbwa6SvwFKZAFKl33q92LeRqBQi6ZMS3fS7CTMY2CS5jk26mBhPEh60PLUsUiMJ1s9sWEXlilR8NY21JIZ+rviYxFxoyjwHZGDAdm2ZuK/3mtFMPrTiZUkiIoPl8UppJiTKeR0J7QwFGOLWFcC3sr5QOmbRg2uGkI/vLLq6ReKvpe0b8vFcrFRRw5ckrOySXxyRUpk1tSITXCySN5Jq/kzXlyXpx352PeuuYsZk7IHzifP71TmKA=</latexit><latexit sha1_base64="t6I76gCURcsujpwdRSE57VSQEHM=">AAACBXicbVA9SwNBEN3zM8avU0stFoNgFe7SaBmwsREi5AuSEPY2c8mSvb1jdy4YjjQ2/hUbC0Vs/Q92/hs3H4UmPhh4vDfDzLwgkcKg5307a+sbm1vbuZ387t7+waF7dFw3cao51HgsY90MmAEpFNRQoIRmooFFgYRGMLyZ+o0RaCNiVcVxAp2I9ZUIBWdopa571kZ4wCDMqpopwziKEdA7poeAk3zXLXhFbwa6SvwFKZAFKl33q92LeRqBQi6ZMS3fS7CTMY2CS5jk26mBhPEh60PLUsUiMJ1s9sWEXlilR8NY21JIZ+rviYxFxoyjwHZGDAdm2ZuK/3mtFMPrTiZUkiIoPl8UppJiTKeR0J7QwFGOLWFcC3sr5QOmbRg2uGkI/vLLq6ReKvpe0b8vFcrFRRw5ckrOySXxyRUpk1tSITXCySN5Jq/kzXlyXpx352PeuuYsZk7IHzifP71TmKA=</latexit><latexit sha1_base64="t6I76gCURcsujpwdRSE57VSQEHM=">AAACBXicbVA9SwNBEN3zM8avU0stFoNgFe7SaBmwsREi5AuSEPY2c8mSvb1jdy4YjjQ2/hUbC0Vs/Q92/hs3H4UmPhh4vDfDzLwgkcKg5307a+sbm1vbuZ387t7+waF7dFw3cao51HgsY90MmAEpFNRQoIRmooFFgYRGMLyZ+o0RaCNiVcVxAp2I9ZUIBWdopa571kZ4wCDMqpopwziKEdA7poeAk3zXLXhFbwa6SvwFKZAFKl33q92LeRqBQi6ZMS3fS7CTMY2CS5jk26mBhPEh60PLUsUiMJ1s9sWEXlilR8NY21JIZ+rviYxFxoyjwHZGDAdm2ZuK/3mtFMPrTiZUkiIoPl8UppJiTKeR0J7QwFGOLWFcC3sr5QOmbRg2uGkI/vLLq6ReKvpe0b8vFcrFRRw5ckrOySXxyRUpk1tSITXCySN5Jq/kzXlyXpx352PeuuYsZk7IHzifP71TmKA=</latexit><latexit sha1_base64="t6I76gCURcsujpwdRSE57VSQEHM=">AAACBXicbVA9SwNBEN3zM8avU0stFoNgFe7SaBmwsREi5AuSEPY2c8mSvb1jdy4YjjQ2/hUbC0Vs/Q92/hs3H4UmPhh4vDfDzLwgkcKg5307a+sbm1vbuZ387t7+waF7dFw3cao51HgsY90MmAEpFNRQoIRmooFFgYRGMLyZ+o0RaCNiVcVxAp2I9ZUIBWdopa571kZ4wCDMqpopwziKEdA7poeAk3zXLXhFbwa6SvwFKZAFKl33q92LeRqBQi6ZMS3fS7CTMY2CS5jk26mBhPEh60PLUsUiMJ1s9sWEXlilR8NY21JIZ+rviYxFxoyjwHZGDAdm2ZuK/3mtFMPrTiZUkiIoPl8UppJiTKeR0J7QwFGOLWFcC3sr5QOmbRg2uGkI/vLLq6ReKvpe0b8vFcrFRRw5ckrOySXxyRUpk1tSITXCySN5Jq/kzXlyXpx352PeuuYsZk7IHzifP71TmKA=</latexit>

Distribution Network<latexit sha1_base64="in+hb/Mv8vlYcW0/BnhADwaUHzY=">AAACB3icbVA9SwNBEN3zM8avU0tBFoNgFe7SaBnQwkoimA9IQtjbzCVL9j7YnVPDkc7Gv2JjoYitf8HOf+NecoUmPhh4vDfDzDwvlkKj43xbS8srq2vrhY3i5tb2zq69t9/QUaI41HkkI9XymAYpQqijQAmtWAELPAlNb3SR+c07UFpE4S2OY+gGbBAKX3CGRurZRx2EB/T89NLsUsJLMpleA95HajQp9uySU3amoIvEzUmJ5Kj17K9OP+JJACFyybRuu06M3ZQpFFzCpNhJNMSMj9gA2oaGLADdTad/TOiJUfrUj5SpEOlU/T2RskDrceCZzoDhUM97mfif107QP++mIowThJDPFvmJpBjRLBTaFwo4yrEhjCthbqV8yBTjaKLLQnDnX14kjUrZdcruTaVUreRxFMghOSanxCVnpEquSI3UCSeP5Jm8kjfryXqx3q2PWeuSlc8ckD+wPn8AqLKZug==</latexit><latexit sha1_base64="in+hb/Mv8vlYcW0/BnhADwaUHzY=">AAACB3icbVA9SwNBEN3zM8avU0tBFoNgFe7SaBnQwkoimA9IQtjbzCVL9j7YnVPDkc7Gv2JjoYitf8HOf+NecoUmPhh4vDfDzDwvlkKj43xbS8srq2vrhY3i5tb2zq69t9/QUaI41HkkI9XymAYpQqijQAmtWAELPAlNb3SR+c07UFpE4S2OY+gGbBAKX3CGRurZRx2EB/T89NLsUsJLMpleA95HajQp9uySU3amoIvEzUmJ5Kj17K9OP+JJACFyybRuu06M3ZQpFFzCpNhJNMSMj9gA2oaGLADdTad/TOiJUfrUj5SpEOlU/T2RskDrceCZzoDhUM97mfif107QP++mIowThJDPFvmJpBjRLBTaFwo4yrEhjCthbqV8yBTjaKLLQnDnX14kjUrZdcruTaVUreRxFMghOSanxCVnpEquSI3UCSeP5Jm8kjfryXqx3q2PWeuSlc8ckD+wPn8AqLKZug==</latexit><latexit sha1_base64="in+hb/Mv8vlYcW0/BnhADwaUHzY=">AAACB3icbVA9SwNBEN3zM8avU0tBFoNgFe7SaBnQwkoimA9IQtjbzCVL9j7YnVPDkc7Gv2JjoYitf8HOf+NecoUmPhh4vDfDzDwvlkKj43xbS8srq2vrhY3i5tb2zq69t9/QUaI41HkkI9XymAYpQqijQAmtWAELPAlNb3SR+c07UFpE4S2OY+gGbBAKX3CGRurZRx2EB/T89NLsUsJLMpleA95HajQp9uySU3amoIvEzUmJ5Kj17K9OP+JJACFyybRuu06M3ZQpFFzCpNhJNMSMj9gA2oaGLADdTad/TOiJUfrUj5SpEOlU/T2RskDrceCZzoDhUM97mfif107QP++mIowThJDPFvmJpBjRLBTaFwo4yrEhjCthbqV8yBTjaKLLQnDnX14kjUrZdcruTaVUreRxFMghOSanxCVnpEquSI3UCSeP5Jm8kjfryXqx3q2PWeuSlc8ckD+wPn8AqLKZug==</latexit><latexit sha1_base64="in+hb/Mv8vlYcW0/BnhADwaUHzY=">AAACB3icbVA9SwNBEN3zM8avU0tBFoNgFe7SaBnQwkoimA9IQtjbzCVL9j7YnVPDkc7Gv2JjoYitf8HOf+NecoUmPhh4vDfDzDwvlkKj43xbS8srq2vrhY3i5tb2zq69t9/QUaI41HkkI9XymAYpQqijQAmtWAELPAlNb3SR+c07UFpE4S2OY+gGbBAKX3CGRurZRx2EB/T89NLsUsJLMpleA95HajQp9uySU3amoIvEzUmJ5Kj17K9OP+JJACFyybRuu06M3ZQpFFzCpNhJNMSMj9gA2oaGLADdTad/TOiJUfrUj5SpEOlU/T2RskDrceCZzoDhUM97mfif107QP++mIowThJDPFvmJpBjRLBTaFwo4yrEhjCthbqV8yBTjaKLLQnDnX14kjUrZdcruTaVUreRxFMghOSanxCVnpEquSI3UCSeP5Jm8kjfryXqx3q2PWeuSlc8ckD+wPn8AqLKZug==</latexit>

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Utility Operation<latexit sha1_base64="JQLh3yD32B84Mnq0Y72IsZz9EVU=">AAACA3icbVC7SgNBFJ31GeNr1U6bwSBYhd00WgZs7IxgHpCEMDu5mwyZnV1m7ophCdj4KzYWitj6E3b+jZNNCk08MHA45x7u3BMkUhj0vG9nZXVtfWOzsFXc3tnd23cPDhsmTjWHOo9lrFsBMyCFgjoKlNBKNLAokNAMRldTv3kP2ohY3eE4gW7EBkqEgjO0Us897iA8YBBmNiwFjulNAjr3Jj235JW9HHSZ+HNSInPUeu5Xpx/zNAKFXDJj2r6XYDdjGgWXMCl2UgMJ4yM2gLalikVgull+w4SeWaVPw1jbp5Dm6u9ExiJjxlFgJyOGQ7PoTcX/vHaK4WU3EypJERSfLQpTSTGm00JoX2jgKMeWMK6F/SvlQ6YZR1tb0ZbgL568TBqVsu+V/dtKqVqe11EgJ+SUnBOfXJAquSY1UiecPJJn8krenCfnxXl3PmajK848c0T+wPn8AejgmD8=</latexit><latexit sha1_base64="JQLh3yD32B84Mnq0Y72IsZz9EVU=">AAACA3icbVC7SgNBFJ31GeNr1U6bwSBYhd00WgZs7IxgHpCEMDu5mwyZnV1m7ophCdj4KzYWitj6E3b+jZNNCk08MHA45x7u3BMkUhj0vG9nZXVtfWOzsFXc3tnd23cPDhsmTjWHOo9lrFsBMyCFgjoKlNBKNLAokNAMRldTv3kP2ohY3eE4gW7EBkqEgjO0Us897iA8YBBmNiwFjulNAjr3Jj235JW9HHSZ+HNSInPUeu5Xpx/zNAKFXDJj2r6XYDdjGgWXMCl2UgMJ4yM2gLalikVgull+w4SeWaVPw1jbp5Dm6u9ExiJjxlFgJyOGQ7PoTcX/vHaK4WU3EypJERSfLQpTSTGm00JoX2jgKMeWMK6F/SvlQ6YZR1tb0ZbgL568TBqVsu+V/dtKqVqe11EgJ+SUnBOfXJAquSY1UiecPJJn8krenCfnxXl3PmajK848c0T+wPn8AejgmD8=</latexit><latexit sha1_base64="JQLh3yD32B84Mnq0Y72IsZz9EVU=">AAACA3icbVC7SgNBFJ31GeNr1U6bwSBYhd00WgZs7IxgHpCEMDu5mwyZnV1m7ophCdj4KzYWitj6E3b+jZNNCk08MHA45x7u3BMkUhj0vG9nZXVtfWOzsFXc3tnd23cPDhsmTjWHOo9lrFsBMyCFgjoKlNBKNLAokNAMRldTv3kP2ohY3eE4gW7EBkqEgjO0Us897iA8YBBmNiwFjulNAjr3Jj235JW9HHSZ+HNSInPUeu5Xpx/zNAKFXDJj2r6XYDdjGgWXMCl2UgMJ4yM2gLalikVgull+w4SeWaVPw1jbp5Dm6u9ExiJjxlFgJyOGQ7PoTcX/vHaK4WU3EypJERSfLQpTSTGm00JoX2jgKMeWMK6F/SvlQ6YZR1tb0ZbgL568TBqVsu+V/dtKqVqe11EgJ+SUnBOfXJAquSY1UiecPJJn8krenCfnxXl3PmajK848c0T+wPn8AejgmD8=</latexit><latexit sha1_base64="JQLh3yD32B84Mnq0Y72IsZz9EVU=">AAACA3icbVC7SgNBFJ31GeNr1U6bwSBYhd00WgZs7IxgHpCEMDu5mwyZnV1m7ophCdj4KzYWitj6E3b+jZNNCk08MHA45x7u3BMkUhj0vG9nZXVtfWOzsFXc3tnd23cPDhsmTjWHOo9lrFsBMyCFgjoKlNBKNLAokNAMRldTv3kP2ohY3eE4gW7EBkqEgjO0Us897iA8YBBmNiwFjulNAjr3Jj235JW9HHSZ+HNSInPUeu5Xpx/zNAKFXDJj2r6XYDdjGgWXMCl2UgMJ4yM2gLalikVgull+w4SeWaVPw1jbp5Dm6u9ExiJjxlFgJyOGQ7PoTcX/vHaK4WU3EypJERSfLQpTSTGm00JoX2jgKMeWMK6F/SvlQ6YZR1tb0ZbgL568TBqVsu+V/dtKqVqe11EgJ+SUnBOfXJAquSY1UiecPJJn8krenCfnxXl3PmajK848c0T+wPn8AejgmD8=</latexit>

TrunkSmart Meter Data

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Trunk

Trunk

RetailMarket

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DistributionControllers

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SystemMeasurements<latexit sha1_base64="hwe1I4lPS4hhWHDzc7E4gKxhnjE=">AAACCnicbVC7TsMwFHXKq5RXgJHFUCF1qpIuMFZiYUEqgj6kJqoc96a16jiR7SBFVWcWfoWFAYRY+QI2/ganzQAtR7J0dM49ur4nSDhT2nG+rdLa+sbmVnm7srO7t39gHx51VJxKCm0a81j2AqKAMwFtzTSHXiKBRAGHbjC5yv3uA0jFYnGvswT8iIwECxkl2kgD+9SjIDRIJkb4LlMaIux5+AaISiVExlIDu+rUnTnwKnELUkUFWgP7yxvGNM3DlBOl+q6TaH9KpGaUw6zipQoSQidkBH1DBYlA+dP5KTN8bpQhDmNpntB4rv5OTEmkVBYFZjIieqyWvVz8z+unOrz0p0wkqQZBF4vClGMd47wXPGQSqOaZIYRKZv6K6ZhIQk05qmJKcJdPXiWdRt116u5to9qsFXWU0Qk6QzXkogvURNeohdqIokf0jF7Rm/VkvVjv1sditGQVmWP0B9bnD0kFmoo=</latexit><latexit sha1_base64="hwe1I4lPS4hhWHDzc7E4gKxhnjE=">AAACCnicbVC7TsMwFHXKq5RXgJHFUCF1qpIuMFZiYUEqgj6kJqoc96a16jiR7SBFVWcWfoWFAYRY+QI2/ganzQAtR7J0dM49ur4nSDhT2nG+rdLa+sbmVnm7srO7t39gHx51VJxKCm0a81j2AqKAMwFtzTSHXiKBRAGHbjC5yv3uA0jFYnGvswT8iIwECxkl2kgD+9SjIDRIJkb4LlMaIux5+AaISiVExlIDu+rUnTnwKnELUkUFWgP7yxvGNM3DlBOl+q6TaH9KpGaUw6zipQoSQidkBH1DBYlA+dP5KTN8bpQhDmNpntB4rv5OTEmkVBYFZjIieqyWvVz8z+unOrz0p0wkqQZBF4vClGMd47wXPGQSqOaZIYRKZv6K6ZhIQk05qmJKcJdPXiWdRt116u5to9qsFXWU0Qk6QzXkogvURNeohdqIokf0jF7Rm/VkvVjv1sditGQVmWP0B9bnD0kFmoo=</latexit><latexit sha1_base64="hwe1I4lPS4hhWHDzc7E4gKxhnjE=">AAACCnicbVC7TsMwFHXKq5RXgJHFUCF1qpIuMFZiYUEqgj6kJqoc96a16jiR7SBFVWcWfoWFAYRY+QI2/ganzQAtR7J0dM49ur4nSDhT2nG+rdLa+sbmVnm7srO7t39gHx51VJxKCm0a81j2AqKAMwFtzTSHXiKBRAGHbjC5yv3uA0jFYnGvswT8iIwECxkl2kgD+9SjIDRIJkb4LlMaIux5+AaISiVExlIDu+rUnTnwKnELUkUFWgP7yxvGNM3DlBOl+q6TaH9KpGaUw6zipQoSQidkBH1DBYlA+dP5KTN8bpQhDmNpntB4rv5OTEmkVBYFZjIieqyWvVz8z+unOrz0p0wkqQZBF4vClGMd47wXPGQSqOaZIYRKZv6K6ZhIQk05qmJKcJdPXiWdRt116u5to9qsFXWU0Qk6QzXkogvURNeohdqIokf0jF7Rm/VkvVjv1sditGQVmWP0B9bnD0kFmoo=</latexit><latexit sha1_base64="hwe1I4lPS4hhWHDzc7E4gKxhnjE=">AAACCnicbVC7TsMwFHXKq5RXgJHFUCF1qpIuMFZiYUEqgj6kJqoc96a16jiR7SBFVWcWfoWFAYRY+QI2/ganzQAtR7J0dM49ur4nSDhT2nG+rdLa+sbmVnm7srO7t39gHx51VJxKCm0a81j2AqKAMwFtzTSHXiKBRAGHbjC5yv3uA0jFYnGvswT8iIwECxkl2kgD+9SjIDRIJkb4LlMaIux5+AaISiVExlIDu+rUnTnwKnELUkUFWgP7yxvGNM3DlBOl+q6TaH9KpGaUw6zipQoSQidkBH1DBYlA+dP5KTN8bpQhDmNpntB4rv5OTEmkVBYFZjIieqyWvVz8z+unOrz0p0wkqQZBF4vClGMd47wXPGQSqOaZIYRKZv6K6ZhIQk05qmJKcJdPXiWdRt116u5to9qsFXWU0Qk6QzXkogvURNeohdqIokf0jF7Rm/VkvVjv1sditGQVmWP0B9bnD0kFmoo=</latexit>

InfrastructureMaintenance<latexit sha1_base64="M1dl+ddCpN1Y6Ug6aP+K16A6XDI=">AAACEXicbVC7SgNBFJ31GeNr1dJmMAipwm4aLQM2WggRzAOyS7g7uZsMmZ1dZmaFEPILNv6KjYUitnZ2/o2TR6GJBwYO59w7M+dEmeDaeN63s7a+sbm1Xdgp7u7tHxy6R8dNneaKYYOlIlXtCDQKLrFhuBHYzhRCEglsRcOrqd96QKV5Ku/NKMMwgb7kMWdgrNR1ywFDaVBx2ac3MlagjcqZyRXSIKC3wK0pQTLsuiWv4s1AV4m/ICWyQL3rfgW9lOWJvZ4J0Lrje5kJx6AMZwInxSDXmAEbQh87lkpIUIfjWaIJPbdKj8apskcaOlN/b4wh0XqURHYyATPQy95U/M/r5Ca+DMdcZrnNxeYPxbmgJqXTemiPK2RGjCwBprj9K2UDUMBsR7poS/CXI6+SZrXiexX/rlqqlRd1FMgpOSNl4pMLUiPXpE4ahJFH8kxeyZvz5Lw4787HfHTNWeyckD9wPn8A6Q6dlg==</latexit><latexit sha1_base64="M1dl+ddCpN1Y6Ug6aP+K16A6XDI=">AAACEXicbVC7SgNBFJ31GeNr1dJmMAipwm4aLQM2WggRzAOyS7g7uZsMmZ1dZmaFEPILNv6KjYUitnZ2/o2TR6GJBwYO59w7M+dEmeDaeN63s7a+sbm1Xdgp7u7tHxy6R8dNneaKYYOlIlXtCDQKLrFhuBHYzhRCEglsRcOrqd96QKV5Ku/NKMMwgb7kMWdgrNR1ywFDaVBx2ac3MlagjcqZyRXSIKC3wK0pQTLsuiWv4s1AV4m/ICWyQL3rfgW9lOWJvZ4J0Lrje5kJx6AMZwInxSDXmAEbQh87lkpIUIfjWaIJPbdKj8apskcaOlN/b4wh0XqURHYyATPQy95U/M/r5Ca+DMdcZrnNxeYPxbmgJqXTemiPK2RGjCwBprj9K2UDUMBsR7poS/CXI6+SZrXiexX/rlqqlRd1FMgpOSNl4pMLUiPXpE4ahJFH8kxeyZvz5Lw4787HfHTNWeyckD9wPn8A6Q6dlg==</latexit><latexit sha1_base64="M1dl+ddCpN1Y6Ug6aP+K16A6XDI=">AAACEXicbVC7SgNBFJ31GeNr1dJmMAipwm4aLQM2WggRzAOyS7g7uZsMmZ1dZmaFEPILNv6KjYUitnZ2/o2TR6GJBwYO59w7M+dEmeDaeN63s7a+sbm1Xdgp7u7tHxy6R8dNneaKYYOlIlXtCDQKLrFhuBHYzhRCEglsRcOrqd96QKV5Ku/NKMMwgb7kMWdgrNR1ywFDaVBx2ac3MlagjcqZyRXSIKC3wK0pQTLsuiWv4s1AV4m/ICWyQL3rfgW9lOWJvZ4J0Lrje5kJx6AMZwInxSDXmAEbQh87lkpIUIfjWaIJPbdKj8apskcaOlN/b4wh0XqURHYyATPQy95U/M/r5Ca+DMdcZrnNxeYPxbmgJqXTemiPK2RGjCwBprj9K2UDUMBsR7poS/CXI6+SZrXiexX/rlqqlRd1FMgpOSNl4pMLUiPXpE4ahJFH8kxeyZvz5Lw4787HfHTNWeyckD9wPn8A6Q6dlg==</latexit><latexit sha1_base64="M1dl+ddCpN1Y6Ug6aP+K16A6XDI=">AAACEXicbVC7SgNBFJ31GeNr1dJmMAipwm4aLQM2WggRzAOyS7g7uZsMmZ1dZmaFEPILNv6KjYUitnZ2/o2TR6GJBwYO59w7M+dEmeDaeN63s7a+sbm1Xdgp7u7tHxy6R8dNneaKYYOlIlXtCDQKLrFhuBHYzhRCEglsRcOrqd96QKV5Ku/NKMMwgb7kMWdgrNR1ywFDaVBx2ac3MlagjcqZyRXSIKC3wK0pQTLsuiWv4s1AV4m/ICWyQL3rfgW9lOWJvZ4J0Lrje5kJx6AMZwInxSDXmAEbQh87lkpIUIfjWaIJPbdKj8apskcaOlN/b4wh0XqURHYyATPQy95U/M/r5Ca+DMdcZrnNxeYPxbmgJqXTemiPK2RGjCwBprj9K2UDUMBsR7poS/CXI6+SZrXiexX/rlqqlRd1FMgpOSNl4pMLUiPXpE4ahJFH8kxeyZvz5Lw4787HfHTNWeyckD9wPn8A6Q6dlg==</latexit>

Microgrid Controland Scheduling

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Third-party DERAuction

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Distributed Gen.Scheduling

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Contro

lSig

nal

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Netw

ork

Sta

tes

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Prosumer Scheduling<latexit sha1_base64="gcKdp/Aqp44yV9wlDFFCwgOrgRU=">AAACJHicbVBJSwMxGM241rpVPXoJFqEHKZPRMhUvBS8eK9oF2qFk0sw0NLOQZIQy9Md48a948eCCBy/+FjPtHLT1QeDx3rfluTFnUpnml7Gyura+sVnYKm7v7O7tlw4O2zJKBKEtEvFIdF0sKWchbSmmOO3GguLA5bTjjq8zv/NAhWRReK8mMXUC7IfMYwQrLQ1KV2l/NqQnfNdJzep5DVl1dGZWrYvLul3TxDRtZNvTpohkElAB78iIDhO9zx+UypmdAS4TlJMyyNEclN77w4joKaEiHEvZQ2asnBQLxQin02I/kTTGZIx92tM0xAGVTjq7bwpPtTKEXiT0CxWcqb87UhxIOQlcXRlgNZKLXib+5/US5dWdlIVxomhI5ou8hEMVwSwxOGSCEsUnmmAimL4VkhEWmCida1GHgBa/vEzaVhWZVXRrlRuVPI4COAYnoAIQsEED3IAmaAECHsEzeAVvxpPxYnwYn/PSFSPvOQJ/YHz/ACkfoWA=</latexit><latexit sha1_base64="gcKdp/Aqp44yV9wlDFFCwgOrgRU=">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</latexit><latexit sha1_base64="gcKdp/Aqp44yV9wlDFFCwgOrgRU=">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</latexit><latexit sha1_base64="gcKdp/Aqp44yV9wlDFFCwgOrgRU=">AAACJHicbVBJSwMxGM241rpVPXoJFqEHKZPRMhUvBS8eK9oF2qFk0sw0NLOQZIQy9Md48a948eCCBy/+FjPtHLT1QeDx3rfluTFnUpnml7Gyura+sVnYKm7v7O7tlw4O2zJKBKEtEvFIdF0sKWchbSmmOO3GguLA5bTjjq8zv/NAhWRReK8mMXUC7IfMYwQrLQ1KV2l/NqQnfNdJzep5DVl1dGZWrYvLul3TxDRtZNvTpohkElAB78iIDhO9zx+UypmdAS4TlJMyyNEclN77w4joKaEiHEvZQ2asnBQLxQin02I/kTTGZIx92tM0xAGVTjq7bwpPtTKEXiT0CxWcqb87UhxIOQlcXRlgNZKLXib+5/US5dWdlIVxomhI5ou8hEMVwSwxOGSCEsUnmmAimL4VkhEWmCida1GHgBa/vEzaVhWZVXRrlRuVPI4COAYnoAIQsEED3IAmaAECHsEzeAVvxpPxYnwYn/PSFSPvOQJ/YHz/ACkfoWA=</latexit>

Distributed Gen.Biddings

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DER O↵ers<latexit sha1_base64="Ctph/NjAq/y/iiqqY5u/hs2ELsQ=">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</latexit><latexit sha1_base64="Ctph/NjAq/y/iiqqY5u/hs2ELsQ=">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</latexit><latexit sha1_base64="Ctph/NjAq/y/iiqqY5u/hs2ELsQ=">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</latexit><latexit sha1_base64="Ctph/NjAq/y/iiqqY5u/hs2ELsQ=">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</latexit>

Figure 1.2. Future coordinated market hierarchy.

4

More specifically, we outline the detailed contributions and achievements of this work as

below.

• Long-term Operation: Coordinated System Planning: We propose a novel

modeling framework and decomposition-based solution strategy combining stochastic

programming (SP) and robust optimization (RO) to deal with multiplex uncertain-

ties in coordinated long-term power system planning. The problem is formulated as

a multi-year generation and transmission planning problem from a system operator’s

perspective to minimize both expansion and operational costs under binary and contin-

uous uncertainties, i.e., system component contingency, and load/generation variation.

N-k contingencies are captured in RO using the reformulated contingency criteria, while

the load/generation uncertainty is considered in SP embedded with RO using operat-

ing scenarios generated from the historical data with spatiotemporal correlations. The

original hybrid model is highly intractable, but the intractability can be relieved by the

proposed decomposition strategy based on the column-and-constraint generation and

L-shaped algorithms. We apply our model to perform long-term system planning un-

der extremely high renewable penetration and investigate the case of 100% renewables

in long-term planning.

• Short-term Operation: Transmission and Distribution Coordinated Mar-

ket: We propose a three-stage unit commitment model for the market operation of

transmission and distribution coordination under the uncertainties of renewable gen-

eration and demand variations. The first stage is for the independent system operator

that determines the commitment decisions of the transmission-level generators and

the distribution-level system reconfiguration; the second stage optimizes the transmis-

sion economic dispatch; then distribution system operators in the third stage perform

their economic dispatch and exchange information with the transmission operator at

the boundary nodes. Between the transmission and distribution networks, not only

conventional thermal generators but renewables and variable demands are considered,

which are tackled via a multi-stage stochastic programming approach. The model

adopts a convexified alternating-current branch flow formulation in the distribution

5

system. We devise a generalized nested L-shaped algorithm to solve the proposed

framework in an efficient manner.

• Electricity Market Co-simulation: General Simulation Tool for User-friendly

Flexibility: With the growing penetration of renewable resources, market operators

call for an evolving market framework with refined operational models considering

the unique contributions from renewable resources and energy storage systems. In

the same vein, real-time system-wide simulations also need to intercommunicate with

power system dynamics. To meet the demand of a general testing environment for

new market mechanisms and requirements, we develop a novel electricity market oper-

ation paradigm released as MIDAS-S, a flexible Python-based simulation platform. In

this framework, MIDAS-S can efficiently build and solve multi-timescale market opti-

mization models considering novel operation constraints while interfacing with system

dynamic simulations performed in commercial software, e.g., PSS®E. Numerical ex-

periments corroborate the proposed platform’s effectiveness by conducting economic

assessments on real-sized systems with real-time dynamics.

We organize the remainder of this dissertation as follows.

Chapter 2 develops a long-term power system planning problem considering the unit

contingencies and the renewable/load uncertainties. The hybrid stochastic and robust plan-

ning problem is solved by a combined algorithm of column-and-constraint generation and

L-shaped method. Chapter 3 quantitatively analyzes the T-D coordinated market frame-

work that takes the local DERs into account with utility-level renewable energy. We devise

a generalized decomposition technique to facilitate the solution of such a complex problem.

Chapter 4 first gives a comprehensive overview for state-of-the-art electricity market oper-

ations in the current industry practice and academia, then presents a general simulation

platform devised for flexible market environment parameterization and use case designs.

Chapter 5 concludes this dissertation with several remarks.

6

Chapter 2

GENERATION AND TRANSMISSION EXPANSION PLANNING

TOWARDS A 100% RENEWABLE FUTURE

2.1. Nomenclature

We provide a detailed notation list for this chapter.

1) Sets:

G /R /L Conventional generator / renewable generator / transmission line

S /V Sending/receiving bus of transmission lines

C Bus mapping of conventional generators, renewable generators, and demand

XG /XR Candidate conventional / renewable generator

XL Candidate transmission line

T Planning horizon (years)

H Operation horizon (hours)

M Planning / operation time index mapping

N Control mapping between units and investable years

2) Indices:

g / r / ` Conventional generator / renewable generator / transmission line

d Demand

n Bus

ω Scenario

7

k Iteration counter for C&CG algorithm

o Iteration counter for L-shaped algorithm

t /h Planning / operating time index. (t, h) ∈M

3) Parameters:

ICGg,t/IC

Rr,t Investment cost for gth conventional / rth renewable generator in tth year [$]

ICL`,t Investment cost for `th transmission line in tth year [$]

OCg,h Operating cost for gth generator in hth hour [$/MWh]

PCd,h Load-shedding cost for unserved load in dth demand in hth hour [$/MWh]

BG /BL Investment budget for generators / transmission lines [$]

Kt] Contingency criterion for system units in tth year

A` Line reactance of `th transmission line under base MVA [p.u.]

FL` Flow capacity of `th transmission line [MW]

PLG

g Capacity limit of gth conventional generator [MW]

∆n /∆n Angle limit of phase angle at nth bus [rad]

4) Uncertain Parameters:

PLR

r,h(ω) Available active power of rth renewable generator in hth hour [MW].

Pd,h(ω) Active demand of dth load in hth hour [MW].

5) Variables:

xGg,t/xRr,t

Binary expansion decision for gth conventional / rth renewable generator in tth year.

x = 1 means built; x = 0 otherwise.

xL`,tBinary expansion decision for `th transmission line. x = 1 means built; x = 0

otherwise.

8

yGg,t/yRr,t

Binary availability indicator for gth conventional / rth renewable generator

in tth year. y = 1 means available; y = 0 otherwise.

yL`,tBinary availability indicator for `th transmission line. y = 1 means available;

y = 0 otherwise.

aGg,t /aRr,t

Binary outage indicator for gth / rth generator. a = 1 means outage; a = 0

otherwise.

aL`,tBinary outage indicator for `th transmission line. a = 1 means outage; a = 0

otherwise.

pGg,h(ω)Scheduled active power of gth conventional generator in hth hour under

scenario ω [MW].

pRr,h(ω)Scheduled active power of rth renewable generator in hth hour under

scenario ω [MW].

rd,h(ω) Scheduled load shedding in dth demand in hth hour under scenario ω [MW].

f`,h(ω)Active power flow through `th transmission line in hth hour under scenario ω

unit: [MW].

δn,h(ω) Phase angle at nth bus in hth hour under scenario ω [rad].

2.2. Overview

The investigation on the operational patterns and economics of a power system with

high renewable penetration attracts tremendous attention from both academia and indus-

try. Despite the clean energy and zero operational cost that renewables have, the stochastic

generation pattern of renewables challenges the reliability of power systems, especially when

the grid requires a high level of reserves for contingencies. Thus, albeit the investment of

renewables is greatly encouraged by the U.S. government and broader community [8], the

system planning under high-level penetration of renewables needs to be further investigated.

Heretofore, researchers have done extensive work on the uncertainty-based system plan-

ning [6,9–11]. On the one hand, generators and lines are the biggest reliability concern from

9

an ISO’s perspective, which has been omnipresent in many operation problems, e.g., unit

commitment problem [12] and transmission planning problem [9]. This type of uncertainty

can be effectively tackled by robust optimization (RO), as described in [9] and [13]. The

basic idea of RO is to find the worst-case scenario and then make preventative decisions,

which in turn also makes the solution highly conservative. On the other hand, the uncertain

nature of renewable energy and elastic demand is another challenge for the system plan-

ner. A significant amount of literature (e.g., [14] and [15]) also adopts RO to deal with this

type of uncertainty by setting conservative boundaries. However, comparatively, stochas-

tic programming (SP) yields less conservative solutions than RO, and we can leverage the

historical data to generate scenarios. Particularly, for a large system with high renewable

penetration, the spatiotemporal correlations of renewables and demand can be accurately

captured by scenario generation, e.g., Monte-Carlo simulation together with a multi-stage

scenario tree [16].

Since the uncertainties are multiplex in today’s transmission networks, e.g., the binary

status of generator/line and continuous generation/load volatility, the combination of SP and

RO is a promising formulation with higher reliability. Recently, many works (e.g., [10, 11]

and [17]) also consider the hybrid SP and RO formulation with only the continuous formula-

tion, which provides a limited evaluation on uncertainties. There are few works considering

the contingencies of power system components and stochastic generation together in a gen-

eration and transmission expansion problem (G&TEP), especially under high renewable

penetration levels.

The co-optimization of G&TEP under multiplex uncertainties is a nontrivial problem

due to its non-convexity and uncertainties involved. A. Moreira et al. [6] integrated binary

contingencies with continuous uncertainties but did not explore the improvement of algo-

rithms to facilitate the solution. L. Gallego et al. [18] used heuristic algorithms to avoid

solving an intricate model but obtained local optima. It is not trivial, however, to evalu-

ate how far this obtained optimum is away from the global optimum. We argue that for a

large-scale economic assessment like the power system planning, which may involve multiple

10

planning years and hence does not need to be solved in real time, it would be better if we

can have a guaranteed global optimum. This can be viewed as an advantage of our proposed

method compared with those intelligent methods. Besides, most of the existing studies solve

single-year planning problems [17,19,20], which do not capture certain aspects of power sys-

tem planning such as the timing of the commissioning or retirement of a generator. In our

study, however, the objective is to investigate whether an ultra-high renewable installation

in the system is beneficial or not. Since the renewable units are widely considered as units

without variable operation costs [21], the economic payback of renewable investment may

outperform the conventional units over the long term. Hence, to ensure a fair comparison

between different generation technologies, we also take the annuitized investment cost into

the consideration and intend to uncover the cost-effectiveness of renewable investment, es-

pecially under an ultra-high renewable penetration case.

In order to investigate a 100% renewable system, the correlations between renewables,

such as wind power in different regions, wind and solar, wind and demand, etc., are also

critical in system operation and planning. At any instant, the system should have enough

generation from renewables to cover the demand. However, the system planner has to take

a system-wide approach considering mutual support between regions and avoid generation

shortage at a particular location. Besides, adequate transmission capacity, even in a degraded

N-k line outage scenario, is paramount for inter-regional energy exchange to maintain the

system energy balance. The final optimal G&TEP investment ought to consider an optimal

generation mix and long-term renewable generation futures. Some researchers (e.g., [20])

argue that 100% is less profitable than a generation mix with conventional units, but this is

actually sensitive to the system settings.

In this work, we propose an effective modeling and solution approach for G&TEP to-

wards 100% penetration of renewables. The column-and-constraint generation (C&CG)

method developed by [22] can decompose our problem to a master problem and a subprob-

lem. This method has been adopted in numerous studies like [10, 12, 17], and become a

mainstream solution methodology for multi-stage RO problems. Besides, as reported in [17]

11

and [6], solving subproblems consumes the majority of the computational time. We propose

to apply the L-shaped method [23] to further facilitate the solution of subproblems, which

also enables parallel computing. In the meantime, we specifically consider the short-term

and long-term spatiotemporal correlations between renewables and load by leveraging the

Monte-Carlo sampling on regional renewables. Figure 2.1 provides a general framework of

our proposed methodology.

Compared with the state-of-the-art research, The main contributions of this work are

three-fold.

1. We analyze the coordinated system planning under an ultra-high level of renewable

penetration and investigate the potential issues of a 100% renewable penetration on

an IEEE test system and the WECC system with specific parameter settings.

2. To tackle multiplex uncertainties in the G&TEP, we propose a three-stage multi-year

robust co-optimization model with a stochastic recourse. The robust counterpart cap-

tures the N-k contingency, whereas sampling-based scenarios considering spatiotem-

poral correlations between renewables and load realize the demand/renewable uncer-

tainty.

3. The C&CG algorithm is leveraged to decompose the overall problem into a master-slave

structure, and the stochastic subproblem is further reformulated based on the dual-

ity theory and decomposed by the L-shaped method. The computational efficiency is

greatly improved, and multi-scale test cases verify the efficacy of the proposed method-

ology.

The rest of the work is organized as follows: Section 2.3 describes the mathematical

formulation of the G&TEP model; Section 2.4 presents the proposed solution strategy and

the detailed flowchart; case studies and a scalability test are analyzed in Section 2.5 and

Section 2.6, respectively; Section 2.7 summarizes this work with several remarks.

2.3. Mathematical Formulation of the Generation and Transmission Planning

12

MultiplexUncertainty-basedCoordinated Planning

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Master Problem<latexit sha1_base64="fb84m6i+NthOFBsP8RCkFUV832c=">AAACAXicbVDLSgMxFM34rPU16kZwEyxCV2WmG10W3LgRKtgHtEPJZO60oZlkSDJCGerGX3HjQhG3/oU7/8a0nYW2Hggczr3n5t4Tppxp43nfztr6xubWdmmnvLu3f3DoHh23tcwUhRaVXKpuSDRwJqBlmOHQTRWQJOTQCcfXs3rnAZRmUtybSQpBQoaCxYwSY6WBe9oXkokIhMG3RBtQuKmkNScDt+LVvDnwKvELUkEFmgP3qx9JmiV2FOVE657vpSbIiTKMcpiW+5mGlNAxGULPUkES0EE+v2CKL6wS4Vgq++wqc/W3IyeJ1pMktJ0JMSO9XJuJ/9V6mYmvgpyJNDMg6OKjOOPYSDyLA0dMATV8YgmhitldMR0RRaiNQpdtCP7yyaukXa/5Xs2/q1ca1SKOEjpD56iKfHSJGugGNVELUfSIntErenOenBfn3flYtK45hecE/YHz+QNXlZa3</latexit><latexit sha1_base64="fb84m6i+NthOFBsP8RCkFUV832c=">AAACAXicbVDLSgMxFM34rPU16kZwEyxCV2WmG10W3LgRKtgHtEPJZO60oZlkSDJCGerGX3HjQhG3/oU7/8a0nYW2Hggczr3n5t4Tppxp43nfztr6xubWdmmnvLu3f3DoHh23tcwUhRaVXKpuSDRwJqBlmOHQTRWQJOTQCcfXs3rnAZRmUtybSQpBQoaCxYwSY6WBe9oXkokIhMG3RBtQuKmkNScDt+LVvDnwKvELUkEFmgP3qx9JmiV2FOVE657vpSbIiTKMcpiW+5mGlNAxGULPUkES0EE+v2CKL6wS4Vgq++wqc/W3IyeJ1pMktJ0JMSO9XJuJ/9V6mYmvgpyJNDMg6OKjOOPYSDyLA0dMATV8YgmhitldMR0RRaiNQpdtCP7yyaukXa/5Xs2/q1ca1SKOEjpD56iKfHSJGugGNVELUfSIntErenOenBfn3flYtK45hecE/YHz+QNXlZa3</latexit><latexit sha1_base64="fb84m6i+NthOFBsP8RCkFUV832c=">AAACAXicbVDLSgMxFM34rPU16kZwEyxCV2WmG10W3LgRKtgHtEPJZO60oZlkSDJCGerGX3HjQhG3/oU7/8a0nYW2Hggczr3n5t4Tppxp43nfztr6xubWdmmnvLu3f3DoHh23tcwUhRaVXKpuSDRwJqBlmOHQTRWQJOTQCcfXs3rnAZRmUtybSQpBQoaCxYwSY6WBe9oXkokIhMG3RBtQuKmkNScDt+LVvDnwKvELUkEFmgP3qx9JmiV2FOVE657vpSbIiTKMcpiW+5mGlNAxGULPUkES0EE+v2CKL6wS4Vgq++wqc/W3IyeJ1pMktJ0JMSO9XJuJ/9V6mYmvgpyJNDMg6OKjOOPYSDyLA0dMATV8YgmhitldMR0RRaiNQpdtCP7yyaukXa/5Xs2/q1ca1SKOEjpD56iKfHSJGugGNVELUfSIntErenOenBfn3flYtK45hecE/YHz+QNXlZa3</latexit><latexit sha1_base64="fb84m6i+NthOFBsP8RCkFUV832c=">AAACAXicbVDLSgMxFM34rPU16kZwEyxCV2WmG10W3LgRKtgHtEPJZO60oZlkSDJCGerGX3HjQhG3/oU7/8a0nYW2Hggczr3n5t4Tppxp43nfztr6xubWdmmnvLu3f3DoHh23tcwUhRaVXKpuSDRwJqBlmOHQTRWQJOTQCcfXs3rnAZRmUtybSQpBQoaCxYwSY6WBe9oXkokIhMG3RBtQuKmkNScDt+LVvDnwKvELUkEFmgP3qx9JmiV2FOVE657vpSbIiTKMcpiW+5mGlNAxGULPUkES0EE+v2CKL6wS4Vgq++wqc/W3IyeJ1pMktJ0JMSO9XJuJ/9V6mYmvgpyJNDMg6OKjOOPYSDyLA0dMATV8YgmhitldMR0RRaiNQpdtCP7yyaukXa/5Xs2/q1ca1SKOEjpD56iKfHSJGugGNVELUfSIntErenOenBfn3flYtK45hecE/YHz+QNXlZa3</latexit>

Subproblem<latexit sha1_base64="K2H5mrYxZL+Y68ZeSdiSTXCKS4U=">AAAB8XicbVC7TgJBFL2LL8QXamkzkZhQkV0aKUlsLDHKI8KGzA6zMGEem5lZE7LhL2wsNMbWv7HzbxxgCwVPMsnJOedm7j1Rwpmxvv/tFba2d3b3ivulg8Oj45Py6VnHqFQT2iaKK92LsKGcSdq2zHLaSzTFIuK0G01vFn73iWrDlHyws4SGAo8lixnB1kmP92mUaOXCYliu+DV/CbRJgpxUIEdrWP4ajBRJBZWWcGxMP/ATG2ZYW0Y4nZcGqaEJJlM8pn1HJRbUhNly4zm6csoIxUq7Jy1aqr8nMiyMmYnIJQW2E7PuLcT/vH5q40aYMZmklkqy+ihOObIKLc5HI6YpsXzmCCaauV0RmWCNiXUllVwJwfrJm6RTrwV+LbirV5rVvI4iXMAlVCGAa2jCLbSgDQQkPMMrvHnGe/HevY9VtODlM+fwB97nD7/CkN0=</latexit><latexit sha1_base64="K2H5mrYxZL+Y68ZeSdiSTXCKS4U=">AAAB8XicbVC7TgJBFL2LL8QXamkzkZhQkV0aKUlsLDHKI8KGzA6zMGEem5lZE7LhL2wsNMbWv7HzbxxgCwVPMsnJOedm7j1Rwpmxvv/tFba2d3b3ivulg8Oj45Py6VnHqFQT2iaKK92LsKGcSdq2zHLaSzTFIuK0G01vFn73iWrDlHyws4SGAo8lixnB1kmP92mUaOXCYliu+DV/CbRJgpxUIEdrWP4ajBRJBZWWcGxMP/ATG2ZYW0Y4nZcGqaEJJlM8pn1HJRbUhNly4zm6csoIxUq7Jy1aqr8nMiyMmYnIJQW2E7PuLcT/vH5q40aYMZmklkqy+ihOObIKLc5HI6YpsXzmCCaauV0RmWCNiXUllVwJwfrJm6RTrwV+LbirV5rVvI4iXMAlVCGAa2jCLbSgDQQkPMMrvHnGe/HevY9VtODlM+fwB97nD7/CkN0=</latexit><latexit sha1_base64="K2H5mrYxZL+Y68ZeSdiSTXCKS4U=">AAAB8XicbVC7TgJBFL2LL8QXamkzkZhQkV0aKUlsLDHKI8KGzA6zMGEem5lZE7LhL2wsNMbWv7HzbxxgCwVPMsnJOedm7j1Rwpmxvv/tFba2d3b3ivulg8Oj45Py6VnHqFQT2iaKK92LsKGcSdq2zHLaSzTFIuK0G01vFn73iWrDlHyws4SGAo8lixnB1kmP92mUaOXCYliu+DV/CbRJgpxUIEdrWP4ajBRJBZWWcGxMP/ATG2ZYW0Y4nZcGqaEJJlM8pn1HJRbUhNly4zm6csoIxUq7Jy1aqr8nMiyMmYnIJQW2E7PuLcT/vH5q40aYMZmklkqy+ihOObIKLc5HI6YpsXzmCCaauV0RmWCNiXUllVwJwfrJm6RTrwV+LbirV5rVvI4iXMAlVCGAa2jCLbSgDQQkPMMrvHnGe/HevY9VtODlM+fwB97nD7/CkN0=</latexit><latexit sha1_base64="K2H5mrYxZL+Y68ZeSdiSTXCKS4U=">AAAB8XicbVC7TgJBFL2LL8QXamkzkZhQkV0aKUlsLDHKI8KGzA6zMGEem5lZE7LhL2wsNMbWv7HzbxxgCwVPMsnJOedm7j1Rwpmxvv/tFba2d3b3ivulg8Oj45Py6VnHqFQT2iaKK92LsKGcSdq2zHLaSzTFIuK0G01vFn73iWrDlHyws4SGAo8lixnB1kmP92mUaOXCYliu+DV/CbRJgpxUIEdrWP4ajBRJBZWWcGxMP/ATG2ZYW0Y4nZcGqaEJJlM8pn1HJRbUhNly4zm6csoIxUq7Jy1aqr8nMiyMmYnIJQW2E7PuLcT/vH5q40aYMZmklkqy+ihOObIKLc5HI6YpsXzmCCaauV0RmWCNiXUllVwJwfrJm6RTrwV+LbirV5rVvI4iXMAlVCGAa2jCLbSgDQQkPMMrvHnGe/HevY9VtODlM+fwB97nD7/CkN0=</latexit>

Sub-Master Problem<latexit sha1_base64="yPD142wCVExpE7ofwIm2llvB/5U=">AAAB+3icbVDLSgMxFL1TX7W+xrp0EyxCN5aZbnRZcONGqGgf0A4lk2ba0GQyJBmxDP0VNy4UceuPuPNvTNtZaOuBwOGce7g3J0w408bzvp3CxubW9k5xt7S3f3B45B6X21qmitAWkVyqbog15SymLcMMp91EUSxCTjvh5Hrudx6p0kzGD2aa0EDgUcwiRrCx0sAt36fhxS3WhirUVNLGxMCteDVvAbRO/JxUIEdz4H71h5KkgsaGcKx1z/cSE2RYGUY4nZX6qaYJJhM8oj1LYyyoDrLF7TN0bpUhiqSyLzZoof5OZFhoPRWhnRTYjPWqNxf/83qpia6CjMVJamhMlouilCMj0bwINGSKEsOnlmCimL0VkTFWmNgqdMmW4K9+eZ206zXfq/l39UqjmtdRhFM4gyr4cAkNuIEmtIDAEzzDK7w5M+fFeXc+lqMFJ8+cwB84nz9065P3</latexit><latexit sha1_base64="yPD142wCVExpE7ofwIm2llvB/5U=">AAAB+3icbVDLSgMxFL1TX7W+xrp0EyxCN5aZbnRZcONGqGgf0A4lk2ba0GQyJBmxDP0VNy4UceuPuPNvTNtZaOuBwOGce7g3J0w408bzvp3CxubW9k5xt7S3f3B45B6X21qmitAWkVyqbog15SymLcMMp91EUSxCTjvh5Hrudx6p0kzGD2aa0EDgUcwiRrCx0sAt36fhxS3WhirUVNLGxMCteDVvAbRO/JxUIEdz4H71h5KkgsaGcKx1z/cSE2RYGUY4nZX6qaYJJhM8oj1LYyyoDrLF7TN0bpUhiqSyLzZoof5OZFhoPRWhnRTYjPWqNxf/83qpia6CjMVJamhMlouilCMj0bwINGSKEsOnlmCimL0VkTFWmNgqdMmW4K9+eZ206zXfq/l39UqjmtdRhFM4gyr4cAkNuIEmtIDAEzzDK7w5M+fFeXc+lqMFJ8+cwB84nz9065P3</latexit><latexit sha1_base64="yPD142wCVExpE7ofwIm2llvB/5U=">AAAB+3icbVDLSgMxFL1TX7W+xrp0EyxCN5aZbnRZcONGqGgf0A4lk2ba0GQyJBmxDP0VNy4UceuPuPNvTNtZaOuBwOGce7g3J0w408bzvp3CxubW9k5xt7S3f3B45B6X21qmitAWkVyqbog15SymLcMMp91EUSxCTjvh5Hrudx6p0kzGD2aa0EDgUcwiRrCx0sAt36fhxS3WhirUVNLGxMCteDVvAbRO/JxUIEdz4H71h5KkgsaGcKx1z/cSE2RYGUY4nZX6qaYJJhM8oj1LYyyoDrLF7TN0bpUhiqSyLzZoof5OZFhoPRWhnRTYjPWqNxf/83qpia6CjMVJamhMlouilCMj0bwINGSKEsOnlmCimL0VkTFWmNgqdMmW4K9+eZ206zXfq/l39UqjmtdRhFM4gyr4cAkNuIEmtIDAEzzDK7w5M+fFeXc+lqMFJ8+cwB84nz9065P3</latexit><latexit sha1_base64="yPD142wCVExpE7ofwIm2llvB/5U=">AAAB+3icbVDLSgMxFL1TX7W+xrp0EyxCN5aZbnRZcONGqGgf0A4lk2ba0GQyJBmxDP0VNy4UceuPuPNvTNtZaOuBwOGce7g3J0w408bzvp3CxubW9k5xt7S3f3B45B6X21qmitAWkVyqbog15SymLcMMp91EUSxCTjvh5Hrudx6p0kzGD2aa0EDgUcwiRrCx0sAt36fhxS3WhirUVNLGxMCteDVvAbRO/JxUIEdz4H71h5KkgsaGcKx1z/cSE2RYGUY4nZX6qaYJJhM8oj1LYyyoDrLF7TN0bpUhiqSyLzZoof5OZFhoPRWhnRTYjPWqNxf/83qpia6CjMVJamhMlouilCMj0bwINGSKEsOnlmCimL0VkTFWmNgqdMmW4K9+eZ206zXfq/l39UqjmtdRhFM4gyr4cAkNuIEmtIDAEzzDK7w5M+fFeXc+lqMFJ8+cwB84nz9065P3</latexit>

Sub-Subproblem<latexit sha1_base64="LuCwYeWkOodmuxloYMWNGycaQys=">AAAB9XicbVC7SgNBFL3rM8ZX1NJmMAhpDLtptAzYWEY0D0jWMDuZTYbMY5mZVcKS/7CxUMTWf7Hzb5wkW2jigQuHc+5h7pwo4cxY3//21tY3Nre2CzvF3b39g8PS0XHLqFQT2iSKK92JsKGcSdq0zHLaSTTFIuK0HY2vZ377kWrDlLy3k4SGAg8lixnB1kkPd2l04SbRygVEv1T2q/4caJUEOSlDjka/9NUbKJIKKi3h2Jhu4Cc2zLC2jHA6LfZSQxNMxnhIu45KLKgJs/nVU3TulAGKlXYjLZqrvxMZFsZMROQ2BbYjs+zNxP+8bmrjqzBjMkktlWTxUJxyZBWaVYAGTFNi+cQRTDRztyIywhoT64oquhKC5S+vklatGvjV4LZWrlfyOgpwCmdQgQAuoQ430IAmENDwDK/w5j15L96797FYXfPyzAn8gff5A2taklw=</latexit><latexit sha1_base64="LuCwYeWkOodmuxloYMWNGycaQys=">AAAB9XicbVC7SgNBFL3rM8ZX1NJmMAhpDLtptAzYWEY0D0jWMDuZTYbMY5mZVcKS/7CxUMTWf7Hzb5wkW2jigQuHc+5h7pwo4cxY3//21tY3Nre2CzvF3b39g8PS0XHLqFQT2iSKK92JsKGcSdq0zHLaSTTFIuK0HY2vZ377kWrDlLy3k4SGAg8lixnB1kkPd2l04SbRygVEv1T2q/4caJUEOSlDjka/9NUbKJIKKi3h2Jhu4Cc2zLC2jHA6LfZSQxNMxnhIu45KLKgJs/nVU3TulAGKlXYjLZqrvxMZFsZMROQ2BbYjs+zNxP+8bmrjqzBjMkktlWTxUJxyZBWaVYAGTFNi+cQRTDRztyIywhoT64oquhKC5S+vklatGvjV4LZWrlfyOgpwCmdQgQAuoQ430IAmENDwDK/w5j15L96797FYXfPyzAn8gff5A2taklw=</latexit><latexit sha1_base64="LuCwYeWkOodmuxloYMWNGycaQys=">AAAB9XicbVC7SgNBFL3rM8ZX1NJmMAhpDLtptAzYWEY0D0jWMDuZTYbMY5mZVcKS/7CxUMTWf7Hzb5wkW2jigQuHc+5h7pwo4cxY3//21tY3Nre2CzvF3b39g8PS0XHLqFQT2iSKK92JsKGcSdq0zHLaSTTFIuK0HY2vZ377kWrDlLy3k4SGAg8lixnB1kkPd2l04SbRygVEv1T2q/4caJUEOSlDjka/9NUbKJIKKi3h2Jhu4Cc2zLC2jHA6LfZSQxNMxnhIu45KLKgJs/nVU3TulAGKlXYjLZqrvxMZFsZMROQ2BbYjs+zNxP+8bmrjqzBjMkktlWTxUJxyZBWaVYAGTFNi+cQRTDRztyIywhoT64oquhKC5S+vklatGvjV4LZWrlfyOgpwCmdQgQAuoQ430IAmENDwDK/w5j15L96797FYXfPyzAn8gff5A2taklw=</latexit><latexit sha1_base64="LuCwYeWkOodmuxloYMWNGycaQys=">AAAB9XicbVC7SgNBFL3rM8ZX1NJmMAhpDLtptAzYWEY0D0jWMDuZTYbMY5mZVcKS/7CxUMTWf7Hzb5wkW2jigQuHc+5h7pwo4cxY3//21tY3Nre2CzvF3b39g8PS0XHLqFQT2iSKK92JsKGcSdq0zHLaSTTFIuK0HY2vZ377kWrDlLy3k4SGAg8lixnB1kkPd2l04SbRygVEv1T2q/4caJUEOSlDjka/9NUbKJIKKi3h2Jhu4Cc2zLC2jHA6LfZSQxNMxnhIu45KLKgJs/nVU3TulAGKlXYjLZqrvxMZFsZMROQ2BbYjs+zNxP+8bmrjqzBjMkktlWTxUJxyZBWaVYAGTFNi+cQRTDRztyIywhoT64oquhKC5S+vklatGvjV4LZWrlfyOgpwCmdQgQAuoQ430IAmENDwDK/w5j15L96797FYXfPyzAn8gff5A2taklw=</latexit>

Binary Programming<latexit sha1_base64="enz1UGg5nifvlSoqcCGNCBMm+fw=">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</latexit><latexit sha1_base64="enz1UGg5nifvlSoqcCGNCBMm+fw=">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</latexit><latexit sha1_base64="enz1UGg5nifvlSoqcCGNCBMm+fw=">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</latexit><latexit sha1_base64="enz1UGg5nifvlSoqcCGNCBMm+fw=">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</latexit>

Binary Programming<latexit sha1_base64="q0rPf1rh8IhzqOhV+AluN6XwehA=">AAACDHicdVDLSgMxFM3UV62vqks3wSJ0NcyMpa27ohuXFWwttKVk0kwbmseQZIQy9APc+CtuXCji1g9w59+YaSuo6IHA4dxzb+49YcyoNp734eRWVtfWN/Kbha3tnd294v5BW8tEYdLCkknVCZEmjArSMtQw0okVQTxk5CacXGT1m1uiNJXi2kxj0udoJGhEMTJWGhRLPSGpGBJhYI/weJyeU4HUFDaVHCnEORWjmXV57lm9GlSq0HM9r+YHfkaCWuW0An2rZCiBJZqD4ntvKHHC7VTMkNZd34tNP0XKUMzIrNBLNIkRnqAR6VoqECe6n86PmcETqwxhJJV9dqu5+r0jRVzrKQ+tkyMz1r9rmfhXrZuYqN5PqYgTQwRefBQlDBoJs2TgkCqCDZtagrCidleIx0ghbGx+BRvC16Xwf9IOXN9z/aug1Cgv48iDI3AMysAHNdAAl6AJWgCDO/AAnsCzc+88Oi/O68Kac5Y9h+AHnLdP4GacEA==</latexit><latexit sha1_base64="q0rPf1rh8IhzqOhV+AluN6XwehA=">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</latexit><latexit sha1_base64="q0rPf1rh8IhzqOhV+AluN6XwehA=">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</latexit><latexit sha1_base64="q0rPf1rh8IhzqOhV+AluN6XwehA=">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</latexit>

Relaxed StochasticLinear Programming

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Mixed-integer StochasticNonlinear Programming

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Mixed-integer StochasticNonlinear Programming

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Investment Decisions<latexit sha1_base64="oD3NlM7q4QgCrmp3QnvwDGrBxbM=">AAACJnicbZBNSwJBGMdnezV72+rYZUkCDyEzHlIPgVCHuhnkC+gis+OjDs6+MDMryOKn6dJX6dLBiOjWR2lWPZT2wMCP//M6fy8SXGmMv6yNza3tnd3MXnb/4PDo2D45bagwlgzqLBShbHlUgeAB1DXXAlqRBOp7Apre6DbNN8cgFQ+DJz2JwPXpIOB9zqg2Ute+STrzIW058NwEFzDGhJCrFEjpGhuoVMpFUp4+BGNQ2odAO3fAeDpQTbt2bt5iwlkHsoQcWkata886vZDF6RgmqFJtgiPtJlRqzgRMs51YQUTZiA6gbTCgPig3mV84dS6N0nP6oTTPnDFXf3ck1Fdq4num0qd6qFZzqfhfrh3rftlNeBDFGgK2WNSPhaNDJ/XM6XEJTIuJAcokN7c6bEglZdo4mzUmkNUvr0OjWCC4QB6LuWp+aUcGnaMLlEcElVAV3aMaqiOGntErmqF368V6sz6sz0XphrXsOUN/wvr+AdI8okU=</latexit><latexit sha1_base64="oD3NlM7q4QgCrmp3QnvwDGrBxbM=">AAACJnicbZBNSwJBGMdnezV72+rYZUkCDyEzHlIPgVCHuhnkC+gis+OjDs6+MDMryOKn6dJX6dLBiOjWR2lWPZT2wMCP//M6fy8SXGmMv6yNza3tnd3MXnb/4PDo2D45bagwlgzqLBShbHlUgeAB1DXXAlqRBOp7Apre6DbNN8cgFQ+DJz2JwPXpIOB9zqg2Ute+STrzIW058NwEFzDGhJCrFEjpGhuoVMpFUp4+BGNQ2odAO3fAeDpQTbt2bt5iwlkHsoQcWkata886vZDF6RgmqFJtgiPtJlRqzgRMs51YQUTZiA6gbTCgPig3mV84dS6N0nP6oTTPnDFXf3ck1Fdq4num0qd6qFZzqfhfrh3rftlNeBDFGgK2WNSPhaNDJ/XM6XEJTIuJAcokN7c6bEglZdo4mzUmkNUvr0OjWCC4QB6LuWp+aUcGnaMLlEcElVAV3aMaqiOGntErmqF368V6sz6sz0XphrXsOUN/wvr+AdI8okU=</latexit><latexit sha1_base64="oD3NlM7q4QgCrmp3QnvwDGrBxbM=">AAACJnicbZBNSwJBGMdnezV72+rYZUkCDyEzHlIPgVCHuhnkC+gis+OjDs6+MDMryOKn6dJX6dLBiOjWR2lWPZT2wMCP//M6fy8SXGmMv6yNza3tnd3MXnb/4PDo2D45bagwlgzqLBShbHlUgeAB1DXXAlqRBOp7Apre6DbNN8cgFQ+DJz2JwPXpIOB9zqg2Ute+STrzIW058NwEFzDGhJCrFEjpGhuoVMpFUp4+BGNQ2odAO3fAeDpQTbt2bt5iwlkHsoQcWkata886vZDF6RgmqFJtgiPtJlRqzgRMs51YQUTZiA6gbTCgPig3mV84dS6N0nP6oTTPnDFXf3ck1Fdq4num0qd6qFZzqfhfrh3rftlNeBDFGgK2WNSPhaNDJ/XM6XEJTIuJAcokN7c6bEglZdo4mzUmkNUvr0OjWCC4QB6LuWp+aUcGnaMLlEcElVAV3aMaqiOGntErmqF368V6sz6sz0XphrXsOUN/wvr+AdI8okU=</latexit><latexit sha1_base64="oD3NlM7q4QgCrmp3QnvwDGrBxbM=">AAACJnicbZBNSwJBGMdnezV72+rYZUkCDyEzHlIPgVCHuhnkC+gis+OjDs6+MDMryOKn6dJX6dLBiOjWR2lWPZT2wMCP//M6fy8SXGmMv6yNza3tnd3MXnb/4PDo2D45bagwlgzqLBShbHlUgeAB1DXXAlqRBOp7Apre6DbNN8cgFQ+DJz2JwPXpIOB9zqg2Ute+STrzIW058NwEFzDGhJCrFEjpGhuoVMpFUp4+BGNQ2odAO3fAeDpQTbt2bt5iwlkHsoQcWkata886vZDF6RgmqFJtgiPtJlRqzgRMs51YQUTZiA6gbTCgPig3mV84dS6N0nP6oTTPnDFXf3ck1Fdq4num0qd6qFZzqfhfrh3rftlNeBDFGgK2WNSPhaNDJ/XM6XEJTIuJAcokN7c6bEglZdo4mzUmkNUvr0OjWCC4QB6LuWp+aUcGnaMLlEcElVAV3aMaqiOGntErmqF368V6sz6sz0XphrXsOUN/wvr+AdI8okU=</latexit>

Worst-case Contingency<latexit sha1_base64="c1cGvZ/SKmGoY8rkZsvDxypBIFE=">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</latexit><latexit sha1_base64="c1cGvZ/SKmGoY8rkZsvDxypBIFE=">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</latexit><latexit sha1_base64="c1cGvZ/SKmGoY8rkZsvDxypBIFE=">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</latexit><latexit sha1_base64="c1cGvZ/SKmGoY8rkZsvDxypBIFE=">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</latexit>

Operation Costs<latexit sha1_base64="HD5aKOufSRweeTOPeStd89X144k=">AAACInicbZDLSgMxFIYz9VbrrerSTbAIXUhJurDtrtCNOyvYVpgOJZNm2tDMhSQjlKHP4sZXceNCUVeCD2NmOgtt/SHw8Z9zkpPfjQRXGqEvq7CxubW9U9wt7e0fHB6Vj0/6KowlZT0ailDeu0QxwQPW01wLdh9JRnxXsIE766T1wQOTiofBnZ5HzPHJJOAep0Qba1RuJcPsEltOXCdBNYQQxvgyBdy4QgZarWYdNxfwJmIyG4KdUGm1GJUrWbsRXAecQwXk6o7KH8NxSGOfBZoKopSNUaSdhEjNqWCL0jBWLCJ0RibMNhgQnyknybZbwAvjjKEXSnMCDTP390RCfKXmvms6faKnarWWmv/V7Fh7TSfhQRRrFtDlQ14soA5hmhccc8moFnMDhEpudoV0SiSh2qRaMiHg1S+vQ79ew6iGb+uVdjWPowjOwDmoAgwaoA2uQRf0AAWP4Bm8gjfryXqx3q3PZWvBymdOwR9Z3z/9S6As</latexit><latexit sha1_base64="HD5aKOufSRweeTOPeStd89X144k=">AAACInicbZDLSgMxFIYz9VbrrerSTbAIXUhJurDtrtCNOyvYVpgOJZNm2tDMhSQjlKHP4sZXceNCUVeCD2NmOgtt/SHw8Z9zkpPfjQRXGqEvq7CxubW9U9wt7e0fHB6Vj0/6KowlZT0ailDeu0QxwQPW01wLdh9JRnxXsIE766T1wQOTiofBnZ5HzPHJJOAep0Qba1RuJcPsEltOXCdBNYQQxvgyBdy4QgZarWYdNxfwJmIyG4KdUGm1GJUrWbsRXAecQwXk6o7KH8NxSGOfBZoKopSNUaSdhEjNqWCL0jBWLCJ0RibMNhgQnyknybZbwAvjjKEXSnMCDTP390RCfKXmvms6faKnarWWmv/V7Fh7TSfhQRRrFtDlQ14soA5hmhccc8moFnMDhEpudoV0SiSh2qRaMiHg1S+vQ79ew6iGb+uVdjWPowjOwDmoAgwaoA2uQRf0AAWP4Bm8gjfryXqx3q3PZWvBymdOwR9Z3z/9S6As</latexit><latexit sha1_base64="HD5aKOufSRweeTOPeStd89X144k=">AAACInicbZDLSgMxFIYz9VbrrerSTbAIXUhJurDtrtCNOyvYVpgOJZNm2tDMhSQjlKHP4sZXceNCUVeCD2NmOgtt/SHw8Z9zkpPfjQRXGqEvq7CxubW9U9wt7e0fHB6Vj0/6KowlZT0ailDeu0QxwQPW01wLdh9JRnxXsIE766T1wQOTiofBnZ5HzPHJJOAep0Qba1RuJcPsEltOXCdBNYQQxvgyBdy4QgZarWYdNxfwJmIyG4KdUGm1GJUrWbsRXAecQwXk6o7KH8NxSGOfBZoKopSNUaSdhEjNqWCL0jBWLCJ0RibMNhgQnyknybZbwAvjjKEXSnMCDTP390RCfKXmvms6faKnarWWmv/V7Fh7TSfhQRRrFtDlQ14soA5hmhccc8moFnMDhEpudoV0SiSh2qRaMiHg1S+vQ79ew6iGb+uVdjWPowjOwDmoAgwaoA2uQRf0AAWP4Bm8gjfryXqx3q3PZWvBymdOwR9Z3z/9S6As</latexit><latexit sha1_base64="HD5aKOufSRweeTOPeStd89X144k=">AAACInicbZDLSgMxFIYz9VbrrerSTbAIXUhJurDtrtCNOyvYVpgOJZNm2tDMhSQjlKHP4sZXceNCUVeCD2NmOgtt/SHw8Z9zkpPfjQRXGqEvq7CxubW9U9wt7e0fHB6Vj0/6KowlZT0ailDeu0QxwQPW01wLdh9JRnxXsIE766T1wQOTiofBnZ5HzPHJJOAep0Qba1RuJcPsEltOXCdBNYQQxvgyBdy4QgZarWYdNxfwJmIyG4KdUGm1GJUrWbsRXAecQwXk6o7KH8NxSGOfBZoKopSNUaSdhEjNqWCL0jBWLCJ0RibMNhgQnyknybZbwAvjjKEXSnMCDTP390RCfKXmvms6faKnarWWmv/V7Fh7TSfhQRRrFtDlQ14soA5hmhccc8moFnMDhEpudoV0SiSh2qRaMiHg1S+vQ79ew6iGb+uVdjWPowjOwDmoAgwaoA2uQRf0AAWP4Bm8gjfryXqx3q3PZWvBymdOwR9Z3z/9S6As</latexit>

C&CGAlgorithm

<latexit sha1_base64="uwYwfeCfW/WOxsDJHdsAajzQ9Qg=">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</latexit><latexit sha1_base64="uwYwfeCfW/WOxsDJHdsAajzQ9Qg=">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</latexit><latexit sha1_base64="uwYwfeCfW/WOxsDJHdsAajzQ9Qg=">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</latexit><latexit sha1_base64="uwYwfeCfW/WOxsDJHdsAajzQ9Qg=">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</latexit>

L-shapedMethod

<latexit sha1_base64="oR3NrcV6a8PFlDmdGGv4Drepv6s=">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</latexit><latexit sha1_base64="oR3NrcV6a8PFlDmdGGv4Drepv6s=">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</latexit><latexit sha1_base64="oR3NrcV6a8PFlDmdGGv4Drepv6s=">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</latexit><latexit sha1_base64="oR3NrcV6a8PFlDmdGGv4Drepv6s=">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</latexit>

Figure 2.1. Framework of the proposed methodology.

We provide the detailed mathematical formulation of the multiplex uncertainty-based

G&TEP in this section. First, we show the deterministic equivalent form (DEF) of the

three-stage formulation of the hybrid SP and RO model in (2-1). The term “stage” we used

here refers to the mathematical structure of the proposed optimization framework.

minxGg,t,x

Rg,t,

xL`,t

T

∑t

⎛⎜⎝

XG

∑g

ICGg,tx

Gg,t +

XR

∑g

ICRr,tx

Rr,t +

XL

∑`

ICL`,tx

L`,t

⎞⎟⎠

+ maxaGg,t,a

L`,t

Eω⎧⎪⎪⎪⎨⎪⎪⎪⎩

minpGg,h,rd,h

H

∑h

[G

∑g

OCg,hpGg,h(ω)+

D

∑d

PCd,hrd,h(ω)]

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(2-1)

subject to

T

∑t

XG

∑g

ICGg,tx

Gg,t +

XR

∑g

ICRr,tx

Rr,t ≤ BG, (2-1a)

13

T

∑t

XL

∑`

ICL`,tx

L`,t ≤ BL, (2-1b)

∀∗ ∈ g, r, `, ∀(∗, t) ∈ N(∗, t) ∶

T

∑t′≠t

x∗,t′ = 0, (2-1c)

T

∑t′≥t

y∗,t′ = (T − t + 1) ⋅ x∗,t, (2-1d)

∑t′<t

y∗,t′ = 0, (2-1e)

G

∑g

aGg,t +R

∑r

aRr,t +L

∑`

aL`,t ≤Kt, ∀t, (2-1f)

∀ω ∶

G

∑g∣C(g)=n

pGg,h(ω) +R

∑r∣C(r)=n

pRr,h(ω) −L

∑`∣S(`)=n

f`,h(ω)+

L

∑`∣V (`)=n

f`,h(ω) =D

∑d∣C(d)=n

Pd,h(ω) − rd,h(ω) ,∀n,∀h, (2-1g)

f`,h(ω) = yL`,t(1 − a

L`,t)A

−1` ⋅ [δn∣S(`)=n,h(ω) − δn∣V (`)=n,h(ω)],

∀` ∈ L,∀(t, h) ∈M, (2-1h)

− yL`,t(1 − aL`,t)FL` ≤ f`,h(ω),∀` ∈X

L,∀(t, h) ∈M, (2-1i)

f`,h(ω) ≤ yL`,t(1 − a

L`,t)FL`,∀` ∈X

L,∀(t, h) ∈M, (2-1j)

− (1 − aL`,t)FL` ≤ f`,h(ω),∀` ∈ L/XL,∀(t, h) ∈M, (2-1k)

f`,h(ω) ≤ (1 − aL`,t)FL`,∀` ∈ L/XL,∀(t, h) ∈M, (2-1l)

pGg,h(ω) ≤ yGg,t(1 − a

Gg,t)PL

G

g ,∀g ∈XG,∀(t, h) ∈M, (2-1m)

pGg,h(ω) ≤ (1 − aGg,t)PLG

g ,∀g ∈ G/XG,∀(t, h) ∈M, (2-1n)

pRr,h(ω) ≤ yRr,t(1 − a

Rr,t)PL

R

r,h(ω),∀r ∈XR,∀(t, h) ∈M, (2-1o)

pRr,h(ω) ≤ (1 − aRr,t)PLR

r,h(ω),∀r ∈ R/XR,∀(t, h) ∈M, (2-1p)

∆n ≤ δn,h(ω) ≤ ∆n,∀n,∀h, (2-1q)

The objective function (2-1) formulates the investment cost of the proposed generator

and transmission line expansions, plus the operating cost of scheduled conventional genera-

14

tors and possible load shedding.

Constraints (2-1a)-(2-1b) model the investment budget of both generators and transmis-

sion lines, which construct the first-stage feasible region. Constraints (2-1c)-(2-1e) represent

that once an investment is made in year t, the component will be available for the rest of the

planning horizon. Control sets N(∗, t) represent the mapping between the candidate units

and planning time, in which we categorize the candidate units based on the investable years.

These constraints formulate the constraint space for the first stage. Constraint (2-1f) shows

the N-k criteria for both generators and transmission lines, where Kt can be adjusted to per-

form different contingency analyses. This constraint is in the second-stage (the second-stage

objective function can be regarded as 0⊺a, in which a is the vector of the outage indica-

tors) formulating the uncertainty set [24]. This uncertainty set is discretely polyhedral.

For the third-stage, we first explain the relationship between t and h. Since we con-

sider a long-term planning problem, traversing 8760 hours for a whole year renders heavy

intractability in this model. Hence, we consider a 24-hour operation for the third stage as an

analysis of hourly dispatch in one typical day. This operation is also adopted in [20]. Then

the time index mapping set M can be described as

M = (t1, h1), ..., (t1, h24), (t2, h25), ...(t2, h48), ....

We also note that this design further generalizes the application of selecting representative

hours/days. When a system planner intends to select more representative hours to perform

the G&TEP study, it is trivial to adjust the set M by increasing the number of hours. This

also shows that our proposed framework is highly flexible and general to be expanded to the

extent the system planner would like to use in G&TEP problems.

Notably, the outage indicator in our formulation is a day-based variable. As we indicated,

the third-stage problem can be regarded as a 24-hour economic dispatch problem. For such

operation problems, the contingency is often considered throughout the operation horizon,

i.e., 24 hours [25]. Practically, 94% of the planned and operational outage have a duration

15

of over 2 hours and 34.6% of them are over 48 hours [26]. In some works of system planning,

the horizon-long contingency is also adopted [11]. For the sake of simplicity, since we choose

one representative day for one year with multiple uncertain scenarios, we slightly abuse our

notation and use a year-based index to represent a day-based variable, e.g., aGg,t. Besides,

for the third-stage operation problem, we choose using one representative day with multiple

scenarios to investigate the short-term correlation between regional renewables and load,

which can be better captured by hourly operations, as also adopted in [20].

Particularly, constraint (2-1g) is the nodal balance constraint, and constraint (2-1h)

defines the DC line flow equations. Note that here, we slightly abuse the notation that the per

unit reactance A` should be normalized under the system base MVA. Constraints (2-1i)-(2-1l)

are line flow limitation constraints for both candidate and existing lines. Constraints (2-1m)-

(2-1p) are the generation capacity constraints for both candidate and existing conventional

and renewable generators. Note that the renewable generators can be dispatched, and thus

we permit the renewable curtailment. Constraint (2-1q) shows the phase angle limitation.

We can find that all of the third-stage variables are associated with the scenario index ω for

different realizations of renewable generation and load. Note that the binary variables and

the variable multiplications such as in (2-1h), (2-1i), (2-1j), (2-1m) and (2-1o) render this

model mixed-integer nonlinear.

The proposed model (2-1) is a very general framework for the hybrid stochastic and

robust optimization-based system planning and can be easily adjusted or expanded to include

more constraints regarding various research directions or industrial applications. In our case

studies, for example, we do not have any existing generator and we can hence disregard

constraints (2-1n) and (2-1p) in the formulation. We also modify (2-1) to incorporate unique

case settings such as the regional N-1 contingency and multiple investments in one candidate

bus, which can be achieved by replacing constraint (2-1f) with (1-1r) and adding constraint

16

(1-1s) respectively as shown below.

Gz

∑g

aGg,t +Rz

∑r

aRr,t +Lz

∑`

aL`,t ≤Kt, ∀t,∀z ∈ Z, (1-1r)

G

∑g∣C(g)=ncg

xGg,t +R

∑r∣C(r)=ncr

xRr,t ≤ 2, ∀t,∀n, (1-1s)

where z denotes the region indicator, Z is the total region set, ncg denotes the candidate bus

for thermal generators, and ncr denotes the candidate bus for renewable generators.

2.4. Problem Decomposition and Solution Strategy

The model described in Section 2.3 presents an intractable three-stage mixed-integer non-

linear problem with stochastic recourse. However, we can decompose the original problem

into a structure of mixed-integer linear programming (MILP) master problem and mixed-

integer bilinear subproblem, which can be solved iteratively based on the theory of the C&CG

algorithm [22]. Linear relaxation techniques can tackle the nonlinearity of the subproblem.

Besides, the subproblem can be further decomposed by the L-shaped method [23].

Figure 2.2 depicts the overall solution workflow. Generally, we adopt the C&CG algo-

rithm to decompose the problem into a master-slave structure, and the L-shaped algorithm

then further decomposes the subproblem by different scenarios. We will explain the proce-

dure and notations of Figure 2.2 in the problem formulation. To be more concise, we use

compact form below.

2.4.1. Master Problem

We formulate the master problem in the C&CG procedure as in (2-2). x denotes the

first-stage variables, z denotes the third-stage variables with iteration index k′, which en-

closes all the past iterations before the current iteration k. IC and PC represent the

vectors of investment costs and penalty costs, whereas pG and r denote the vectors of gen-

erator dispatch and load shedding. φ is an auxiliary variable that formulates a relaxed lower

bound of the second-stage problem. The constraints for third-stage variables are (2-2d).

17

Initialization

k k + 1

o o + 1

k 0o 0

End

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NO<latexit sha1_base64="7viRbuBd4JGrayd3CYPkdXEypP4=">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</latexit><latexit sha1_base64="7viRbuBd4JGrayd3CYPkdXEypP4=">AAACG3icdVBLSwMxGMz6rPVV9eglWBQPUrLrtl1vBS+etIJ9QLuUbJq2odkHSVYoy/4PL/4VLx4U8SR48N+YbSuo6EBgmJkv+TJexJlUCH0YC4tLyyurubX8+sbm1nZhZ7cpw1gQ2iAhD0Xbw5JyFtCGYorTdiQo9j1OW974PPNbt1RIFgY3ahJR18fDgA0YwUpLvYKVdKeXdMTQcxNUQg5yLOdEkyk0OSsjZJXTrhfyfnJ5laa9QlGrTsWyKzDLVU3LzIhVtU9taM4ni2COeq/w1u2HJPZpoAjHUnZMFCk3wUIxwmma78aSRpiM8ZB2NA2wT6WbTNdK4aFW+nAQCn0CBafq94kE+1JOfE8nfaxG8reXiX95nVgNHDdhQRQrGpDZQ4OYQxXCrCjYZ4ISxSeaYCKY3hWSERaYKF1nXpfw9VP4P2laJROVzGu7WDua15ED++AAHAMTVEENXIA6aAAC7sADeALPxr3xaLwYr7PogjGf2QM/YLx/Ajf7nbY=</latexit><latexit sha1_base64="7viRbuBd4JGrayd3CYPkdXEypP4=">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</latexit><latexit sha1_base64="7viRbuBd4JGrayd3CYPkdXEypP4=">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</latexit>

YES<latexit sha1_base64="7tRwvyNw+f6EwH6HNJITMq+WVD4=">AAACHHicdVBLSwMxGMz6rPVV9eglWBQPUrLb13oriOCxon1Iu5Rsmrah2QdJVijL/hAv/hUvHhTx4kHw35htK6joQGCYmS/5Mm7ImVQIfRgLi0vLK6uZtez6xubWdm5ntymDSBDaIAEPRNvFknLm04ZiitN2KCj2XE5b7vgs9Vu3VEgW+NdqElLHw0OfDRjBSku9XDHuTi/piKHrxKiAbGRb9okmU2hyWkbIKiddN+D9+Ob8Kkl6ubyW7YpVqsA0WDUtMyVWtVQsQXM+mgdz1Hu5t24/IJFHfUU4lrJjolA5MRaKEU6TbDeSNMRkjIe0o6mPPSqdeLpXAg+10oeDQOjjKzhVv0/E2JNy4rk66WE1kr+9VPzL60RqYDsx88NIUZ/MHhpEHKoApk3BPhOUKD7RBBPB9K6QjLDAROk+s7qEr5/C/0nTKpioYF6W8rWjeR0ZsA8OwDEwQRXUwAWogwYg4A48gCfwbNwbj8aL8TqLLhjzmT3wA8b7J+tbnhQ=</latexit><latexit sha1_base64="7tRwvyNw+f6EwH6HNJITMq+WVD4=">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</latexit><latexit sha1_base64="7tRwvyNw+f6EwH6HNJITMq+WVD4=">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</latexit><latexit sha1_base64="7tRwvyNw+f6EwH6HNJITMq+WVD4=">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</latexit>

NO<latexit sha1_base64="7viRbuBd4JGrayd3CYPkdXEypP4=">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</latexit><latexit sha1_base64="7viRbuBd4JGrayd3CYPkdXEypP4=">AAACG3icdVBLSwMxGMz6rPVV9eglWBQPUrLrtl1vBS+etIJ9QLuUbJq2odkHSVYoy/4PL/4VLx4U8SR48N+YbSuo6EBgmJkv+TJexJlUCH0YC4tLyyurubX8+sbm1nZhZ7cpw1gQ2iAhD0Xbw5JyFtCGYorTdiQo9j1OW974PPNbt1RIFgY3ahJR18fDgA0YwUpLvYKVdKeXdMTQcxNUQg5yLOdEkyk0OSsjZJXTrhfyfnJ5laa9QlGrTsWyKzDLVU3LzIhVtU9taM4ni2COeq/w1u2HJPZpoAjHUnZMFCk3wUIxwmma78aSRpiM8ZB2NA2wT6WbTNdK4aFW+nAQCn0CBafq94kE+1JOfE8nfaxG8reXiX95nVgNHDdhQRQrGpDZQ4OYQxXCrCjYZ4ISxSeaYCKY3hWSERaYKF1nXpfw9VP4P2laJROVzGu7WDua15ED++AAHAMTVEENXIA6aAAC7sADeALPxr3xaLwYr7PogjGf2QM/YLx/Ajf7nbY=</latexit><latexit sha1_base64="7viRbuBd4JGrayd3CYPkdXEypP4=">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</latexit><latexit sha1_base64="7viRbuBd4JGrayd3CYPkdXEypP4=">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</latexit>

Solve MP(k)

obtain x(k) and lowerbound LB(k) = MP(k)

UB(k) = minUB0(k), UB(k 1)

C&CG

L-shaped

Calculate ↵(k, o) =P!

Prob(!)(k, o,!)>r(!)<latexit sha1_base64="KxRSEBWmKv9XWG7bOnwGNlUj/0I=">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</latexit><latexit sha1_base64="KxRSEBWmKv9XWG7bOnwGNlUj/0I=">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</latexit><latexit sha1_base64="KxRSEBWmKv9XWG7bOnwGNlUj/0I=">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</latexit><latexit sha1_base64="KxRSEBWmKv9XWG7bOnwGNlUj/0I=">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</latexit>

Obtain dual solutions (k, o,!)<latexit sha1_base64="Om9CctetcE45r+IExxfeoTA7vpE=">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</latexit><latexit sha1_base64="Om9CctetcE45r+IExxfeoTA7vpE=">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</latexit><latexit sha1_base64="Om9CctetcE45r+IExxfeoTA7vpE=">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</latexit><latexit sha1_base64="Om9CctetcE45r+IExxfeoTA7vpE=">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</latexit>

(k, o) = P!

Prob(!)C(!)>(k, o,!)<latexit sha1_base64="znHIfyH/hpUVS0MP1T9lcsROd3g=">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</latexit><latexit sha1_base64="znHIfyH/hpUVS0MP1T9lcsROd3g=">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</latexit><latexit sha1_base64="znHIfyH/hpUVS0MP1T9lcsROd3g=">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</latexit><latexit sha1_base64="znHIfyH/hpUVS0MP1T9lcsROd3g=">AAACXHicbVFBSyMxFM7Mdndru+7WFfbiZbAuVOiWGS96EQQvHitYLTS1ZNLXGppMhuSNWIb5k968+Fc0nQ6LVh+EfPne915evsSpFBbD8Mnzv9S+fvte32o0f2z//NXa+X1tdWY4DLiW2gxjZkGKBAYoUMIwNcBULOEmXpyv8jf3YKzQyRUuUxgrNk/ETHCGjpq07AGNtZzapXJbTmNAVnQWXX14+o/aTFEplEA7yalWMGcFRXjAvG90XHTW1GFZn5//P99S1Om7pg+ibNmtBAeTVjvshWUEH0FUgTapoj9pPdKp5pmCBLlk1o6iMMVxzgwKLqFo0MxCyviCzWHkYMIU2HFemlMEfx0zDWbauJVgULJvK3Km7GpSp1QM7+xmbkV+lhtlODsZ5yJJM4SEry+aZTJAHaycDqbCAEe5dIBxI9ysAb9jhnF0/9FwJkSbT/4Iro96UdiLLo/aZ93KjjrZI/ukQyJyTM7IBemTAeHkibx4dW/Le/ZrftPfXkt9r6rZJe/C//MKFN+3Cw==</latexit>

Add one constraint↵(k, o) + (k, o)>a

<latexit sha1_base64="uvizuizWyPppTdScNGQcFQ6gvrw=">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</latexit><latexit sha1_base64="uvizuizWyPppTdScNGQcFQ6gvrw=">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</latexit><latexit sha1_base64="uvizuizWyPppTdScNGQcFQ6gvrw=">AAACbHicdVBdb9MwFHUyPkb5WBk8gCYki45pE1PlhG4Nb0O88Dgkuk2qQ3XjuK1Vxw62g1RFeeIf8sZP4IXfgJMVCRBcyfLROffj3JuVUlhHyLcg3Lpx89bt7Tu9u/fuP9jpP9y9sLoyjE+YltpcZWC5FIpPnHCSX5WGQ5FJfpmt3rb65WdurNDqg1uXPC1gocRcMHCemvW/UMaV40aoRU27dlOzyNKaDElCkjg59qALD16fEBKfNPhNnmOtOGZaWWdAKEdpb5+CLJdwuDrWRy9ppmVu14X/appxB03Hf6ROl51WQ0Ml/0S9tN/M+gPfPDmNR6e4HTeO4qgF8Xj0aoSjjYEB2sT5rP+V5ppVhXfOJFg7jUjp0hqME0zypkcry0tgK1jwqYcKCm7TutuuwS88k+O5Nv4phzv294oaCtta95kFuKX9W2vJf2nTys2TtBaqrBxX7HrQvJLYadxeHufCcObk2gNgRnivmC3BAPPntz1/hF+b4v+Di3gYkWH0Ph6cHWzOsY320HN0iCI0RmfoHTpHE8TQ92AneBI8DX6Ej8O98Nl1ahhsah6hPyI8+AnxiLe9</latexit><latexit sha1_base64="uvizuizWyPppTdScNGQcFQ6gvrw=">AAACbHicdVBdb9MwFHUyPkb5WBk8gCYki45pE1PlhG4Nb0O88Dgkuk2qQ3XjuK1Vxw62g1RFeeIf8sZP4IXfgJMVCRBcyfLROffj3JuVUlhHyLcg3Lpx89bt7Tu9u/fuP9jpP9y9sLoyjE+YltpcZWC5FIpPnHCSX5WGQ5FJfpmt3rb65WdurNDqg1uXPC1gocRcMHCemvW/UMaV40aoRU27dlOzyNKaDElCkjg59qALD16fEBKfNPhNnmOtOGZaWWdAKEdpb5+CLJdwuDrWRy9ppmVu14X/appxB03Hf6ROl51WQ0Ml/0S9tN/M+gPfPDmNR6e4HTeO4qgF8Xj0aoSjjYEB2sT5rP+V5ppVhXfOJFg7jUjp0hqME0zypkcry0tgK1jwqYcKCm7TutuuwS88k+O5Nv4phzv294oaCtta95kFuKX9W2vJf2nTys2TtBaqrBxX7HrQvJLYadxeHufCcObk2gNgRnivmC3BAPPntz1/hF+b4v+Di3gYkWH0Ph6cHWzOsY320HN0iCI0RmfoHTpHE8TQ92AneBI8DX6Ej8O98Nl1ahhsah6hPyI8+AnxiLe9</latexit>

Solve Sub M(k, o)

Solve Sub S(k, o,!)

Calculate the upper boundUB0(k) = u(k) + f(x(k)),

u(k, o) 0

Update and delivera(k, o)

<latexit sha1_base64="t+T39a81XXKFuNDpC3nRiJS1HvA=">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</latexit><latexit sha1_base64="t+T39a81XXKFuNDpC3nRiJS1HvA=">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</latexit><latexit sha1_base64="t+T39a81XXKFuNDpC3nRiJS1HvA=">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</latexit><latexit sha1_base64="t+T39a81XXKFuNDpC3nRiJS1HvA=">AAACRHicdVBNb9QwEHXKV1m+lnLkYrEFFVStnLDtprdKXDgWiW0rJWE1cWa31jp2ZDuVVlF+HJf+gN74Bb1wACGuCGe7SIDgSZae3sy8Gb+8ksI6xj4FGzdu3rp9Z/Nu7979Bw8f9R9vHVtdG44TrqU2pzlYlELhxAkn8bQyCGUu8SRfvOnqJ+dorNDqvVtWmJUwV2ImODgvTftJylE5NELNaZOu/BIzz7OGDVnM4ije9WQFTw72GIv22klVgEMKqqCFX+zt07RHt9Ncy6KB9sOrncWufrndTvsDPxPvR6N92rmMwyjsSDQevR7RcO07IGscTfuXaaF5XfqLuARrk5BVLmvAOMEltr20tlgBX8AcE08VlGizZnV0S597paAzbfxTjq7U3ycaKK1dlrnvLMGd2b9rnfivWlK7WZw1QlW1Q8WvF81qSZ2mXaK0EAa5k0tPgBvhb6X8DAxwH6vt+RB+/ZT+nxxHw5ANw3fR4PDFOo5N8pQ8IzskJGNySN6SIzIhnHwkV+QL+RpcBJ+Db8H369aNYD3zhPyB4MdP0jCtFw==</latexit>

Add one more seriesof constraints (2c)-(2d)

<latexit sha1_base64="/udw9s23KSwJLo9QXqxPb0kVBoU=">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</latexit><latexit sha1_base64="/udw9s23KSwJLo9QXqxPb0kVBoU=">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</latexit><latexit sha1_base64="/udw9s23KSwJLo9QXqxPb0kVBoU=">AAACTHicdVBNixNBEO2Juq7xK65HL4VByYKGnja7GW8rXjyuYHYXMkPo6anJNtvTPXT3LIQhP9CLB2/+Ci8eFBHsyUZQ0YKGx6t6r6pfXivpPKWfot616zd2bu7e6t++c/fe/cGDvRNnGitwJowy9iznDpXUOPPSKzyrLfIqV3iaX7zu+qeXaJ00+p1f1ZhVfKllKQX3gVoMRCpQe7RSL6FNN35zu8yzlo5pQhOWPAtgUwG8PKCUHaxfFQUYjVAZi+CCFl2aQt+UIIx23nKpvYMRE/vPR6zYXy8Gw6BNDtnkEDq3acziDrDp5MUE4q3/kGzreDH4mBZGNFW4TCju3Dymtc9abr0UCtf9tHFYc3HBlzgPUPMKXdZujl/Dk8AUUBobnvawYX9XtLxyblXlYbLi/tz93evIf/XmjS+TrJW6bjxqcbWobBR4A12yUEiLwqtVAFxYGW4Fcc4tFyFe1w8h/Pop/B+csHFMx/FbNjx6uo1jlzwij8mIxGRKjsgbckxmRJD35DP5Sr5FH6Iv0ffox9VoL9pqHpI/qrfzE6FJrto=</latexit><latexit sha1_base64="/udw9s23KSwJLo9QXqxPb0kVBoU=">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</latexit>

Check if|u(k,o)u(k,o1)

u(k,o) | "1?<latexit sha1_base64="b8ZBhlFiizEMnSnutbs5nhT9fe8=">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</latexit><latexit sha1_base64="b8ZBhlFiizEMnSnutbs5nhT9fe8=">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</latexit><latexit sha1_base64="b8ZBhlFiizEMnSnutbs5nhT9fe8=">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</latexit><latexit sha1_base64="b8ZBhlFiizEMnSnutbs5nhT9fe8=">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</latexit>

Check|UB(k) LB(k)| "2?

<latexit sha1_base64="24XwiihzbxqYXgzof2Tqu+4ieuM=">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</latexit><latexit sha1_base64="24XwiihzbxqYXgzof2Tqu+4ieuM=">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</latexit><latexit sha1_base64="24XwiihzbxqYXgzof2Tqu+4ieuM=">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</latexit><latexit sha1_base64="24XwiihzbxqYXgzof2Tqu+4ieuM=">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</latexit>

Denote the optimalobjective value as u(k, o)

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Figure 2.2. Workflow of the proposed algorithm.

18

Note that the second-stage variables a are fixed here, which are delivered from the subprob-

lem. Hence, the master problem now becomes a deterministic MILP problem that can be

efficiently tackled via off-the-shelf solvers. Here, x includes the first-stage variables, i.e.,

xGg,t, xRr,t, x

L`,t, y

Gg,t, y

Rr,t, y

L`,t and z includes the third-stage variables, i.e., pGg,h, p

Rr,h, rd,h, f`,h, δn,h.

MP = minx,z

f(x) + φ (2-2)

subject to

Constraints (2-1a)-(2-1e) (2-2a)

f(x) = IC⊺x, (2-2b)

φ ≥ EωOC⊺pG +PC⊺r, ∀k′ ≤ k, (2-2c)

Constraints (2-1g)-(2-1q), ∀k′ ≤ k (2-2d)

2.4.2. Subproblem

The subproblem of the C&CG procedure is constructed from the second- and third-stage

problems, i.e., the problem determining the worst-case scenario of contingency. Here, a

includes the second-stage variables, i.e., aGg,t, aRr,t, a

L`,t.

maxa

Eωminp,r

OC⊺pG +PC⊺r (2-3)

subject to

Constraints (2-1f)-(2-1q). (3a)

Note that the first-stage variables in the constraints become fixed parameters obtained

from the master problem in the previous iteration. Since the inner minimization problem

of (2-3) has linear programming characteristics, according to the strong duality theory, it

19

is equivalent to rewrite (2-3) to its dual form (2-4), after giving an objective handle for the

second-stage.

Sub = maxa

0⊺a +Eω maxπ

Q(a,π) (2-4)

subject to

Q(a,π) ∈ Γa,π, (2-4a)

where Q(a,π) represents the dualized objective function, π is the vector of all dual variables

in (2-3) and Γa,π consists of the constraint space. For better demonstration, problem (2-4) is

further altered to a minimization problem (2-5) where we rewrite Γa,π in the form of (2-5a)-

(2-5c). Specifically, (2-5a) constructs the uncertainty set, (2-5b) formulates the operational

constraints, and (2-5c) ensures the dual feasibility of the Karush–Kuhn–Tucker condition.

Sub = mina

−0⊺a + Eω minπ

−Q(a,π) (2-5)

subject to

Constraint (1f), (2-5a)

D(ω)π ≤ r(ω) −C(ω)a, (2-5b)

π ≥ 0. (2-5c)

Note that in the objective function, Q(a,π) includes the first-stage decisions x∗ de-

termined from the master problem. This subproblem shows a typical two-stage stochastic

mixed-integer bilinear minimization structure. Furthermore, linear relaxation techniques,

e.g., the Big M method (see Appendix A), can effectively relax the bilinear parts without

sacrificing accuracy. As the binary nature of the integer variables tightens the convex re-

laxation, we can ensure the accuracy of the obtained solution. According to the stochastic

20

L-shaped method [23], the two-stage mixed-integer stochastic program with linear recourse

and finite support is also decomposable, as shown in the following.

2.4.2.1. L-shaped Master Problem

Sub −M = mina

−0⊺a + η (2-6)

subject to

Constraint (1f), (2-6a)

αo′ +β⊺o′a ≤ η, ∀o′ ≤ o (2-6b)

In the L-shaped master problem, η is an auxiliary variable, αo′ and βo′ are subgradients

computed from the dual of the L-shaped subproblem, which will be discussed in the next

subsection. Note that this formulation is for the single-cut L-shaped algorithm.

2.4.2.2. L-shaped Subproblem

Sub −S(ω) = minπ

−Q(a∗,π) (2-7)

subject to

D(ω)π ≤ r(ω) −C(ω)a∗ ∶ ξ(ω), (2-7a)

π ≥ 0. (2-7b)

When the L-shaped master problem yields an optimal solution of a∗, the subproblem

receives this solution and solves the operation problem. Let ξ∗(ω) denote the optimal dual

21

solution of the operation constraints (7a). After solving ω individual L-shaped subproblems,

which can be done in parallel, the subgradients for each iteration can be computed as follows.

αo =∑ω

Prob(ω)ξ∗o(ω)⊺r(ω),

βo = −∑ω

Prob(ω)C(ω)⊺ξ∗o(ω).

Afterwards, the L-shaped master problem receives an optimality cut (2-6b). Finally, the

inner iteration loop for the L-shaped procedure determines the final worst-case contingency

a∗ and sends it back to the C&CG master problem. As we enable load shedding, the

feasibility of all problems is guaranteed, and the feasibility cut is therefore negligible.

The proposed algorithm guarantees its convergence by the finite extreme points of the

uncertainty set and finite support in the second-stage stochastic recourse according to the

convergence analysis [22] and [23]. For the convergence speed, according to Figure 2.2,

the C&CG procedure does not influence the convergence of the embedded L-shaped method,

which means we still retain the fast convergence of the C&CG method [22]. The convergence

of the C&CG part relies on the convergence of the L-shaped algorithm, which is guaranteed

by the finite support in the stochastic recourse. The motivation of leveraging the L-shaped

method for the subproblem is that solving the original subproblem consumes the majority of

the computational time, as reported in [6] and [17]. We enjoy the merit of parallel computing

to facilitate solving subproblems when the L-shaped method is applied.

2.5. Case Studies

This section provides results and discussions for case studies on a modified IEEE 30-bus

system based on [27]. We aim to investigate the benefit of ultra-high renewable penetration

towards 100% in a long-term planning problem. We implement all of the experiments in

GAMS 25.0.3 [28] with CPLEX 12.8 and run it on a 2.60GHz Windows PC with a 6-core

Intel i7 CPU and 8GB RAM. We also leverage a GAMS-embedded parallel computing tool,

i.e., Gather-Update-Solve-Scatter (GUSS) [29], for the L-shaped subproblem to improve the

22

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Figure 2.3. System topology of the modified IEEE 30-bus system.

computational efficiency.

To illustrate our proposed methodology, we modify the IEEE 30-bus system to have

reduced transmission lines and no existing generators. Figure 2.3 provides the detailed

topology, where the blue and orange buses indicate the candidate locations for building

wind and solar generators, respectively. Thermal generators can be invested in any load bus.

The dashed lines indicate the candidate transmission lines. Table 2.1 shows the investment

information, whose reference can be found in [30–32]. The modified IEEE 30-bus system has

a total daily peak power demand as 2,000 MW, distributed to the three regions. And we set

the load shedding penalty cost as $1,000/MWh.

23

Table 2.1. Data for Candidate Generation Units.

GenerationType

Operatingcost

[$/MWh]

Maximumcapacity

[MW]

Overnightcapital cost

[M$]

Wind 0 300 390

Solar 0 150 170

Thermal 41.2 300 67.8

Trans. Line 0 600 50

2.5.1. Scenario Generation & Reduction

There are three regions in the system, where different scenario sets of renewable and

demand are applied. To verify the investment performance considering the spatiotemporal

correlation between renewables and load, we create the scenario set for each region indi-

vidually by using Monte-Carlo simulation on three different datasets of renewable output

from [33]. Figure 2.4 depicts the regional 24-hour sequential wind, solar, and demand sce-

narios. The average capacity factors of wind and solar are 45.14% and 28.86%, respectively.

We also assume that the two neighbored wind buses in Figure 2.3 share the same wind out-

put time series, and one can easily simulate scenarios with respect to different time series

for more detailed spatiotemporal studies.

For the spatiotemporal correlations between renewables and load, after sampling on three

different datasets with respect to three different regions, we apply the Fast Backward/For-

ward method embedded in the GAMS SCENRED toolbox to reduce the scenario number

according to the balance between the number of scenarios and the solution accuracy. This

strategy is generally an approximation of the complete scenario tree benchmarked by dif-

ferent criteria such as the Fortet-Mourier metric [34] and Lr-distance [35], which have been

widely adopted in many stochastic system planning works e.g. [36] and [37]. To capture the

spatiotemporal correlations between renewables and demand, we use three temporal datasets

for real renewable outputs and demand in three forecast zones in the ERCOT area as a basis

24

1 2 3 4 5 6 7 8 9 101112131415161718192021222324

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.P

ower

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ower

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ower

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ower

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)

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d(M

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d(M

W)

1 2 3 4 5 6 7 8 9 101112131415161718192021222324

Hour

0

200

400

600

Loa

d(M

W)

Figure 2.4. Regional scenario (10 for each).

for our Monte-Carlo sampling to create the scenario sets. With both the demand and re-

newable uncertainties being taken into consideration, the proposed G&TEP framework can

yield planning results according to the correlations between multiplex uncertainties. It is

also imperative to indicate that a more accurate and efficient scenario generation/reduction

technique concerning the spatiotemporal correlation between uncertainties is of great future

research interest, to which our proposed framework in this work can easily adapt.

To balance the tradeoff between the accuracy and tractability, we test different scenario

sets by running a one-year planning problem where two respective investments on solar and

wind generators are mandatory, as shown in Figure 2.5, and the highest computational time

stands for the highest time of solving subproblems among all iterations. Since the differ-

ence in objective function values between 100 scenarios and 1000 scenarios is within 2%, we

argue that the set of 100 scenarios is accurate enough to perform the following analyses.

Besides, Figure 2.5 also validates the necessity of using decomposition techniques for solving

25

Figure 2.5. Comparison for different sizes of scenarios.

stochastic subproblems, as the computational burden grows exponentially with the increas-

ing number of scenarios in the DEF problem as in [10] and [17], but grows linearly in the

L-shaped problem in our work.

2.5.2. Case Studies: Towards 100% Renewable Penetration

We design three cases to show the pathways to achieve 100% renewable penetration. We

apply the N-1 criterion to each region. Each candidate bus can build two generators in each

year. The algorithmic convergence gap ε2 is set to be 0.1%, and the solver’s MIP gap is set

as 0.01%.

• Case 1. $900M generation investment budget (6yrs);

• Case 2. $9,000M generation investment budget (6yrs);

• Case 3. $9,000M generation investment budget. Thermal units can only be installed

26

in the first year. All the thermal units will phase out by 20% capacity per year (6yrs)

with a salvage income.

To be more practical, for thermal units, we consider a 5% annual inflation rate of the fuel

price. For renewable energy, the investment cost has a discount factor of 5% per year as

the technology develops. The salvage price of phased out thermal generators is 40% of the

investment cost. The total system demand also increases by 5% per year. Table 2.2 shows

the investment decisions, costs and system information obtained from the three cases.

For the salvage income, we directly add a salvage income term in the objective function

of (2-1). The salvage of generators includes the sales of the salvageable parts of the unit,

recycling worn-out equipment, and re-utilizing the designated real estate [37]. According

to [38], the salvage value for generators is calculated based on a linear relationship with the

proportion of the used life and the remaining life, resulting the following equation:

S = Creplace ⋅Rremain

Rcomponent

in which Creplace is the replacement cost that is about 80% of the initial investment cost,

Rremain is the remaining life and Rcomponent is the component lifetime. We also refer to [39]

for the lifetime of a pulverized coal power plant as 30 years. Thus, in year 6, the remaining

lifetime of such unit is 24 years, and thus the salvage income should be 80% ⋅ (24/30) = 64%

of the investment cost. Considering the demolition cost and the personnel cost, we set the

salvage value for a thermal unit in our study as 40% of the investment cost. We will revisit

Case 1 and Case 2 in subsection E by considering the investment cost annuitization to fur-

ther assess the economic aspects of renewable installations.

1). General investment plan and operation

For the generation investment, on the one hand, since we consider the N-1 criterion in

each region, the system planner invests in a large amount of generation capacity, especially

when the renewable is installed (e.g., the total capacity in Case 3 is 152.9% higher than

27

Table 2.2. Investment Portfolio Report for the 30-bus System.

Gen. Investment*

Case 1 Case 2 Case 3

Year 1 2, 4, 6, 8, 10, 12, 14, 15, 18,

24, 29, 30

2, 4, 5, 5, 10, 9, 14, 15, 18, 21, 22, 24, 27,

29

4, 5, 5, 9, 9, 11, 11, 16, 21, 21, 22, 22, 25, 25, 27, 27,

30

Year 2 None 21 9, 13, 21, 22, 25, 25

Year 3 None None None




Line Investment Case 1 Case 2 Case 3

Year 1 4-11, 7-8, 6-22,

9-10, 15-26, 22-24,

4-11, 7-8, 6-22, 9-10, 15-18, 15-26, 22-

24

4-11, 7-8, 6-22, 9-10, 15-18, 15-24,

15-26, 22-24

Year 2-6 None None None

Gen. Invest. Cost $813.6M $2,812.9M $7,237.4M

Gen. Opera. Cost $5,976.81M $3,542.70M $168.93M

Line Invest. Cost $300M $350M $400M

Load Shed Cost 0 0 $1,079.43M

Salvage Income 0 0 $81.36M

Total Cost $7,090.41M $6,705.60M $8,754.40M

Avg. Load Shed Portion 0% 0% 0.96%

Avg. Renewable Curtailment 0% 7.75% 12.11%

Final Renewable Percentage

0% 42.86% 100%

* The numbers indicate the buses built with thermal, wind and solar.

28

the peak load). While the system planner has already known the demand increase rate for

the later years, most of the new generation capacity is invested in the first year to meet

the peak demand growth in the following years. However, since the renewable installation is

limited compared with the thermal installation, and considering the yearly increasing system

demand and the unit contingencies every year, there are still investments after the first year

especially in Case 2 and Case 3 to ensure the system reliability. Another noteworthy issue

is renewable curtailment. The peak curtailment in Case 3 is 18.29% when the average cur-

tailment across the scenarios also reaches 12.11%. They are comparatively small since the

excess renewable power can be used to support other regions. Future works considering using

the curtailed energy to provide ancillary services and storage charging can be envisioned.

On the other hand, for the transmission investment, new transmission lines tend to be

built in the first year to prepare for the worst-case contingency in the subsequent years. The

new transmission build-out is mostly to accommodate the injections from the new genera-

tors. Notably, lines 6-8, 12-15, and 6-28 are reported to be the ones with the most frequent

outage under the worst-case contingency. Thus, in the investment portfolio, lines 7-8, 6-22,

and 22-24 are always built in the three cases to aid power transmission. The system finally

achieves the 100% renewable penetration level in the 6th year in Case 3. The 100% is based

on the generation capacity percentage.

By comparing Case 1 and Case 2 in Table 2.2, we can find that if we have a sufficient

investment budget at the beginning, it is more profitable to invest in a proper generation

mix of thermal, wind and solar technologies from the system perspective. Since we consider

a long-term investment plan, the operation cost for 6 years in Case 1 far outnumbers the

investment cost, whereas the renewable generators only have a one-time cost for installation.

By comparing Case 2 and Case 3 in Table 2.2, we find that the operation cost is further

greatly decreased in Case 3, but the investment cost and the load shedding cost increase sig-

nificantly, which results in a higher total cost. However, as the investment cost of renewable

technology decreases yearly, in the long-term planning, the system will tend to have higher

renewable penetration approaching 100%, which will be detailed in subsection 2.5.4.

29

Under the current setting, we do not consider the energy storage and the time-shifting

demand response since we would like to investigate the cost-effectiveness of renewables only

from their own economic aspects. However, it is imperative to state that considering 100%

renewable energy without any energy storage or demand response is not beneficial for all

circumstances. It can be envisioned that energy storage and demand response can greatly

reduce the need for over-installation of renewable generation.

2). Interregional support with spatiotemporal correlations

The spatiotemporal correlations between renewables and regional differences of the un-

certainty also influence the investment portfolio of renewables. When we do not consider

any storage device or time-shifting demand response in our model, there could be a large

amount of load shedding in a particular region when the renewable generation in that region

is very low. However, such a situation can be largely mitigated with adequate transmission

capacity with the other regions since the variation of renewable generation output from each

region tends to cancel out over a large geographical area to provide a more stable output.

Hence, by using the proposed scenario generation and the optimization framework, the new

generators are built in the way that they can provide interregional energy support when

needed, based on their spatiotemporal correlations.

On the one hand, from the long-term investment perspective, in Figure 2.4, the diurnal

wind output in Region 1 is small, which incentivizes the investment of the solar unit at

bus 5 and reduces the investment of wind units in the region. Besides, in Case 3, the lines

connecting Region 2 and Region 3, i.e., line 15-18 and line 22-24, are all constructed to help

Region 3 cover its demand as the wind output in Region 3 in a long run is relatively lower

compared with the other regions.

On the other hand, from the short-term operation perspective, in the 6th year of

Case 3, we particularly analyze one operation scenario. Figure 2.6 depicts the active power

flow in the tie-lines with Region 1, i.e., lines 9-10, and 6-22. It can be seen that the power

support from the other regions reaches the peak value at hour 10 due to the lack of local

30

Figure 2.6. Power transfer from/to Region 1: one scenario.

wind power in Region 1, but Region 1 can export active power to the other regions at night

when the wind output increases. This phenomenon validates the interregional support when

the spatiotemporal correlations between renewables and load are present.

2.5.3. Case Studies: Contingency Criterion

Next, we elaborate on the contingency criterion and the impact of the worst-case contin-

gency by comparing the following Case 4 with the previous Case 3.

• Case 4. We apply no contingency criterion in all regions, i.e., the robust counterpart

in the model is omitted. The other settings are the same as Case 3.

Figure 2.7 demonstrates the generator’s installation details of these two cases. It can be

seen from the results that when the N-1 contingency criterion is applied for each region in

Case 3, the system needs to have more renewable generators installed to secure the demand.

Particularly for the worst-case contingency, the system is prone to increase the investment

in generators to provide more contingency reserves, which is also one of the reasons of the

huge installations in Case 3.

2.5.4. Case Studies: Long-term Cost-effectiveness

31

Figure 2.7. Comparison in the contingency criterion.

To further elaborate the cost-effectiveness of long-term investment in renewables, es-

pecially for the 100% renewable penetration, we carry out two case studies of a 12-year

expansion in the following:

• Case 5. $12,000M generation investment budget (12yrs);

• Case 6. $12,000M generation investment budget. Thermal units can only be installed

in the first year. All the thermal units phase 10% capacity out per year (12yrs).

It is straightforward that the 100% renewable penetration is accomplished in the 11th

year in Case 6. We enable the transmission expansion between any two connected buses

for the annually increasing demand, and the other system settings are the same as in the

previous case studies. Figure 2.8 depicts the investment plans and system costs. Accordingly,

the salvage value of conventional generators is set as 25% of the investment cost in this case.

32

Figure 2.8. Investment plans and system cost of the long-term planning.

In Case 5, the installation of one thermal unit is mainly to balance the load shedding

caused by stochastic generation, even when the interregional support from renewables has

already largely mitigated this issue. And in Case 6 when the condition changes that the

thermal units are no longer able to fully provide energy, the system planner is still prone

to invest one thermal unit due to the high load shedding cost. Though the 100% renewable

case, i.e., Case 6, still has a slightly higher cost than Case 5, we argue that the final cost

difference highly depends on how we choose the system parameters. Nonetheless, from these

two cases, we can see the potential of renewable generation’s long-term cost-effectiveness.

In Case 5 when the phasing-out is not allowed, the system in the 12th year still reaches

96.43% renewable penetration, which implies the long-term economic payoff is higher than

conventional units. It should also be noted that different settings of renewable-based G&TEP

33

studies could lead to different findings (e.g., [20]) but the proposed general framework still

applies.

2.5.5. Case Studies: Investment Annuitization

In many investment studies, investors often consider annuitization of the investment cost

to distribute the investment over the planning horizon. Hence, in this section, we illustrate

the effect of considering the annuitization of the investment cost with the same settings of

Case 1 and Case 2. We keep the same investment budget as in Case 1 and Case 2, and

consider a general annual interest rate i of the investment as 6.04% computed from the

weighted average cost of capital (WACC) of 5.7% [40]. We adopt the investment annuitizing

method from Appendix A in [40] as we first calculate the annual discount factor (DFt) by

DFt =1

(1 + i)t,

then the annualized capital cost (ACCg) and the discounted investment cost (ICg,t) used in

the objective function can be computed as

ACCg = OCCg ⋅i ⋅ (1 + i)LTg

(1 + i)n − 1,

ICg,t = ACCg ⋅ ∑t′≤minLTg ,T remain

DFt′ ,

where OCCg denotes the overnight capital cost, LTg denotes the generator lifetime, and

T remain denotes the remaining time of the planning horizon. The annual increase in thermal

units’ fuel cost and demand still applies.

Table 2.3 tabulates the results of the revisited Case 1 and Case 2 with considering

the annuitized investment cost. In both cases, there is no load shedding. Thanks to the

annuitization, the investment of generators can now be distributed to later planning years.

Compared with the previous Case 1 and Case 2, we find that both total costs do not

differ much, and the results in Case 1 remain nearly the same except for some investments

34

Table 2.3. Investment Portfolio with the Investment Annuitization.

Investment* Revisited Case 1 Revisited Case 2

Year 1 10 0 0 6 5 6 1 6 Year 2 0 0 0 0 0 3 0 1 Year 3 1 0 0 0 0 0 1 0 Year 4 1 0 0 0 1 0 0 0 Year 5 0 0 0 0 0 1 0 0 Year 6 0 0 0 0 0 0 0 0

6-yr Invest. Payment $762.67M $4,153.67M

Line Invest. Cost $300M $350M Gen. Opera. Cost $5,978.10M $2,163.77M

Total Cost $7,040.78M $6,667.44M * The integrals denote the numbers of invested thermal, wind, solar generators, and line.

distributed to later years. But in Case 2, the renewable installation is more prioritized, as

the investment annuitization makes the renewable investment more competitive with thermal

investments. However, we can still draw similar conclusions that a proper generation mix

of conventional and renewable technologies renders a more cost-effective planning portfolio

if the budget allows for renewable installations. As the annuitization method has already

taken the generators’ lifetime into account [40], we do not apply the technique of considering

salvage incomes [37] in the revisited cases. Leveraging the annuitization method can make

the investment in multi-year planning more comparable, which should be included in a more

realistic G&TEP study. Also, it is noteworthy that the performance is highly sensitive to

the selected WACC determined by the tax rate and expected returns on equity and debt.

2.6. Scalability Test

To validate the scalability of the proposed framework, we modify the WECC 243-bus

system based on [41] for G&TEP studies and carry out the scalability test. Four genera-

tion technologies are included, i.e., coal, combined-cycle, wind, and solar. We carry out a

35

Figure 2.9. Installed capacity for the WECC system.

regionally N-1 contingency-constrained 5-year planning problem with the hourly operation.

The number of combinatorial scenarios, in this case, is further reduced to 10 to reduce the

computational burden. And we enable the phasing-out of conventional units as in Case 3.

Figure 2.9 depicts the scheduled installed capacity for each generation technology and

transmission line, where we can observe that the investment in solar and wind generators

grows sharply in the first year and keeps increasing, due to the 25% annual phasing-out

rate of the conventional generators. The computational time of the WECC simulation is 27

hours, which is comparably reasonable concerning the scale of the multi-year stochastic and

robust planning problem.

2.7. Discussions

To clarify the scope of our work and how it should be used as a reference for both

36

academia and industry, we provide several discussions on this work.

1. Though the day-based contingencies have already been adopted in this work and other

works, e.g., [11, 25], the contingencies of power system components can be evaluated

in a practically smaller time resolution.

2. Using only a few representative days has been found to be not enough, and more

representative days should be included in more realistic studies of power systems with

large amounts of renewable [42].

3. In the case studies, multiple economic factors can affect the optimized planning portfo-

lio, e.g., the increasing fuel cost, decreasing renewable investment cost, annual interest

rate, and the salvage income. Different economic settings may lead to a different result,

but the conclusion of the renewable-involved G&TEP study should hold similarly as

discussed in the work. Besides, when we consider a longer term of planning, renewables

will begin to show the potential of higher cost-effectiveness in the G&TEP.

4. Other practical factors should be taken into account in future research to make more

precise investment decisions for investors, including but not limited to the employment

of energy storage devices, more types of renewables, demand response, and ancillary

services.

This work aims at providing a general framework and a solution algorithm, which can

be applied to different uncertainty-based G&TEP studies. The case studies in this work

are carried out under simplified simulation settings and demonstrate different renewable

investment scenarios under specific parameters. The observations of a more complete and

comprehensive long-term G&TEP study for realistic large-scale power systems may vary

from this work as the findings are dependent upon many sensitive parameters including

cost, discount rate and other detailed operational constraints.

2.8. Summary

This work introduces a novel modeling and solution strategy for the generation and trans-

mission expansion planning under ultra-high renewable penetration, which allows analyses

37

of discrete and continuous uncertainties. Based on the theoretical derivation and numerical

experiments, several remarks are in order:

• We propose a general hybrid stochastic and robust model that can accurately capture

the uncertainties in the modern power grid, whose discrete and continuous features are

taken into consideration with high flexibility.

• We propose a combinatorial solution strategy leveraging the state-of-the-art C&CG and

L-shaped algorithms that can efficiently tackle intractable and multiplex uncertainty-

based planning problems.

• We investigate the long-term renewable cost-effectiveness in the test results. A proper

portfolio of generation mix of the conventional and renewable generation is shown to

be beneficial under our specific problem settings, and we also pave a way for future

discussions on the 100% renewable penetration potentials by investigating the long-

term cost-effectiveness of the renewable generation.

38

Chapter 3

STOCHASTIC MARKET OPERATION FOR COORDINATED

TRANSMISSION AND DISTRIBUTION SYSTEMS

3.1. Nomenclature

We provide a detailed notation list for this chapter.

1) Sets:

T /D Transmission (T) / distribution (D) system

GT /GD Conventional thermal generator of T / D system

E / I Renewable generator / connected bus

Q Network reconfiguration set

LT /LD Load of T / D system

F T /FD Line of T / D system

AD /ADT Bus of D system / with substation

FDQ Configurable line of D system

S /N Sending / receiving bus of line

C Generator or injection mapping with bus

H Time horizon

O Iteration

2) Indices:

gT / gD Indices for conventional generators in T/D system

39

rT / rD Indices for renewable generators in T/D system

`T / `D Indices for loads in T/D system

fT Indices for transmission lines

mn Indices for connected distribution branches

nT /nD Indices for nodes in T/D system

i Indices for boundary buses

ωT /ωD Indices for operating scenarios in T/D system

o Indices for iteration counter

h Indices for time (hours)

c Indices for multiple D systems

q Indices for configurable swtich in D system

3) Parameters (in both T and D, otherwise specified):

PC Penalty cost [$/MWh]

R /X /B Resistance / reactance / susceptance [p.u.]

RD /RU Ramp-down / up limits [MW/h]

SD /SU Shut-down / start-up cost [$]

MO /MD Minimum ON / OFF time limits [h]

P T,maxf Active power flow limits [MW]

Y D,maxmn Upper limit of the branch current [A]

V min /V max Lower/Upper limit of the nodal voltage [kV]

Pming /Pmax

g Active power output limits of generators [MW]

40

4) Uncertainties (in both T and D):

Pmaxr,h (ω) Maximum available renewable power [MW]

L`,h(ω) Nodal demand [MW]

Prω Probability for scenario ω

5) Binary Variables:

uTg,h /dTg,h Start-up/shut-down indicator for generators

iTg,h Unit commitment indicator for generators

zq,h,c Distribution configuration indicator

6) Continuous Variables:

p(T,D)

(g,r)/pi Active generation output / power injection

qD(g,r)

/ qi Reactive generation output / power injection

pDn / qDn Active / reactive node injection

pTf Active line flow

pDmn / qDmn Active / reactive feeder flow

xDmn Line orientation variable

vn / δTn Squared nodal voltage magnitude / angle

yDmn Squared branch current

s` Active load shedding

3.2. Overview

Unit commitment (UC) and economic dispatch (ED) have been of capital importance in

the market operations for system operators in the world. The coordination between trans-

mission and distribution systems (T-D coordination) in the market operation has attacked

41

great attention. The primary motivation for T-D coordination is based on the fact that

the distribution system has become active due to the increasing deployment of smart grid

technologies including distributed energy resources (DERs), electric vehicles, microgrids, etc.

The system dynamics within the distribution networks can propagate to the transmission

side and vice versa [43]. A recent report from California Independent System Operator

(CAISO) [44] supports this motivation. It indicates that, due to the growing penetration of

the DERs, Independent System Operator (ISO), which regulates the upstream market in the

transmission level, needs to attain more visibility on the downstream markets to reinforce

its decision-making.

In the conventional market hierarchy, ISOs and utilities carry out UC and ED problems

within their respective territories. The upstream operators make scheduling decisions ac-

cording to the estimated information from the downstream aggregators [45]. Without the

T-D coordination, the power mismatch at the boundary node between transmission and dis-

tribution systems proliferates as distribution systems become more and more active [46]. To

cope with the issue of power mismatches, Z. Li et al. [47] proposed a distributed paradigm

that ISO delivers its optimal locational marginal price (LMP) to distribution system op-

erators (DSOs), whereas DSOs return their optimal energy demand to ISO. The mutually

observable information in the T-D coordination should be limited to protect the confiden-

tiality of stakeholders [47].

Recently, based on this decentralized framework, many papers have considered more ele-

ments and more efficient algorithms. P. Li et al. [48] built a distributionally robust optimiza-

tion framework considering system uncertainties and adopted an analytical target cascading

algorithm to decentralize the problem. R. Nejad et al. [49] and J. Zhao et al. [50] focused

on the integrated system restoration using the alternating direction of multipliers method

(ADMM) to decentralize the problem. M. Arpanahi et al. [51] proposed a decentralized

framework for T-D coordination that considers robust optimization for the uncertainties,

while using an enhanced ADMM algorithm to solve. A. Nawaz et al. [52] proposed a prob-

abilistic coordination method to select one stochastic scenario to solve the T-D coordinated

42

problem. However, most of the works focus on ED or OPF, instead of the UC, which could

not serve as a general reference for electricity market operations. It is also reasonable that

the algorithms proposed above lie in the category of the augmented Lagrangian method,

which would fail if integer variables are present in the problem.

For the modeling of T-D coordination, recently, research on this topic gains prevalence,

and potential modeling frameworks from the perspective of DSOs can be categorized into

three types, i.e., centralized, decentralized, and transactive models, which are summarized

in [7]. Among the models discussed therein, Model 3, i.e., the distribution node utility model,

acts as a transitional model from the centralized operation to the decentralized operation,

where the ISO and DSOs take the responsibilities of market operations in the transmission

and distribution level, respectively, and they exchange information in the boundary substa-

tion. The natural compatibility with the current market practice is the major advantage of

this model [7]. Based on this framework, we propose a three-stage stochastic co-optimization

for the T-D coordinated UC and ED. In this scheme, the ISO regulates the coordination be-

tween the transmission and distribution systems, where the transmission system operation is

under control of the ISO and the distribution system operation is under control of the DSO.

As in the current industry’s practice [53], the ISO carries out the transmission-level market

operations including UC and ED with the distribution-level inputs from DSOs. DSOs are

utilities that own the distribution network with aggregated DERs. It is also noteworthy

that the current research on decentralized or transactive models is fruitful and few are in-

vestigating centralized frameworks due to the computational hurdle. However, before the

fully decentralized distribution operations, the industry needs a transitional model that al-

lows the ISOs to take the local distribution information into account while regulating the

coordination [54]. Hence, in the proposed market hierarchy, the ISO performs the day-ahead

transmission-level UC and ED with the aggregated distribution information, and DSOs per-

form the distribution-level ED. The ISO does not need to attain detailed distribution system

information, which protects the local utilities’ confidentiality.

Apart from the scheduling, the distribution-level network reconfiguration drastically

43

Optimal Dispatch<latexit sha1_base64="2+8j9hLkXAHh+EANbQfx6hieGgo=">AAAB+XicbVBNSwMxEM3Wr1q/Vj16CRahp7Lbix4LevBmBfsB7VKyadqGJtmQzBbK0n/ixYMiXv0n3vw3pu0etPXBwOO9GWbmxVpwC0Hw7RW2tnd294r7pYPDo+MT//SsZZPUUNakiUhMJyaWCa5YEzgI1tGGERkL1o4ntwu/PWXG8kQ9wUyzSJKR4kNOCTip7/sPGrgkAt9xqwnQcd8vB9VgCbxJwpyUUY5G3//qDRKaSqaACmJtNww0RBkxwKlg81IvtUwTOiEj1nVUEclslC0vn+MrpwzwMDGuFOCl+nsiI9LamYxdpyQwtuveQvzP66YwvIkyrnQKTNHVomEqMCR4EQMecMMoiJkjhBrubsV0TAyh4MIquRDC9Zc3SatWDYNq+Fgr1yt5HEV0gS5RBYXoGtXRPWqgJqJoip7RK3rzMu/Fe/c+Vq0FL585R3/gff4AScqTVQ==</latexit><latexit sha1_base64="2+8j9hLkXAHh+EANbQfx6hieGgo=">AAAB+XicbVBNSwMxEM3Wr1q/Vj16CRahp7Lbix4LevBmBfsB7VKyadqGJtmQzBbK0n/ixYMiXv0n3vw3pu0etPXBwOO9GWbmxVpwC0Hw7RW2tnd294r7pYPDo+MT//SsZZPUUNakiUhMJyaWCa5YEzgI1tGGERkL1o4ntwu/PWXG8kQ9wUyzSJKR4kNOCTip7/sPGrgkAt9xqwnQcd8vB9VgCbxJwpyUUY5G3//qDRKaSqaACmJtNww0RBkxwKlg81IvtUwTOiEj1nVUEclslC0vn+MrpwzwMDGuFOCl+nsiI9LamYxdpyQwtuveQvzP66YwvIkyrnQKTNHVomEqMCR4EQMecMMoiJkjhBrubsV0TAyh4MIquRDC9Zc3SatWDYNq+Fgr1yt5HEV0gS5RBYXoGtXRPWqgJqJoip7RK3rzMu/Fe/c+Vq0FL585R3/gff4AScqTVQ==</latexit><latexit sha1_base64="2+8j9hLkXAHh+EANbQfx6hieGgo=">AAAB+XicbVBNSwMxEM3Wr1q/Vj16CRahp7Lbix4LevBmBfsB7VKyadqGJtmQzBbK0n/ixYMiXv0n3vw3pu0etPXBwOO9GWbmxVpwC0Hw7RW2tnd294r7pYPDo+MT//SsZZPUUNakiUhMJyaWCa5YEzgI1tGGERkL1o4ntwu/PWXG8kQ9wUyzSJKR4kNOCTip7/sPGrgkAt9xqwnQcd8vB9VgCbxJwpyUUY5G3//qDRKaSqaACmJtNww0RBkxwKlg81IvtUwTOiEj1nVUEclslC0vn+MrpwzwMDGuFOCl+nsiI9LamYxdpyQwtuveQvzP66YwvIkyrnQKTNHVomEqMCR4EQMecMMoiJkjhBrubsV0TAyh4MIquRDC9Zc3SatWDYNq+Fgr1yt5HEV0gS5RBYXoGtXRPWqgJqJoip7RK3rzMu/Fe/c+Vq0FL585R3/gff4AScqTVQ==</latexit><latexit sha1_base64="2+8j9hLkXAHh+EANbQfx6hieGgo=">AAAB+XicbVBNSwMxEM3Wr1q/Vj16CRahp7Lbix4LevBmBfsB7VKyadqGJtmQzBbK0n/ixYMiXv0n3vw3pu0etPXBwOO9GWbmxVpwC0Hw7RW2tnd294r7pYPDo+MT//SsZZPUUNakiUhMJyaWCa5YEzgI1tGGERkL1o4ntwu/PWXG8kQ9wUyzSJKR4kNOCTip7/sPGrgkAt9xqwnQcd8vB9VgCbxJwpyUUY5G3//qDRKaSqaACmJtNww0RBkxwKlg81IvtUwTOiEj1nVUEclslC0vn+MrpwzwMDGuFOCl+nsiI9LamYxdpyQwtuveQvzP66YwvIkyrnQKTNHVomEqMCR4EQMecMMoiJkjhBrubsV0TAyh4MIquRDC9Zc3SatWDYNq+Fgr1yt5HEV0gS5RBYXoGtXRPWqgJqJoip7RK3rzMu/Fe/c+Vq0FL585R3/gff4AScqTVQ==</latexit>

UC Decisions<latexit sha1_base64="2jloJ7zD5svzF/ZbHNcKwDf6Ee8=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahp5L0osdCPXisYFqhDWWznbRLN5uwuxFK6d/w4kERr/4Zb/4bN20O2vpg4PHeDDPzwlRwbVz32yltbe/s7pX3KweHR8cn1dOzrk4yxdBniUjUY0g1Ci7RN9wIfEwV0jgU2Aun7dzvPaHSPJEPZpZiENOx5BFn1Fhp4LfJLTKe23pYrbkNdwmySbyC1KBAZ1j9GowSlsUoDRNU677npiaYU2U4E7ioDDKNKWVTOsa+pZLGqIP58uYFubLKiESJsiUNWaq/J+Y01noWh7Yzpmai171c/M/rZya6CeZcpplByVaLokwQk5A8ADLiCpkRM0soU9zeStiEKsqMjaliQ/DWX94k3WbDcxvefbPWqhdxlOECLqEOHlxDC+6gAz4wSOEZXuHNyZwX5935WLWWnGLmHP7A+fwBaneRLw==</latexit><latexit sha1_base64="2jloJ7zD5svzF/ZbHNcKwDf6Ee8=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahp5L0osdCPXisYFqhDWWznbRLN5uwuxFK6d/w4kERr/4Zb/4bN20O2vpg4PHeDDPzwlRwbVz32yltbe/s7pX3KweHR8cn1dOzrk4yxdBniUjUY0g1Ci7RN9wIfEwV0jgU2Aun7dzvPaHSPJEPZpZiENOx5BFn1Fhp4LfJLTKe23pYrbkNdwmySbyC1KBAZ1j9GowSlsUoDRNU677npiaYU2U4E7ioDDKNKWVTOsa+pZLGqIP58uYFubLKiESJsiUNWaq/J+Y01noWh7Yzpmai171c/M/rZya6CeZcpplByVaLokwQk5A8ADLiCpkRM0soU9zeStiEKsqMjaliQ/DWX94k3WbDcxvefbPWqhdxlOECLqEOHlxDC+6gAz4wSOEZXuHNyZwX5935WLWWnGLmHP7A+fwBaneRLw==</latexit><latexit sha1_base64="2jloJ7zD5svzF/ZbHNcKwDf6Ee8=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahp5L0osdCPXisYFqhDWWznbRLN5uwuxFK6d/w4kERr/4Zb/4bN20O2vpg4PHeDDPzwlRwbVz32yltbe/s7pX3KweHR8cn1dOzrk4yxdBniUjUY0g1Ci7RN9wIfEwV0jgU2Aun7dzvPaHSPJEPZpZiENOx5BFn1Fhp4LfJLTKe23pYrbkNdwmySbyC1KBAZ1j9GowSlsUoDRNU677npiaYU2U4E7ioDDKNKWVTOsa+pZLGqIP58uYFubLKiESJsiUNWaq/J+Y01noWh7Yzpmai171c/M/rZya6CeZcpplByVaLokwQk5A8ADLiCpkRM0soU9zeStiEKsqMjaliQ/DWX94k3WbDcxvefbPWqhdxlOECLqEOHlxDC+6gAz4wSOEZXuHNyZwX5935WLWWnGLmHP7A+fwBaneRLw==</latexit><latexit sha1_base64="2jloJ7zD5svzF/ZbHNcKwDf6Ee8=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahp5L0osdCPXisYFqhDWWznbRLN5uwuxFK6d/w4kERr/4Zb/4bN20O2vpg4PHeDDPzwlRwbVz32yltbe/s7pX3KweHR8cn1dOzrk4yxdBniUjUY0g1Ci7RN9wIfEwV0jgU2Aun7dzvPaHSPJEPZpZiENOx5BFn1Fhp4LfJLTKe23pYrbkNdwmySbyC1KBAZ1j9GowSlsUoDRNU677npiaYU2U4E7ioDDKNKWVTOsa+pZLGqIP58uYFubLKiESJsiUNWaq/J+Y01noWh7Yzpmai171c/M/rZya6CeZcpplByVaLokwQk5A8ADLiCpkRM0soU9zeStiEKsqMjaliQ/DWX94k3WbDcxvefbPWqhdxlOECLqEOHlxDC+6gAz4wSOEZXuHNyZwX5935WLWWnGLmHP7A+fwBaneRLw==</latexit>

DSO<latexit sha1_base64="6Rv6dViNrtQtKDzoKfDWu80pQ/U=">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</latexit><latexit sha1_base64="6Rv6dViNrtQtKDzoKfDWu80pQ/U=">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</latexit><latexit sha1_base64="6Rv6dViNrtQtKDzoKfDWu80pQ/U=">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</latexit><latexit sha1_base64="6Rv6dViNrtQtKDzoKfDWu80pQ/U=">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</latexit>

ISO<latexit sha1_base64="JZvFGLKBDoVDoUYuOzNvAtmGEh0=">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</latexit><latexit sha1_base64="JZvFGLKBDoVDoUYuOzNvAtmGEh0=">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</latexit><latexit sha1_base64="JZvFGLKBDoVDoUYuOzNvAtmGEh0=">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</latexit><latexit sha1_base64="JZvFGLKBDoVDoUYuOzNvAtmGEh0=">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</latexit>

Uncertainty-based<latexit sha1_base64="x6yhdchilzHkgkRhOgcw2H/NubY=">AAAB+nicbVA9T8MwEHX4LOUrhZHFokLqQpV0gbESC2ORSFuprSrHcVqrjhPZF1AU+lNYGECIlV/Cxr/BaTNAy5NOenrvzr57fiK4Bsf5tjY2t7Z3dit71f2Dw6Nju3bS1XGqKPNoLGLV94lmgkvmAQfB+oliJPIF6/mzm8LvPTCleSzvIUvYKCITyUNOCRhpbNc8SZkCwiVkl8VDwdiuO01nAbxO3JLUUYnO2P4aBjFNIyaBCqL1wHUSGOVEAaeCzavDVLOE0BmZsIGhkkRMj/LF6nN8YZQAh7EyJQEv1N8TOYm0ziLfdEYEpnrVK8T/vEEK4fUo5zJJgUm6/ChMBYYYFznggCtGQWSGEKq42RXTKVGEgkmrakJwV09eJ91W03Wa7l2r3m6UcVTQGTpHDeSiK9RGt6iDPETRI3pGr+jNerJerHfrY9m6YZUzp+gPrM8fZDqT+w==</latexit><latexit sha1_base64="x6yhdchilzHkgkRhOgcw2H/NubY=">AAAB+nicbVA9T8MwEHX4LOUrhZHFokLqQpV0gbESC2ORSFuprSrHcVqrjhPZF1AU+lNYGECIlV/Cxr/BaTNAy5NOenrvzr57fiK4Bsf5tjY2t7Z3dit71f2Dw6Nju3bS1XGqKPNoLGLV94lmgkvmAQfB+oliJPIF6/mzm8LvPTCleSzvIUvYKCITyUNOCRhpbNc8SZkCwiVkl8VDwdiuO01nAbxO3JLUUYnO2P4aBjFNIyaBCqL1wHUSGOVEAaeCzavDVLOE0BmZsIGhkkRMj/LF6nN8YZQAh7EyJQEv1N8TOYm0ziLfdEYEpnrVK8T/vEEK4fUo5zJJgUm6/ChMBYYYFznggCtGQWSGEKq42RXTKVGEgkmrakJwV09eJ91W03Wa7l2r3m6UcVTQGTpHDeSiK9RGt6iDPETRI3pGr+jNerJerHfrY9m6YZUzp+gPrM8fZDqT+w==</latexit><latexit sha1_base64="x6yhdchilzHkgkRhOgcw2H/NubY=">AAAB+nicbVA9T8MwEHX4LOUrhZHFokLqQpV0gbESC2ORSFuprSrHcVqrjhPZF1AU+lNYGECIlV/Cxr/BaTNAy5NOenrvzr57fiK4Bsf5tjY2t7Z3dit71f2Dw6Nju3bS1XGqKPNoLGLV94lmgkvmAQfB+oliJPIF6/mzm8LvPTCleSzvIUvYKCITyUNOCRhpbNc8SZkCwiVkl8VDwdiuO01nAbxO3JLUUYnO2P4aBjFNIyaBCqL1wHUSGOVEAaeCzavDVLOE0BmZsIGhkkRMj/LF6nN8YZQAh7EyJQEv1N8TOYm0ziLfdEYEpnrVK8T/vEEK4fUo5zJJgUm6/ChMBYYYFznggCtGQWSGEKq42RXTKVGEgkmrakJwV09eJ91W03Wa7l2r3m6UcVTQGTpHDeSiK9RGt6iDPETRI3pGr+jNerJerHfrY9m6YZUzp+gPrM8fZDqT+w==</latexit><latexit sha1_base64="x6yhdchilzHkgkRhOgcw2H/NubY=">AAAB+nicbVA9T8MwEHX4LOUrhZHFokLqQpV0gbESC2ORSFuprSrHcVqrjhPZF1AU+lNYGECIlV/Cxr/BaTNAy5NOenrvzr57fiK4Bsf5tjY2t7Z3dit71f2Dw6Nju3bS1XGqKPNoLGLV94lmgkvmAQfB+oliJPIF6/mzm8LvPTCleSzvIUvYKCITyUNOCRhpbNc8SZkCwiVkl8VDwdiuO01nAbxO3JLUUYnO2P4aBjFNIyaBCqL1wHUSGOVEAaeCzavDVLOE0BmZsIGhkkRMj/LF6nN8YZQAh7EyJQEv1N8TOYm0ziLfdEYEpnrVK8T/vEEK4fUo5zJJgUm6/ChMBYYYFznggCtGQWSGEKq42RXTKVGEgkmrakJwV09eJ91W03Wa7l2r3m6UcVTQGTpHDeSiK9RGt6iDPETRI3pGr+jNerJerHfrY9m6YZUzp+gPrM8fZDqT+w==</latexit>

Net

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kR

econ

fig.

<latexit sha1_base64="YquDlGIEF1W1Yvq5RRpzUxUrxks=">AAAB+nicbVA9SwNBEN3zM8avi5Y2i0FIFe7SaBmwsZIo5gOSI+xt5pIle7vH7p4hnPkpNhaK2PpL7Pw3bpIrNPHBwOO9GWbmhQln2njet7OxubW9s1vYK+4fHB4du6WTlpapotCkkkvVCYkGzgQ0DTMcOokCEocc2uH4eu63H0FpJsWDmSYQxGQoWMQoMVbqu6VbMBOpxvgeqBQRG1b7btmregvgdeLnpIxyNPruV28gaRqDMJQTrbu+l5ggI8owymFW7KUaEkLHZAhdSwWJQQfZ4vQZvrDKAEdS2RIGL9TfExmJtZ7Goe2MiRnpVW8u/ud1UxNdBRkTSWpA0OWiKOXYSDzPAQ+YAmr41BJCFbO3YjoiilBj0yraEPzVl9dJq1b1vap/VyvXK3kcBXSGzlEF+egS1dENaqAmomiCntErenOenBfn3flYtm44+cwp+gPn8wfVW5Oe</latexit><latexit sha1_base64="YquDlGIEF1W1Yvq5RRpzUxUrxks=">AAAB+nicbVA9SwNBEN3zM8avi5Y2i0FIFe7SaBmwsZIo5gOSI+xt5pIle7vH7p4hnPkpNhaK2PpL7Pw3bpIrNPHBwOO9GWbmhQln2njet7OxubW9s1vYK+4fHB4du6WTlpapotCkkkvVCYkGzgQ0DTMcOokCEocc2uH4eu63H0FpJsWDmSYQxGQoWMQoMVbqu6VbMBOpxvgeqBQRG1b7btmregvgdeLnpIxyNPruV28gaRqDMJQTrbu+l5ggI8owymFW7KUaEkLHZAhdSwWJQQfZ4vQZvrDKAEdS2RIGL9TfExmJtZ7Goe2MiRnpVW8u/ud1UxNdBRkTSWpA0OWiKOXYSDzPAQ+YAmr41BJCFbO3YjoiilBj0yraEPzVl9dJq1b1vap/VyvXK3kcBXSGzlEF+egS1dENaqAmomiCntErenOenBfn3flYtm44+cwp+gPn8wfVW5Oe</latexit><latexit sha1_base64="YquDlGIEF1W1Yvq5RRpzUxUrxks=">AAAB+nicbVA9SwNBEN3zM8avi5Y2i0FIFe7SaBmwsZIo5gOSI+xt5pIle7vH7p4hnPkpNhaK2PpL7Pw3bpIrNPHBwOO9GWbmhQln2njet7OxubW9s1vYK+4fHB4du6WTlpapotCkkkvVCYkGzgQ0DTMcOokCEocc2uH4eu63H0FpJsWDmSYQxGQoWMQoMVbqu6VbMBOpxvgeqBQRG1b7btmregvgdeLnpIxyNPruV28gaRqDMJQTrbu+l5ggI8owymFW7KUaEkLHZAhdSwWJQQfZ4vQZvrDKAEdS2RIGL9TfExmJtZ7Goe2MiRnpVW8u/ud1UxNdBRkTSWpA0OWiKOXYSDzPAQ+YAmr41BJCFbO3YjoiilBj0yraEPzVl9dJq1b1vap/VyvXK3kcBXSGzlEF+egS1dENaqAmomiCntErenOenBfn3flYtm44+cwp+gPn8wfVW5Oe</latexit><latexit sha1_base64="YquDlGIEF1W1Yvq5RRpzUxUrxks=">AAAB+nicbVA9SwNBEN3zM8avi5Y2i0FIFe7SaBmwsZIo5gOSI+xt5pIle7vH7p4hnPkpNhaK2PpL7Pw3bpIrNPHBwOO9GWbmhQln2njet7OxubW9s1vYK+4fHB4du6WTlpapotCkkkvVCYkGzgQ0DTMcOokCEocc2uH4eu63H0FpJsWDmSYQxGQoWMQoMVbqu6VbMBOpxvgeqBQRG1b7btmregvgdeLnpIxyNPruV28gaRqDMJQTrbu+l5ggI8owymFW7KUaEkLHZAhdSwWJQQfZ4vQZvrDKAEdS2RIGL9TfExmJtZ7Goe2MiRnpVW8u/ud1UxNdBRkTSWpA0OWiKOXYSDzPAQ+YAmr41BJCFbO3YjoiilBj0yraEPzVl9dJq1b1vap/VyvXK3kcBXSGzlEF+egS1dENaqAmomiCntErenOenBfn3flYtm44+cwp+gPn8wfVW5Oe</latexit>





Economic Dispatch<latexit sha1_base64="aU7bGxHXfxGxEt6SVD3vEuhkioY=">AAAB+nicbVBNSwMxEJ2tX7V+bfXoJViEnspuL3osqOCxgv2AdinZNNuGZpMlySpl7U/x4kERr/4Sb/4b03YP2vpg4PHeDDPzwoQzbTzv2ylsbG5t7xR3S3v7B4dHbvm4rWWqCG0RyaXqhlhTzgRtGWY47SaK4jjktBNOruZ+54EqzaS4N9OEBjEeCRYxgo2VBm75hkghY0bQNdMJNmQ8cCtezVsArRM/JxXI0Ry4X/2hJGlMhSEca93zvcQEGVaGEU5npX6qaYLJBI9oz1KBY6qDbHH6DJ1bZYgiqWwJgxbq74kMx1pP49B2xtiM9ao3F//zeqmJLoOMiSQ1VJDloijlyEg0zwENmaLE8KklmChmb0VkjBUmxqZVsiH4qy+vk3a95ns1/65eaVTzOIpwCmdQBR8uoAG30IQWEHiEZ3iFN+fJeXHenY9la8HJZ07gD5zPH/guk7Y=</latexit><latexit sha1_base64="aU7bGxHXfxGxEt6SVD3vEuhkioY=">AAAB+nicbVBNSwMxEJ2tX7V+bfXoJViEnspuL3osqOCxgv2AdinZNNuGZpMlySpl7U/x4kERr/4Sb/4b03YP2vpg4PHeDDPzwoQzbTzv2ylsbG5t7xR3S3v7B4dHbvm4rWWqCG0RyaXqhlhTzgRtGWY47SaK4jjktBNOruZ+54EqzaS4N9OEBjEeCRYxgo2VBm75hkghY0bQNdMJNmQ8cCtezVsArRM/JxXI0Ry4X/2hJGlMhSEca93zvcQEGVaGEU5npX6qaYLJBI9oz1KBY6qDbHH6DJ1bZYgiqWwJgxbq74kMx1pP49B2xtiM9ao3F//zeqmJLoOMiSQ1VJDloijlyEg0zwENmaLE8KklmChmb0VkjBUmxqZVsiH4qy+vk3a95ns1/65eaVTzOIpwCmdQBR8uoAG30IQWEHiEZ3iFN+fJeXHenY9la8HJZ07gD5zPH/guk7Y=</latexit><latexit sha1_base64="aU7bGxHXfxGxEt6SVD3vEuhkioY=">AAAB+nicbVBNSwMxEJ2tX7V+bfXoJViEnspuL3osqOCxgv2AdinZNNuGZpMlySpl7U/x4kERr/4Sb/4b03YP2vpg4PHeDDPzwoQzbTzv2ylsbG5t7xR3S3v7B4dHbvm4rWWqCG0RyaXqhlhTzgRtGWY47SaK4jjktBNOruZ+54EqzaS4N9OEBjEeCRYxgo2VBm75hkghY0bQNdMJNmQ8cCtezVsArRM/JxXI0Ry4X/2hJGlMhSEca93zvcQEGVaGEU5npX6qaYLJBI9oz1KBY6qDbHH6DJ1bZYgiqWwJgxbq74kMx1pP49B2xtiM9ao3F//zeqmJLoOMiSQ1VJDloijlyEg0zwENmaLE8KklmChmb0VkjBUmxqZVsiH4qy+vk3a95ns1/65eaVTzOIpwCmdQBR8uoAG30IQWEHiEZ3iFN+fJeXHenY9la8HJZ07gD5zPH/guk7Y=</latexit><latexit sha1_base64="aU7bGxHXfxGxEt6SVD3vEuhkioY=">AAAB+nicbVBNSwMxEJ2tX7V+bfXoJViEnspuL3osqOCxgv2AdinZNNuGZpMlySpl7U/x4kERr/4Sb/4b03YP2vpg4PHeDDPzwoQzbTzv2ylsbG5t7xR3S3v7B4dHbvm4rWWqCG0RyaXqhlhTzgRtGWY47SaK4jjktBNOruZ+54EqzaS4N9OEBjEeCRYxgo2VBm75hkghY0bQNdMJNmQ8cCtezVsArRM/JxXI0Ry4X/2hJGlMhSEca93zvcQEGVaGEU5npX6qaYLJBI9oz1KBY6qDbHH6DJ1bZYgiqWwJgxbq74kMx1pP49B2xtiM9ao3F//zeqmJLoOMiSQ1VJDloijlyEg0zwENmaLE8KklmChmb0VkjBUmxqZVsiH4qy+vk3a95ns1/65eaVTzOIpwCmdQBR8uoAG30IQWEHiEZ3iFN+fJeXHenY9la8HJZ07gD5zPH/guk7Y=</latexit>



Stage<latexit sha1_base64="Up/35XAuaZZJDy+hxcUBwsG6liE=">AAACF3icdVDLSsNAFJ34rPUVdelmsAhdSEhiaeuu4MZlRfuANJTJdNIOnTyYmQgl5C/c+CtuXCjiVnf+jZO2ARU9MHA499zHHC9mVEjT/NRWVtfWNzZLW+Xtnd29ff3gsCuihGPSwRGLeN9DgjAako6kkpF+zAkKPEZ63vQyr/fuCBc0Cm/lLCZugMYh9SlGUklD3UgH8yEOH3tuahoXzbpdq5+Zhmk2LNvKid2ondeyG4nGJBvqlcIDCw8sPNBSSo4KWKI91D8GowgnAQklZkgIxzJj6aaIS4oZycqDRJAY4aka7igaooAIN52flMFTpYygH3H1Qgnn6veOFAVCzAJPOQMkJ+J3LRf/qjmJ9JtuSsM4kSTEi0V+wqCMYB4SHFFOsGQzRRDmVN0K8QRxhKWKsqxCKH4K/ydd27BMw7q2K63qMo4SOAYnoAos0AAtcAXaoAMwuAeP4Bm8aA/ak/aqvS2sK9qy5wj8gPb+BRXUnAM=</latexit><latexit sha1_base64="Up/35XAuaZZJDy+hxcUBwsG6liE=">AAACF3icdVDLSsNAFJ34rPUVdelmsAhdSEhiaeuu4MZlRfuANJTJdNIOnTyYmQgl5C/c+CtuXCjiVnf+jZO2ARU9MHA499zHHC9mVEjT/NRWVtfWNzZLW+Xtnd29ff3gsCuihGPSwRGLeN9DgjAako6kkpF+zAkKPEZ63vQyr/fuCBc0Cm/lLCZugMYh9SlGUklD3UgH8yEOH3tuahoXzbpdq5+Zhmk2LNvKid2ondeyG4nGJBvqlcIDCw8sPNBSSo4KWKI91D8GowgnAQklZkgIxzJj6aaIS4oZycqDRJAY4aka7igaooAIN52flMFTpYygH3H1Qgnn6veOFAVCzAJPOQMkJ+J3LRf/qjmJ9JtuSsM4kSTEi0V+wqCMYB4SHFFOsGQzRRDmVN0K8QRxhKWKsqxCKH4K/ydd27BMw7q2K63qMo4SOAYnoAos0AAtcAXaoAMwuAeP4Bm8aA/ak/aqvS2sK9qy5wj8gPb+BRXUnAM=</latexit><latexit sha1_base64="Up/35XAuaZZJDy+hxcUBwsG6liE=">AAACF3icdVDLSsNAFJ34rPUVdelmsAhdSEhiaeuu4MZlRfuANJTJdNIOnTyYmQgl5C/c+CtuXCjiVnf+jZO2ARU9MHA499zHHC9mVEjT/NRWVtfWNzZLW+Xtnd29ff3gsCuihGPSwRGLeN9DgjAako6kkpF+zAkKPEZ63vQyr/fuCBc0Cm/lLCZugMYh9SlGUklD3UgH8yEOH3tuahoXzbpdq5+Zhmk2LNvKid2ondeyG4nGJBvqlcIDCw8sPNBSSo4KWKI91D8GowgnAQklZkgIxzJj6aaIS4oZycqDRJAY4aka7igaooAIN52flMFTpYygH3H1Qgnn6veOFAVCzAJPOQMkJ+J3LRf/qjmJ9JtuSsM4kSTEi0V+wqCMYB4SHFFOsGQzRRDmVN0K8QRxhKWKsqxCKH4K/ydd27BMw7q2K63qMo4SOAYnoAos0AAtcAXaoAMwuAeP4Bm8aA/ak/aqvS2sK9qy5wj8gPb+BRXUnAM=</latexit><latexit sha1_base64="Up/35XAuaZZJDy+hxcUBwsG6liE=">AAACF3icdVDLSsNAFJ34rPUVdelmsAhdSEhiaeuu4MZlRfuANJTJdNIOnTyYmQgl5C/c+CtuXCjiVnf+jZO2ARU9MHA499zHHC9mVEjT/NRWVtfWNzZLW+Xtnd29ff3gsCuihGPSwRGLeN9DgjAako6kkpF+zAkKPEZ63vQyr/fuCBc0Cm/lLCZugMYh9SlGUklD3UgH8yEOH3tuahoXzbpdq5+Zhmk2LNvKid2ondeyG4nGJBvqlcIDCw8sPNBSSo4KWKI91D8GowgnAQklZkgIxzJj6aaIS4oZycqDRJAY4aka7igaooAIN52flMFTpYygH3H1Qgnn6veOFAVCzAJPOQMkJ+J3LRf/qjmJ9JtuSsM4kSTEi0V+wqCMYB4SHFFOsGQzRRDmVN0K8QRxhKWKsqxCKH4K/ydd27BMw7q2K63qMo4SOAYnoAos0AAtcAXaoAMwuAeP4Bm8aA/ak/aqvS2sK9qy5wj8gPb+BRXUnAM=</latexit>


Stage<latexit sha1_base64="wHz7l9VSWzaz7OKX6lstQyVw/Po=">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</latexit><latexit sha1_base64="wHz7l9VSWzaz7OKX6lstQyVw/Po=">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</latexit><latexit sha1_base64="wHz7l9VSWzaz7OKX6lstQyVw/Po=">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</latexit><latexit sha1_base64="wHz7l9VSWzaz7OKX6lstQyVw/Po=">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</latexit>

Stage<latexit sha1_base64="wHz7l9VSWzaz7OKX6lstQyVw/Po=">AAACF3icdVBJSwMxGM3UrdZt1KOXYBF6kDIZa5dbwYvHinaBdiiZNG1DMwtJRihD/4UX/4oXD4p41Zv/xsx0BBV9EHi89215bsiZVJb1YeRWVtfWN/Kbha3tnd09c/+gI4NIENomAQ9Ez8WScubTtmKK014oKPZcTrvu7CLxu7dUSBb4N2oeUsfDE5+NGcFKS0OzHA/SIX0xcZ3YKlu18waqnmqSQpOGjRqosbhWeEIXQ7OolXrVrlRhUlNDNkqIXaucVSDKuoogQ2tovg9GAYk86ivCsZR9ZIXKibFQjHC6KAwiSUNMZnp4X1Mfe1Q6cXrSAp5oZQTHgdDPVzBVv3fE2JNy7rm60sNqKn97ifiX14/UuO7EzA8jRX2yXDSOOFQBTEKCIyYoUXyuCSaC6VshmWKBidJRFnQIXz+F/5OOXUZWGV3ZxWYpiyMPjsAxKAEEaqAJLkELtAEBd+ABPIFn4954NF6M12Vpzsh6DsEPGG+fCgeb/A==</latexit><latexit sha1_base64="wHz7l9VSWzaz7OKX6lstQyVw/Po=">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</latexit><latexit sha1_base64="wHz7l9VSWzaz7OKX6lstQyVw/Po=">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</latexit><latexit sha1_base64="wHz7l9VSWzaz7OKX6lstQyVw/Po=">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</latexit>

Optimal Dispatc

h

<latexit sha1_base64="2+8j9hLkXAHh+EANbQfx6hieGgo=">AAAB+XicbVBNSwMxEM3Wr1q/Vj16CRahp7Lbix4LevBmBfsB7VKyadqGJtmQzBbK0n/ixYMiXv0n3vw3pu0etPXBwOO9GWbmxVpwC0Hw7RW2tnd294r7pYPDo+MT//SsZZPUUNakiUhMJyaWCa5YEzgI1tGGERkL1o4ntwu/PWXG8kQ9wUyzSJKR4kNOCTip7/sPGrgkAt9xqwnQcd8vB9VgCbxJwpyUUY5G3//qDRKaSqaACmJtNww0RBkxwKlg81IvtUwTOiEj1nVUEclslC0vn+MrpwzwMDGuFOCl+nsiI9LamYxdpyQwtuveQvzP66YwvIkyrnQKTNHVomEqMCR4EQMecMMoiJkjhBrubsV0TAyh4MIquRDC9Zc3SatWDYNq+Fgr1yt5HEV0gS5RBYXoGtXRPWqgJqJoip7RK3rzMu/Fe/c+Vq0FL585R3/gff4AScqTVQ==</latexit><latexit sha1_base64="2+8j9hLkXAHh+EANbQfx6hieGgo=">AAAB+XicbVBNSwMxEM3Wr1q/Vj16CRahp7Lbix4LevBmBfsB7VKyadqGJtmQzBbK0n/ixYMiXv0n3vw3pu0etPXBwOO9GWbmxVpwC0Hw7RW2tnd294r7pYPDo+MT//SsZZPUUNakiUhMJyaWCa5YEzgI1tGGERkL1o4ntwu/PWXG8kQ9wUyzSJKR4kNOCTip7/sPGrgkAt9xqwnQcd8vB9VgCbxJwpyUUY5G3//qDRKaSqaACmJtNww0RBkxwKlg81IvtUwTOiEj1nVUEclslC0vn+MrpwzwMDGuFOCl+nsiI9LamYxdpyQwtuveQvzP66YwvIkyrnQKTNHVomEqMCR4EQMecMMoiJkjhBrubsV0TAyh4MIquRDC9Zc3SatWDYNq+Fgr1yt5HEV0gS5RBYXoGtXRPWqgJqJoip7RK3rzMu/Fe/c+Vq0FL585R3/gff4AScqTVQ==</latexit><latexit sha1_base64="2+8j9hLkXAHh+EANbQfx6hieGgo=">AAAB+XicbVBNSwMxEM3Wr1q/Vj16CRahp7Lbix4LevBmBfsB7VKyadqGJtmQzBbK0n/ixYMiXv0n3vw3pu0etPXBwOO9GWbmxVpwC0Hw7RW2tnd294r7pYPDo+MT//SsZZPUUNakiUhMJyaWCa5YEzgI1tGGERkL1o4ntwu/PWXG8kQ9wUyzSJKR4kNOCTip7/sPGrgkAt9xqwnQcd8vB9VgCbxJwpyUUY5G3//qDRKaSqaACmJtNww0RBkxwKlg81IvtUwTOiEj1nVUEclslC0vn+MrpwzwMDGuFOCl+nsiI9LamYxdpyQwtuveQvzP66YwvIkyrnQKTNHVomEqMCR4EQMecMMoiJkjhBrubsV0TAyh4MIquRDC9Zc3SatWDYNq+Fgr1yt5HEV0gS5RBYXoGtXRPWqgJqJoip7RK3rzMu/Fe/c+Vq0FL585R3/gff4AScqTVQ==</latexit><latexit sha1_base64="2+8j9hLkXAHh+EANbQfx6hieGgo=">AAAB+XicbVBNSwMxEM3Wr1q/Vj16CRahp7Lbix4LevBmBfsB7VKyadqGJtmQzBbK0n/ixYMiXv0n3vw3pu0etPXBwOO9GWbmxVpwC0Hw7RW2tnd294r7pYPDo+MT//SsZZPUUNakiUhMJyaWCa5YEzgI1tGGERkL1o4ntwu/PWXG8kQ9wUyzSJKR4kNOCTip7/sPGrgkAt9xqwnQcd8vB9VgCbxJwpyUUY5G3//qDRKaSqaACmJtNww0RBkxwKlg81IvtUwTOiEj1nVUEclslC0vn+MrpwzwMDGuFOCl+nsiI9LamYxdpyQwtuveQvzP66YwvIkyrnQKTNHVomEqMCR4EQMecMMoiJkjhBrubsV0TAyh4MIquRDC9Zc3SatWDYNq+Fgr1yt5HEV0gS5RBYXoGtXRPWqgJqJoip7RK3rzMu/Fe/c+Vq0FL585R3/gff4AScqTVQ==</latexit>

BoundaryProc

urement

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Optim

alD

ispatch

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Two-stage Unit Commitment<latexit sha1_base64="D2w9U7HAqcYT5QbbVNXN5716ugY=">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</latexit><latexit sha1_base64="D2w9U7HAqcYT5QbbVNXN5716ugY=">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</latexit><latexit sha1_base64="D2w9U7HAqcYT5QbbVNXN5716ugY=">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</latexit><latexit sha1_base64="D2w9U7HAqcYT5QbbVNXN5716ugY=">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</latexit>

Commitment Dispatch

First<latexit sha1_base64="UDemKxiffvMstRyBiDC3hBtHpBQ=">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</latexit><latexit sha1_base64="UDemKxiffvMstRyBiDC3hBtHpBQ=">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</latexit><latexit sha1_base64="UDemKxiffvMstRyBiDC3hBtHpBQ=">AAACF3icdVBLSwMxGMzWV62vqkcvwSL0ICW79rHeCoJ4rGAfsF1KNk3b0OyDJCuUZf+FF/+KFw+KeNWb/8ZsW0FFBwLDzHzJl/EizqRC6MPIrayurW/kNwtb2zu7e8X9g44MY0Fom4Q8FD0PS8pZQNuKKU57kaDY9zjtetOLzO/eUiFZGNyoWURdH48DNmIEKy0NipWkP7/EEWPPTVAF2ci27FNN5tDkvIaQVUsvmZAqHRRLWrHrVrUOs0zDtMyMWI3qWRWay6kSWKI1KL73hyGJfRoowrGUjoki5SZYKEY4TQv9WNIIkykeU0fTAPtUusl8pRSeaGUIR6HQJ1Bwrn6fSLAv5cz3dNLHaiJ/e5n4l+fEamS7CQuiWNGALB4axRyqEGYlwSETlCg+0wQTwfSukEywwETpKgu6hK+fwv9Jx6qYqGJeW6VmeVlHHhyBY1AGJmiAJrgCLdAGBNyBB/AEno1749F4MV4X0ZyxnDkEP2C8fQIViJwE</latexit><latexit sha1_base64="UDemKxiffvMstRyBiDC3hBtHpBQ=">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</latexit>

Second<latexit sha1_base64="qsD8Gw5wap5/tq6BKJ+jNEDXwGk=">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</latexit><latexit sha1_base64="qsD8Gw5wap5/tq6BKJ+jNEDXwGk=">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</latexit><latexit sha1_base64="qsD8Gw5wap5/tq6BKJ+jNEDXwGk=">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</latexit><latexit sha1_base64="qsD8Gw5wap5/tq6BKJ+jNEDXwGk=">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</latexit>

Third<latexit sha1_base64="6ooPtLawLcg+uGTGgm7sjzchTvw=">AAACF3icdVDNS8MwHE3n15xfVY9egkPYQUpbxzZvAy8eJ2xzsJWRpukWln6QpMIo/S+8+K948aCIV73535huK6jog8DjvfdLfnluzKiQpvmpldbWNza3ytuVnd29/QP98KgvooRj0sMRi/jARYIwGpKepJKRQcwJClxGbt3ZVe7f3hEuaBR25TwmToAmIfUpRlJJY91IR4tLhnziOqlpXLYadr1xbhqm2bRsKyd2s35Rz7pTyr1srFeLDCwysMhASyk5qmCFzlj/GHkRTgISSsyQEEPLjKWTIi4pZiSrjBJBYoRnaEKGioYoIMJJFytl8EwpHvQjrk4o4UL9PpGiQIh54KpkgORU/PZy8S9vmEi/5aQ0jBNJQrx8yE8YlBHMS4Ie5QRLNlcEYU7VrhBPEUdYqiorqoTip/B/0rcNyzSsG7varq3qKIMTcApqwAJN0AbXoAN6AIN78AiewYv2oD1pr9rbMlrSVjPH4Ae09y8gcpwK</latexit><latexit sha1_base64="6ooPtLawLcg+uGTGgm7sjzchTvw=">AAACF3icdVDNS8MwHE3n15xfVY9egkPYQUpbxzZvAy8eJ2xzsJWRpukWln6QpMIo/S+8+K948aCIV73535huK6jog8DjvfdLfnluzKiQpvmpldbWNza3ytuVnd29/QP98KgvooRj0sMRi/jARYIwGpKepJKRQcwJClxGbt3ZVe7f3hEuaBR25TwmToAmIfUpRlJJY91IR4tLhnziOqlpXLYadr1xbhqm2bRsKyd2s35Rz7pTyr1srFeLDCwysMhASyk5qmCFzlj/GHkRTgISSsyQEEPLjKWTIi4pZiSrjBJBYoRnaEKGioYoIMJJFytl8EwpHvQjrk4o4UL9PpGiQIh54KpkgORU/PZy8S9vmEi/5aQ0jBNJQrx8yE8YlBHMS4Ie5QRLNlcEYU7VrhBPEUdYqiorqoTip/B/0rcNyzSsG7varq3qKIMTcApqwAJN0AbXoAN6AIN78AiewYv2oD1pr9rbMlrSVjPH4Ae09y8gcpwK</latexit><latexit sha1_base64="6ooPtLawLcg+uGTGgm7sjzchTvw=">AAACF3icdVDNS8MwHE3n15xfVY9egkPYQUpbxzZvAy8eJ2xzsJWRpukWln6QpMIo/S+8+K948aCIV73535huK6jog8DjvfdLfnluzKiQpvmpldbWNza3ytuVnd29/QP98KgvooRj0sMRi/jARYIwGpKepJKRQcwJClxGbt3ZVe7f3hEuaBR25TwmToAmIfUpRlJJY91IR4tLhnziOqlpXLYadr1xbhqm2bRsKyd2s35Rz7pTyr1srFeLDCwysMhASyk5qmCFzlj/GHkRTgISSsyQEEPLjKWTIi4pZiSrjBJBYoRnaEKGioYoIMJJFytl8EwpHvQjrk4o4UL9PpGiQIh54KpkgORU/PZy8S9vmEi/5aQ0jBNJQrx8yE8YlBHMS4Ie5QRLNlcEYU7VrhBPEUdYqiorqoTip/B/0rcNyzSsG7varq3qKIMTcApqwAJN0AbXoAN6AIN78AiewYv2oD1pr9rbMlrSVjPH4Ae09y8gcpwK</latexit><latexit sha1_base64="6ooPtLawLcg+uGTGgm7sjzchTvw=">AAACF3icdVDNS8MwHE3n15xfVY9egkPYQUpbxzZvAy8eJ2xzsJWRpukWln6QpMIo/S+8+K948aCIV73535huK6jog8DjvfdLfnluzKiQpvmpldbWNza3ytuVnd29/QP98KgvooRj0sMRi/jARYIwGpKepJKRQcwJClxGbt3ZVe7f3hEuaBR25TwmToAmIfUpRlJJY91IR4tLhnziOqlpXLYadr1xbhqm2bRsKyd2s35Rz7pTyr1srFeLDCwysMhASyk5qmCFzlj/GHkRTgISSsyQEEPLjKWTIi4pZiSrjBJBYoRnaEKGioYoIMJJFytl8EwpHvQjrk4o4UL9PpGiQIh54KpkgORU/PZy8S9vmEi/5aQ0jBNJQrx8yE8YlBHMS4Ie5QRLNlcEYU7VrhBPEUdYqiorqoTip/B/0rcNyzSsG7varq3qKIMTcApqwAJN0AbXoAN6AIN78AiewYv2oD1pr9rbMlrSVjPH4Ae09y8gcpwK</latexit>

Clearing<latexit sha1_base64="+ohb/Gfdtg8gV/El+kq/Isto3ao=">AAAB73icbVA9SwNBEJ2LXzF+RW0Em8UgWIW7NLEMpLGMYD4gOcLeZi5Zsrd37u4J4cifsLFQxNa/Y+e/cZNcoYkPBh7vzTAzL0gE18Z1v53C1vbO7l5xv3RweHR8Uj496+g4VQzbLBax6gVUo+AS24Ybgb1EIY0Cgd1g2lz43SdUmsfywcwS9CM6ljzkjBor9ZoCqeJyPCxX3Kq7BNkkXk4qkKM1LH8NRjFLI5SGCap133MT42dUGc4EzkuDVGNC2ZSOsW+ppBFqP1veOyfXVhmRMFa2pCFL9fdERiOtZ1FgOyNqJnrdW4j/ef3UhLd+xmWSGpRstShMBTExWTxPRlwhM2JmCWWK21sJm1BFmbERlWwI3vrLm6RTq3pu1buvVRoXeRxFuIQruAEP6tCAO2hBGxgIeIZXeHMenRfn3flYtRacfOYc/sD5/AH0TI/D</latexit><latexit sha1_base64="+ohb/Gfdtg8gV/El+kq/Isto3ao=">AAAB73icbVA9SwNBEJ2LXzF+RW0Em8UgWIW7NLEMpLGMYD4gOcLeZi5Zsrd37u4J4cifsLFQxNa/Y+e/cZNcoYkPBh7vzTAzL0gE18Z1v53C1vbO7l5xv3RweHR8Uj496+g4VQzbLBax6gVUo+AS24Ybgb1EIY0Cgd1g2lz43SdUmsfywcwS9CM6ljzkjBor9ZoCqeJyPCxX3Kq7BNkkXk4qkKM1LH8NRjFLI5SGCap133MT42dUGc4EzkuDVGNC2ZSOsW+ppBFqP1veOyfXVhmRMFa2pCFL9fdERiOtZ1FgOyNqJnrdW4j/ef3UhLd+xmWSGpRstShMBTExWTxPRlwhM2JmCWWK21sJm1BFmbERlWwI3vrLm6RTq3pu1buvVRoXeRxFuIQruAEP6tCAO2hBGxgIeIZXeHMenRfn3flYtRacfOYc/sD5/AH0TI/D</latexit><latexit sha1_base64="+ohb/Gfdtg8gV/El+kq/Isto3ao=">AAAB73icbVA9SwNBEJ2LXzF+RW0Em8UgWIW7NLEMpLGMYD4gOcLeZi5Zsrd37u4J4cifsLFQxNa/Y+e/cZNcoYkPBh7vzTAzL0gE18Z1v53C1vbO7l5xv3RweHR8Uj496+g4VQzbLBax6gVUo+AS24Ybgb1EIY0Cgd1g2lz43SdUmsfywcwS9CM6ljzkjBor9ZoCqeJyPCxX3Kq7BNkkXk4qkKM1LH8NRjFLI5SGCap133MT42dUGc4EzkuDVGNC2ZSOsW+ppBFqP1veOyfXVhmRMFa2pCFL9fdERiOtZ1FgOyNqJnrdW4j/ef3UhLd+xmWSGpRstShMBTExWTxPRlwhM2JmCWWK21sJm1BFmbERlWwI3vrLm6RTq3pu1buvVRoXeRxFuIQruAEP6tCAO2hBGxgIeIZXeHMenRfn3flYtRacfOYc/sD5/AH0TI/D</latexit><latexit sha1_base64="+ohb/Gfdtg8gV/El+kq/Isto3ao=">AAAB73icbVA9SwNBEJ2LXzF+RW0Em8UgWIW7NLEMpLGMYD4gOcLeZi5Zsrd37u4J4cifsLFQxNa/Y+e/cZNcoYkPBh7vzTAzL0gE18Z1v53C1vbO7l5xv3RweHR8Uj496+g4VQzbLBax6gVUo+AS24Ybgb1EIY0Cgd1g2lz43SdUmsfywcwS9CM6ljzkjBor9ZoCqeJyPCxX3Kq7BNkkXk4qkKM1LH8NRjFLI5SGCap133MT42dUGc4EzkuDVGNC2ZSOsW+ppBFqP1veOyfXVhmRMFa2pCFL9fdERiOtZ1FgOyNqJnrdW4j/ef3UhLd+xmWSGpRstShMBTExWTxPRlwhM2JmCWWK21sJm1BFmbERlWwI3vrLm6RTq3pu1buvVRoXeRxFuIQruAEP6tCAO2hBGxgIeIZXeHMenRfn3flYtRacfOYc/sD5/AH0TI/D</latexit>

Scheduling<latexit sha1_base64="FVh0GCzLTttjwlu2xa7KpZWztno=">AAAB8XicbVDLSsNAFL3xWeur6kZwM1gEVyXpRpcFNy4r2ge2oUwmN+3QySTMTIQS+hduXCji1r9x5984bbPQ1gMDh3PuYe49QSq4Nq777aytb2xubZd2yrt7+weHlaPjtk4yxbDFEpGobkA1Ci6xZbgR2E0V0jgQ2AnGNzO/84RK80Q+mEmKfkyHkkecUWOlx3s2wjCz4eGgUnVr7hxklXgFqUKB5qDy1Q8TlsUoDRNU657npsbPqTKcCZyW+5nGlLIxHWLPUklj1H4+33hKLqwSkihR9klD5urvRE5jrSdxYCdjakZ62ZuJ/3m9zETXfs5lmhmUbPFRlAliEjI7n4RcITNiYgllittdCRtRRZmxJZVtCd7yyaukXa95bs27q1cbp0UdJTiDc7gED66gAbfQhBYwkPAMr/DmaOfFeXc+FqNrTpE5gT9wPn8AmmmQuA==</latexit><latexit sha1_base64="FVh0GCzLTttjwlu2xa7KpZWztno=">AAAB8XicbVDLSsNAFL3xWeur6kZwM1gEVyXpRpcFNy4r2ge2oUwmN+3QySTMTIQS+hduXCji1r9x5984bbPQ1gMDh3PuYe49QSq4Nq777aytb2xubZd2yrt7+weHlaPjtk4yxbDFEpGobkA1Ci6xZbgR2E0V0jgQ2AnGNzO/84RK80Q+mEmKfkyHkkecUWOlx3s2wjCz4eGgUnVr7hxklXgFqUKB5qDy1Q8TlsUoDRNU657npsbPqTKcCZyW+5nGlLIxHWLPUklj1H4+33hKLqwSkihR9klD5urvRE5jrSdxYCdjakZ62ZuJ/3m9zETXfs5lmhmUbPFRlAliEjI7n4RcITNiYgllittdCRtRRZmxJZVtCd7yyaukXa95bs27q1cbp0UdJTiDc7gED66gAbfQhBYwkPAMr/DmaOfFeXc+FqNrTpE5gT9wPn8AmmmQuA==</latexit><latexit sha1_base64="FVh0GCzLTttjwlu2xa7KpZWztno=">AAAB8XicbVDLSsNAFL3xWeur6kZwM1gEVyXpRpcFNy4r2ge2oUwmN+3QySTMTIQS+hduXCji1r9x5984bbPQ1gMDh3PuYe49QSq4Nq777aytb2xubZd2yrt7+weHlaPjtk4yxbDFEpGobkA1Ci6xZbgR2E0V0jgQ2AnGNzO/84RK80Q+mEmKfkyHkkecUWOlx3s2wjCz4eGgUnVr7hxklXgFqUKB5qDy1Q8TlsUoDRNU657npsbPqTKcCZyW+5nGlLIxHWLPUklj1H4+33hKLqwSkihR9klD5urvRE5jrSdxYCdjakZ62ZuJ/3m9zETXfs5lmhmUbPFRlAliEjI7n4RcITNiYgllittdCRtRRZmxJZVtCd7yyaukXa95bs27q1cbp0UdJTiDc7gED66gAbfQhBYwkPAMr/DmaOfFeXc+FqNrTpE5gT9wPn8AmmmQuA==</latexit><latexit sha1_base64="FVh0GCzLTttjwlu2xa7KpZWztno=">AAAB8XicbVDLSsNAFL3xWeur6kZwM1gEVyXpRpcFNy4r2ge2oUwmN+3QySTMTIQS+hduXCji1r9x5984bbPQ1gMDh3PuYe49QSq4Nq777aytb2xubZd2yrt7+weHlaPjtk4yxbDFEpGobkA1Ci6xZbgR2E0V0jgQ2AnGNzO/84RK80Q+mEmKfkyHkkecUWOlx3s2wjCz4eGgUnVr7hxklXgFqUKB5qDy1Q8TlsUoDRNU657npsbPqTKcCZyW+5nGlLIxHWLPUklj1H4+33hKLqwSkihR9klD5urvRE5jrSdxYCdjakZ62ZuJ/3m9zETXfs5lmhmUbPFRlAliEjI7n4RcITNiYgllittdCRtRRZmxJZVtCd7yyaukXa95bs27q1cbp0UdJTiDc7gED66gAbfQhBYwkPAMr/DmaOfFeXc+FqNrTpE5gT9wPn8AmmmQuA==</latexit>

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G1 G2 G3 Hour1 ON ON ON Hour2 ON ON OFF Hour3 OFF ON OFF … … … …

Figure 3.1. Proposed coordinated market paradigm.

influences the market operation in the distribution level, and the influence will propagate

to the transmission level [55]. Since the ISO monitors the T-D coordination, the topology

changes in the active distribution network (ADN) should be taken into consideration in the

ISO’s operation. Besides, utilities are using an hourly decision-making process to evalu-

ate and perform the reconfiguration [56]. Considering that the reconfiguration cannot be

changed frequently in one day and should be determined in a day-ahead manner [56], in this

work, we add the network reconfiguration decisions in the first stage together with the UC

decisions.

44

In summary, we propose a market paradigm for the T-D coordinated power system,

which is depicted in Figure 3.1. Mathematically, the ISO needs to solve a mixed-integer

linear programming (MILP) model for transmission-level UC with distribution network re-

configurations, when DSOs need to solve convex programming models for distribution-level

ED. In the first stage, the ISO performs the transmission UC and determines the distribution

network reconfiguration, then delivers the information to the subsequent stages. Then, the

ED operation is carried out in the second stage and delivers the optimal boundary offer to

the third stage. The structure of the first and second stages forms a two-stage transmission-

level stochastic UC [57]. Finally, DSOs perform their ED operations in the third stage based

on the determined configuration from the first stage and the energy offer from the second

stage.

For the solution strategy, there has been a significant number of studies towards the

efficient solution of large-scale SP problems. One of the most commonly used strategies is

the Benders decomposition or the L-shaped method [58]. For multi-stage SP as in this work,

scenario-wise decomposition like the progressive hedging [59] and stage-wise decomposition

like the approximate dynamic programming [60] are discussed in the literature. Particu-

larly for the dynamic programming category, stochastic dual dynamic integer programming

(SDDiP) has been an effective method for multi-stage stochastic optimization [61], with

various recent applications on power systems. J. Zhou et al. [62] proposed a SDDiP model

for multi-stage stochastic UC and enhanced the algorithm by a regularized level approxima-

tion for Lagrangian cuts. M. Hjelmeland et al. [63] adopted SDDiP solving a hydropower

scheduling problem with binary expansion on state variables and found that the strength-

ened Benders cut presented the highest performance score. T. Ding et al. [64] leveraged the

SDDiP method in the distribution system planning with a multi-stage nested decomposition

algorithm. Though it is well recognized that SDDiP attains fast convergence and accuracy,

it could not solve the proposed T-D coordinated market operation. The reasons are twofold:

1) Each time stage in the SDDiP needs to assume the complete information of both trans-

mission and distribution networks, which compromises the stakeholders’ privacy; 2) There is

45

no evidence that the SDDiP retains global convergence if recourse problems are nonlinear.

Thus, since our work is both scenario-wise and stage-wise complex, we employ the nested

L-shaped method [58] for the proposed T-D coordinated market operation. Furthermore, we

enhance the conventional nested L-shaped method and algorithmically extend it to a multi-

stage mixed-integer convex programming. To the best of the authors’ knowledge, this is

the first work implementing a practical T-D market operation considering accurate network

modeling/reconfiguration and uncertainties.

Based on the state-of-the-art research, the main contribution of this work is summarized

below.

• We formulate the T-D coordinated UC&ED problem considering the distribution net-

work reconfiguration. System-independent scenario sets capture the uncertainty of

renewable generation and elastic demand for each system.

• We propose a market paradigm for the T-D coordinated UC&ED. This paradigm

considers the coordination between different market stakeholders while protecting their

confidentiality, which ISOs can easily adopt.

• We extend the generalized Benders decomposition method to a nested and stochastic

version, which can efficiently solve the proposed T-D coordinated market operation

model. We also theoretically prove and analyze the algorithmic convergence.

3.3. Mathematical Formulation of T-D Coordinated Market Operations

We detail the three-stage stochastic formulation and the convexified AC branch flow

for the distribution part in this section. The overall formulation of the stochastic T-D

coordinated UC & ED is separated and distributed in the following subsections. Note that

the DC power flow is adopted in the transmission level in order to mimic the industrial

practice. In contrast, the second-order cone programming (SOCP)-based convexified AC

branch flow formulation is adopted in the distribution network to retain accuracy.

46

3.3.1. The First Stage: ISO’s UC Problem

The first stage solves a conventional transmission UC problem with the optimal distri-

bution network reconfiguration, which is performed by the ISO.

K1 = minuTg,h,d

Tg,h,

iTg,h,zq,h,c

H

∑h

SUTg u

Tg,h + SD

Tg d

Tg,h

+EωT K2(x∗1, ω

T ), (3-1)

subject to

uTg,h + dTg,h ≤ 1, ∀gT ∈ GT ,∀h ∈H, (3-1a)

uTg,h − dTg,h = i

Tg,h − i

Tg,h−1, ∀gT ∈ GT ,∀h ∈H, (3-1b)

∑h

h−MOg+1uTg,h ≤ i

Tg,h, ∀gT ∈ GT ,∀h ∈H, (3-1c)

∑h

h−MDg+1dTg,h ≤ 1 − iTg,h, ∀gT ∈ GT ,∀h ∈H, (3-1d)

uTg,h, dTg,h, i

Tg,h, zq,h,c ∈ 0,1, (3-1e)

where K2(x∗1, ωT ) denotes the second-stage recourse and x∗1 = [uT∗g,h, d

T∗g,h, i

T∗g,h]. “*” stands for

obtained and fixed decisions. Constraints (3-1a) and (3-1b) enforce the binary exclusiveness

of the start-up and shut-down indicators; constraints (3-1c) and (3-1d) model the minimum

ON/OFF time; constraint (3-1e) indicates the binary nature of the variables. Note that there

is no distribution configuration variable, i.e., zq,h,c, in the objective function and constraints,

which is determined based on the information sent back from the DSO’s stage.

3.3.2. The Second Stage: ISO’s ED Problem

The second stage formulates a transmission-level ED problem based on the UC decisions

47

delivered from the first stage, which is detailed in (3-2). For each ωT :

K2(x∗1, ω

T ) = minpTg,h,sT

`,h

H

∑h

GT

∑gTfT (pTg,h) +

LT

∑`TPCT

` sT`,h

+∑c

wc ⋅EωDcK3(x

∗2,y

∗, ωDc ) (3-2)

subject to

GT

∑gT ∣C(gT )=nT

pTg,h −I

∑ic∣C(ic)=nT

pic,h + ∑rT ∣C(r)=nT

pTr,h−

FT

∑fT ∣S(fT )=nT

pTf,h +FT

∑fT ∣N(fT )=nT

pTf,h =

L

∑`T ∣C(`T )=nT

LT`,h(ωT ) − sT`,h , ∀nT ,∀h, (3-2a)

pTf,h =X−1fT [δ

Tn∣S(nT )=fT ,h − δ

Tn∣N(nT )=fT ,h], ∀fT ,∀h, (3-2b)

− P T,maxf,h ≤ pTf,h ≤ P

T,maxf,h , ∀fT ,∀h, (3-2c)

iT,∗g,hPT,ming,h ≤ pTg,h ≤ i

T,∗g,hP

T,maxg,h , ∀gT ,∀h, (3-2d)

0 ≤ pTr,h ≤ PT,maxr,h (ωT ), ∀rT ,∀h, (3-2e)

pTg,h − pTg,h−1 ≤ RU

Tg (1 − uT,∗g,h ) + u

T,∗g,hP

T,ming,h , ∀gT ,∀h, (3-2f)

pTg,h−1 − pTg,h ≤ RD

Tg (1 − d

T∗g,h) + d

T,∗g,hP

T,ming,h , ∀gT ,∀h, (3-2g)

where K3(x∗2,y∗, ωDc ) denotes the third-stage recourse, x∗2 = [z∗q,h,c], and y∗ = [p∗ic,h]. wc

stands for the weight of different distribution systems pertaining to the importance of the

corresponding system. For instance, hospitals and military regions require higher priority of

electricity supply, which results in higher weight wc in this model (wc ∈ (0,1] and ∑wc = 1).

We also adopt the piecewise-linear thermal generation cost function as fT (⋅) and enable

load shedding to retain feasibility. Constraint (3-2a) enforces the active power balance, and

constraint (3-2b) represents the DC power flow equation in the transmission lines. Line flow

constraints and active power generation constraints for thermal generators are described in

constraints (3-2c) and (3-2d), respectively. In constraint (3-2e), the renewable generation is

48

limited within the uncertain upper bound w.r.t. the scenario index ωT . For brevity, we only

give the scenario index to uncertain parameters, but the variables should be associated with

the scenario index as well. Besides, constraints (3-2f) and (3-2g) represent the ramp-up/down

limits, respectively.

3.3.3. The Third Stage: DSO’s ED Problem

From DSOs’ perspective, the third stage establishes a distribution-level ED, as shown

in (3-3). Since DER generators are fast-responsive, their UC decisions are not considered.

However, the distribution network is reconfigurable [56] and controlled by the ISO in the

proposed T-D coordinated market operation. Note that ωDc for each c in the third stage is

independent of each other and ωT in the second stage. For each c and ωDc :

K3(x∗2,y

∗, ωDc ) = minpDg,h,c

,sD`,h,c

H

∑h

GD

∑gDfD(pDg,h,c)

+LD

∑`DPCD

`,csD`,h,c (3-3)

subject to

PD,ming,h,c ≤ pDg,h,c ≤ P

D,maxg,h,c , ∀gD,∀h, (3-3a)

−RDDg,c ≤ p

Dg,h+1,c − p

Dg,h,c ≤ RU

Dg,c, ∀gD,∀h, (3-3b)

0 ≤ pDr,h,c ≤ PD,maxr,h,c (ωDc ), ∀rD,∀h,∀ωDc , (3-3c)

0 ≤ yDmn,h,c ≤ xDmn,h,cY

D,maxmn,h,c , ∀(m,n) ∈ FD

Q ,∀h, (3-3d)

0 ≤ yDmn,h,c ≤ YD,maxmn,h,c , ∀(m,n) ∈ FD/FD

Q ,∀h, (3-3e)

(pDmn,h,c)2 + (qDmn,h,c)

2 ≤ yDmn,h,cvm,h,c, ∀(m,n) ∈ FD,∀h, (3-3f)

xDmn,h,c ≥ 0, ∀(m,n) ∈ FD,∀h, (3-3g)

xDmn,h,c = 0, ∀n ∈ ADT ,∀h, (3-3h)

xDmn,h,c + xDnm,h,c = 1, ∀n ∈ FD/FD

Q ,∀h, (3-3i)

49

xDmn,h,c + xDnm,h,c = z

∗q,h,c, ∀n ∈ FD

Q ,∀h, (3-3j)

∑n∶(m,n)∈FD

xmn,h,c = 1, ∀m ∈ AD/ADT ,∀h, (3-3k)

∀nD,∀h ∶

pDn,h,c = ∑k∶n→k

pDnk,h,c − ∑j∶j→n

(pDnj,h,c −RnjyDnj,h,c), (3-3l)

qDn,h,c = ∑k∶n→k

qDnk,h,c − ∑j∶j→n

(qDnj,h,c −XnjyDnj,h,c), (3-3m)

vn,h,c − vm,h,c ≤M(1 − xDnm,h,c) + (R2mn +X

2mn)y

Dmn,h,c−

2(RmnpDmn,h,c +Xmnq

Dmn,h,c), (3-3n)

vn,h,c − vm,h,c ≥ −M(1 − xDnm,h,c) + (R2mn +X

2mn)y

Dmn,h,c−

2(RmnpDmn,h,c +Xmnq

Dmn,h,c), (3-3o)

pDn,h,c = ∑gD ∣C(gD)=n

pDg,h,c + ∑ic∣C(ic)=n

p∗ic,h+

∑rD ∣C(rD)=n

pDr,h,c − ∑`D ∣C(`D)=n

LD`,h,c(ωDc ) − sD`,h,c, (3-3p)

qDn,h,c = ∑gD ∣C(gD)=n

qDg,h,c + ∑ic∣C(ic)=n

qic,h+

∑rDD∣C(rD)=n

qDr,h,c − ∑`D ∣C(`D)=n

qD`,h,c(ωDc ), (3-3q)

V minn,h,c ≤ vn,h,c ≤ V

maxn,h,c, (3-3r)

where fD(⋅) represents the piecewise-linear cost function. The constraints for non-reconfigurable

branches are identical to the transmission lines in the second-stage problem, while the con-

straints for reconfigurable branches are with the reconfiguration decisions, as shown in con-

straints (3-3d) and (3-3e).

We cast a SOCP relaxation for power flow modeling. Constraint (3-3f) is the convex SOC

constraint. Note that we neglect shunt impedances for simplicity. k, j,m are notations for

connected buses. We also assume that the distribution networks can well control the voltage

and reactive power performance. Hence, we only consider the reactive power in the distribu-

tion systems and no reactive power exchange will happen [65]. Future works include that the

50

distribution systems can provide reactive power support to the transmission system, where

our proposed model can also be adopted. Also, note that we employ the widely-adopted

branch flow model for the SOCP relaxation [66] of the AC power flow. However, other

SOCP relaxation techniques, including the bus injection model and its variants [67], could

also be adopted in the proposed framework, given that the feasibility region is convex.

To ensure the distribution network’s radiality, we also enforce constraints (3-3g) - (3-3k).

Note that xmn are continuous variables and will be either zero or one to keep arbores-

cence [68], which retains model (3-3)’s convexity. Upon using this, we restrict the distribution

network information in the third stage, which protects the confidentiality of distribution util-

ities. Other constraints are with the typical branch flow model under SOCP relaxation [69].

3.4. Decomposition-based Solution Strategy for the Three-Stage Market Op-

eration

In this section, we provide an efficient solution strategy towards the multi-scale and

multi-stage SP problem for T-D coordinated market operation. As the original formulation

follows a stage-decomposable structure, it can be readily tackled by the L-shaped method, as

adopted in multiple works [58, 70]. However, since the proposed T-D coordinated hierarchy

is a three-stage problem, the L-shaped method needs to be nested. In this work, we tailor

a generalized nested decomposition based on the nested L-shaped method to facilitate the

solution upon different and individual scenario sets. To be more concise, we use the compact

form for equations in this section, and similarly, we only put the scenario indices (ωT ) and

(ωDc ) with the uncertain parameters.

3.4.1. ISO’s Master Problem

The ISO’s UC problem (3-1), i.e., the first-stage problem, serves as the master problem.

The complicating variables in the master problem include the unit commitment decisions, i.e.

ug,h, dg,h, ig,h (denoted by vector x1), and the reconfiguration decisions, i.e. zq,h,c (denoted

51

by vector x2). A compact form for the master problem (3-1) is represented in (3-4).

M = minx1,x2

c⊺1x1 + 0⊺x2 + S∗t , (3-4)

subject to

A1x1 +B1x2 ≤ b1, (3-4a)

A2x1 +B2x2 = b2, (3-4b)

S∗t = maxo∈O

αot + (βot1)⊺x1 + (βot2)

⊺x2, (3-4c)

where S∗t denotes the maximum of cuts returned from the second-stage subproblem, and

constraint (3-4c) includes the optimality cuts returned from the second-stage problem with

the information of the second and third stages for all iterations.

3.4.2. ISO’s Subproblem

The second-stage problem (2), i.e., the ISO’s subproblem (St), models the transmission-

level ED, as shown in (3-5).

∀ωT ∶ St(ωT ) = min

yc⊺2y + S

∗d , (3-5)

subject to

K1y = r1(ωT ) ∶ γ, (3-5a)

H1x∗1 +A3y ≤ b3 ∶ φ, (3-5b)

S∗d = maxo∈O,c

wc ⋅ [αod + (βod1)

⊺y + (βod2)⊺x∗2] ∶ µ, (3-5c)

where x∗1 and x∗2 are delivered from the master problem, y is the second-stage variable

vector, including the generation dispatch, load curtailment and line flows, etc. γ, φ and

µ are the dual vectors, whereas auxiliary variable S∗d and constraint (3-5c) formulate the

52

relaxed counterpart of the third-stage problem. To update the optimality cut (3-4c) in the

master problem, we calculate the subgradients as follows.

αot =∑ωT

PrωT γ⊺r1(ωT ) +φ⊺b3 −µ

⊺[αod + (βod2)⊺x∗2 − S

∗d /wc]

βot1 = −∑ωT

PrωT H⊺1φ; βot2 = −∑

ωT

PrωTβ⊺d2µ.

3.4.3. DSO’s Subproblem

The third-stage problem (3-3), i.e., the DSO’s subproblem (Sd), which is a convex SOCP

model, formulates the distribution-level ED, as shown in the compact form (3-6). Without

loss of generality, we group the affine constraints with the SOC one in constraint (3-6a), as

affine functions are a special case of SOC when K2 is a zero matrix.

∀c,∀ωDc ∶ Sd(wc, ωDc ) = min

zc⊺3z, (3-6)

subject to

∥H2y∗ +K2z + e∥2 ≤ q⊺y∗ + p⊺z +H3x

∗2 + r2(ω

Dc ), (3-6a)

where x∗2 and y∗ are decisions delivered from the first-stage problem and the second-stage

problem, respectively. Then we write the dual form of (3-6) in (3-7).

Sduald (wc, ω

Dc ) = max

u,v[u⊺(H2y

∗ + e) − v(q⊺y∗ +H⊺3x

∗2

+ r2(ωDc ))], (3-7)

53

subject to

K⊺2u + v⊺p = c3, (3-7a)

∥u∥2 ≤ v, (3-7b)

where u and v are the dual vectors of the Euclidean norm and the SOC, respectively. Note

that this dual problem is also a convex SOC and only satisfies the weak duality. However,

given the assumption that the nodal voltage constraint is not binding, which is a mild

assumption in the realistic power system, there exists an interior in the primal and dual

feasibility regions and hence Slater’s condition holds, which means the strong duality can be

retained [71]. Then the subgradients can be calculated for the optimality cut in the ISO’s

subproblem.

αod =∑ωDc

PrωDc[u⊺e − v ⋅ r2(ω

Dc )];

βod1 =∑ωDc

PrωDc[u⊺H2 − v ⋅ q]; βod2 = −∑

ωDc

PrωDcv ⋅H⊺

3;

3.4.4. Generalized Nested Decomposition Algorithm

To cope with the proposed T-D coordinated hierarchy, we devise a generalized nested

decomposition (GND) algorithm. The general procedure is demonstrated in Figure 3.2. In

particular, with an initial feasible point of the complicating variables determined from the

initial master problem, the problem in each stage can deliver the optimal complicating vari-

ables back to the previous stage problem via creating the lower bounding affine cuts by

subgradients. If any recourse is infeasible, a feasibility cut will be returned to the upper

stages and repeat the loop in Figure 3.2 [72]. For more details about the feasibility check,

please see Appendix B. These cuts formulate the relaxed counterparts of the respective sub-

problems and hence decompose the overall problem. Each subsystem in the second-stage and

54

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No<latexit sha1_base64="vrrdIFAq92Bq6xRkyKxYFuzfhnM=">AAAB6XicbVA9SwNBEJ2LXzF+RS1tFoOQKtyl0TJgYyVRTCIkR9jbzCVL9naP3T0hHPkHNhaK2PqP7Pw3bj4KTXww8Hhvhpl5USq4sb7/7RU2Nre2d4q7pb39g8Oj8vFJ26hMM2wxJZR+jKhBwSW2LLcCH1ONNIkEdqLx9czvPKE2XMkHO0kxTOhQ8pgzap10f6v65Ypf8+cg6yRYkgos0eyXv3oDxbIEpWWCGtMN/NSGOdWWM4HTUi8zmFI2pkPsOippgibM55dOyYVTBiRW2pW0ZK7+nshpYswkiVxnQu3IrHoz8T+vm9n4Ksy5TDOLki0WxZkgVpHZ22TANTIrJo5Qprm7lbAR1ZRZF07JhRCsvrxO2vVa4NeCu3qlUV3GUYQzOIcqBHAJDbiBJrSAQQzP8Apv3th78d69j0VrwVvOnMIfeJ8/a5yNLw==</latexit><latexit sha1_base64="vrrdIFAq92Bq6xRkyKxYFuzfhnM=">AAAB6XicbVA9SwNBEJ2LXzF+RS1tFoOQKtyl0TJgYyVRTCIkR9jbzCVL9naP3T0hHPkHNhaK2PqP7Pw3bj4KTXww8Hhvhpl5USq4sb7/7RU2Nre2d4q7pb39g8Oj8vFJ26hMM2wxJZR+jKhBwSW2LLcCH1ONNIkEdqLx9czvPKE2XMkHO0kxTOhQ8pgzap10f6v65Ypf8+cg6yRYkgos0eyXv3oDxbIEpWWCGtMN/NSGOdWWM4HTUi8zmFI2pkPsOippgibM55dOyYVTBiRW2pW0ZK7+nshpYswkiVxnQu3IrHoz8T+vm9n4Ksy5TDOLki0WxZkgVpHZ22TANTIrJo5Qprm7lbAR1ZRZF07JhRCsvrxO2vVa4NeCu3qlUV3GUYQzOIcqBHAJDbiBJrSAQQzP8Apv3th78d69j0VrwVvOnMIfeJ8/a5yNLw==</latexit><latexit sha1_base64="vrrdIFAq92Bq6xRkyKxYFuzfhnM=">AAAB6XicbVA9SwNBEJ2LXzF+RS1tFoOQKtyl0TJgYyVRTCIkR9jbzCVL9naP3T0hHPkHNhaK2PqP7Pw3bj4KTXww8Hhvhpl5USq4sb7/7RU2Nre2d4q7pb39g8Oj8vFJ26hMM2wxJZR+jKhBwSW2LLcCH1ONNIkEdqLx9czvPKE2XMkHO0kxTOhQ8pgzap10f6v65Ypf8+cg6yRYkgos0eyXv3oDxbIEpWWCGtMN/NSGOdWWM4HTUi8zmFI2pkPsOippgibM55dOyYVTBiRW2pW0ZK7+nshpYswkiVxnQu3IrHoz8T+vm9n4Ksy5TDOLki0WxZkgVpHZ22TANTIrJo5Qprm7lbAR1ZRZF07JhRCsvrxO2vVa4NeCu3qlUV3GUYQzOIcqBHAJDbiBJrSAQQzP8Apv3th78d69j0VrwVvOnMIfeJ8/a5yNLw==</latexit><latexit sha1_base64="vrrdIFAq92Bq6xRkyKxYFuzfhnM=">AAAB6XicbVA9SwNBEJ2LXzF+RS1tFoOQKtyl0TJgYyVRTCIkR9jbzCVL9naP3T0hHPkHNhaK2PqP7Pw3bj4KTXww8Hhvhpl5USq4sb7/7RU2Nre2d4q7pb39g8Oj8vFJ26hMM2wxJZR+jKhBwSW2LLcCH1ONNIkEdqLx9czvPKE2XMkHO0kxTOhQ8pgzap10f6v65Ypf8+cg6yRYkgos0eyXv3oDxbIEpWWCGtMN/NSGOdWWM4HTUi8zmFI2pkPsOippgibM55dOyYVTBiRW2pW0ZK7+nshpYswkiVxnQu3IrHoz8T+vm9n4Ksy5TDOLki0WxZkgVpHZ22TANTIrJo5Qprm7lbAR1ZRZF07JhRCsvrxO2vVa4NeCu3qlUV3GUYQzOIcqBHAJDbiBJrSAQQzP8Apv3th78d69j0VrwVvOnMIfeJ8/a5yNLw==</latexit>

Optimality<latexit sha1_base64="8Hdb0teegmaIsq7//z8KOJ22Q+w=">AAAB+nicbVBNS8NAEN34WetXqkcvi0XoqSS96LHgxZsV7Ae0oWy2m3bp7ibsTpQQ+1O8eFDEq7/Em//GbZuDtj4YeLw3w8y8MBHcgOd9OxubW9s7u6W98v7B4dGxWznpmDjVlLVpLGLdC4lhgivWBg6C9RLNiAwF64bT67nffWDa8FjdQ5awQJKx4hGnBKw0dCsDJpNJfpsAl0RwyGZDt+rVvQXwOvELUkUFWkP3azCKaSqZAiqIMX3fSyDIiQZOBZuVB6lhCaFTMmZ9SxWRzAT54vQZvrDKCEextqUAL9TfEzmRxmQytJ2SwMSsenPxP6+fQnQV5FwlKTBFl4uiVGCI8TwHPOKaURCZJYRqbm/FdEI0oWDTKtsQ/NWX10mnUfe9un/XqDZrRRwldIbOUQ356BI10Q1qoTai6BE9o1f05jw5L86787Fs3XCKmVP0B87nDwNClGM=</latexit><latexit sha1_base64="8Hdb0teegmaIsq7//z8KOJ22Q+w=">AAAB+nicbVBNS8NAEN34WetXqkcvi0XoqSS96LHgxZsV7Ae0oWy2m3bp7ibsTpQQ+1O8eFDEq7/Em//GbZuDtj4YeLw3w8y8MBHcgOd9OxubW9s7u6W98v7B4dGxWznpmDjVlLVpLGLdC4lhgivWBg6C9RLNiAwF64bT67nffWDa8FjdQ5awQJKx4hGnBKw0dCsDJpNJfpsAl0RwyGZDt+rVvQXwOvELUkUFWkP3azCKaSqZAiqIMX3fSyDIiQZOBZuVB6lhCaFTMmZ9SxWRzAT54vQZvrDKCEextqUAL9TfEzmRxmQytJ2SwMSsenPxP6+fQnQV5FwlKTBFl4uiVGCI8TwHPOKaURCZJYRqbm/FdEI0oWDTKtsQ/NWX10mnUfe9un/XqDZrRRwldIbOUQ356BI10Q1qoTai6BE9o1f05jw5L86787Fs3XCKmVP0B87nDwNClGM=</latexit><latexit sha1_base64="8Hdb0teegmaIsq7//z8KOJ22Q+w=">AAAB+nicbVBNS8NAEN34WetXqkcvi0XoqSS96LHgxZsV7Ae0oWy2m3bp7ibsTpQQ+1O8eFDEq7/Em//GbZuDtj4YeLw3w8y8MBHcgOd9OxubW9s7u6W98v7B4dGxWznpmDjVlLVpLGLdC4lhgivWBg6C9RLNiAwF64bT67nffWDa8FjdQ5awQJKx4hGnBKw0dCsDJpNJfpsAl0RwyGZDt+rVvQXwOvELUkUFWkP3azCKaSqZAiqIMX3fSyDIiQZOBZuVB6lhCaFTMmZ9SxWRzAT54vQZvrDKCEextqUAL9TfEzmRxmQytJ2SwMSsenPxP6+fQnQV5FwlKTBFl4uiVGCI8TwHPOKaURCZJYRqbm/FdEI0oWDTKtsQ/NWX10mnUfe9un/XqDZrRRwldIbOUQ356BI10Q1qoTai6BE9o1f05jw5L86787Fs3XCKmVP0B87nDwNClGM=</latexit><latexit sha1_base64="8Hdb0teegmaIsq7//z8KOJ22Q+w=">AAAB+nicbVBNS8NAEN34WetXqkcvi0XoqSS96LHgxZsV7Ae0oWy2m3bp7ibsTpQQ+1O8eFDEq7/Em//GbZuDtj4YeLw3w8y8MBHcgOd9OxubW9s7u6W98v7B4dGxWznpmDjVlLVpLGLdC4lhgivWBg6C9RLNiAwF64bT67nffWDa8FjdQ5awQJKx4hGnBKw0dCsDJpNJfpsAl0RwyGZDt+rVvQXwOvELUkUFWkP3azCKaSqZAiqIMX3fSyDIiQZOBZuVB6lhCaFTMmZ9SxWRzAT54vQZvrDKCEextqUAL9TfEzmRxmQytJ2SwMSsenPxP6+fQnQV5FwlKTBFl4uiVGCI8TwHPOKaURCZJYRqbm/FdEI0oWDTKtsQ/NWX10mnUfe9un/XqDZrRRwldIbOUQ356BI10Q1qoTai6BE9o1f05jw5L86787Fs3XCKmVP0B87nDwNClGM=</latexit>

Yes<latexit sha1_base64="3b9GCAdhDkS3SLFnMYLvOC5pWHg=">AAAB6nicbVA9TwJBEJ3DL8Qv1NJmIzGhInc0UpLYWGKUDwMXsrfMwYa9vcvungm58BNsLDTG1l9k579xgSsUfMkkL+/NZGZekAiujet+O4Wt7Z3dveJ+6eDw6PikfHrW0XGqGLZZLGLVC6hGwSW2DTcCe4lCGgUCu8H0ZuF3n1BpHssHM0vQj+hY8pAzaqx0/4h6WK64NXcJskm8nFQgR2tY/hqMYpZGKA0TVOu+5ybGz6gynAmclwapxoSyKR1j31JJI9R+tjx1Tq6sMiJhrGxJQ5bq74mMRlrPosB2RtRM9Lq3EP/z+qkJG37GZZIalGy1KEwFMTFZ/E1GXCEzYmYJZYrbWwmbUEWZsemUbAje+subpFOveW7Nu6tXmtU8jiJcwCVUwYNraMIttKANDMbwDK/w5gjnxXl3PlatBSefOYc/cD5/AEEqja0=</latexit><latexit sha1_base64="3b9GCAdhDkS3SLFnMYLvOC5pWHg=">AAAB6nicbVA9TwJBEJ3DL8Qv1NJmIzGhInc0UpLYWGKUDwMXsrfMwYa9vcvungm58BNsLDTG1l9k579xgSsUfMkkL+/NZGZekAiujet+O4Wt7Z3dveJ+6eDw6PikfHrW0XGqGLZZLGLVC6hGwSW2DTcCe4lCGgUCu8H0ZuF3n1BpHssHM0vQj+hY8pAzaqx0/4h6WK64NXcJskm8nFQgR2tY/hqMYpZGKA0TVOu+5ybGz6gynAmclwapxoSyKR1j31JJI9R+tjx1Tq6sMiJhrGxJQ5bq74mMRlrPosB2RtRM9Lq3EP/z+qkJG37GZZIalGy1KEwFMTFZ/E1GXCEzYmYJZYrbWwmbUEWZsemUbAje+subpFOveW7Nu6tXmtU8jiJcwCVUwYNraMIttKANDMbwDK/w5gjnxXl3PlatBSefOYc/cD5/AEEqja0=</latexit><latexit sha1_base64="3b9GCAdhDkS3SLFnMYLvOC5pWHg=">AAAB6nicbVA9TwJBEJ3DL8Qv1NJmIzGhInc0UpLYWGKUDwMXsrfMwYa9vcvungm58BNsLDTG1l9k579xgSsUfMkkL+/NZGZekAiujet+O4Wt7Z3dveJ+6eDw6PikfHrW0XGqGLZZLGLVC6hGwSW2DTcCe4lCGgUCu8H0ZuF3n1BpHssHM0vQj+hY8pAzaqx0/4h6WK64NXcJskm8nFQgR2tY/hqMYpZGKA0TVOu+5ybGz6gynAmclwapxoSyKR1j31JJI9R+tjx1Tq6sMiJhrGxJQ5bq74mMRlrPosB2RtRM9Lq3EP/z+qkJG37GZZIalGy1KEwFMTFZ/E1GXCEzYmYJZYrbWwmbUEWZsemUbAje+subpFOveW7Nu6tXmtU8jiJcwCVUwYNraMIttKANDMbwDK/w5gjnxXl3PlatBSefOYc/cD5/AEEqja0=</latexit><latexit sha1_base64="3b9GCAdhDkS3SLFnMYLvOC5pWHg=">AAAB6nicbVA9TwJBEJ3DL8Qv1NJmIzGhInc0UpLYWGKUDwMXsrfMwYa9vcvungm58BNsLDTG1l9k579xgSsUfMkkL+/NZGZekAiujet+O4Wt7Z3dveJ+6eDw6PikfHrW0XGqGLZZLGLVC6hGwSW2DTcCe4lCGgUCu8H0ZuF3n1BpHssHM0vQj+hY8pAzaqx0/4h6WK64NXcJskm8nFQgR2tY/hqMYpZGKA0TVOu+5ybGz6gynAmclwapxoSyKR1j31JJI9R+tjx1Tq6sMiJhrGxJQ5bq74mMRlrPosB2RtRM9Lq3EP/z+qkJG37GZZIalGy1KEwFMTFZ/E1GXCEzYmYJZYrbWwmbUEWZsemUbAje+subpFOveW7Nu6tXmtU8jiJcwCVUwYNraMIttKANDMbwDK/w5gjnxXl3PlatBSefOYc/cD5/AEEqja0=</latexit>

Update bounds<latexit sha1_base64="fCh6HOGbf9duTDzFuSO43tfFJgw=">AAAB9HicbVBNT8JAEJ36ifiFevSykZhwIi0XPZJ48YiJBRJoyHa7hQ3bbd2dkhDC7/DiQWO8+mO8+W9coAcFXzLJy3szmZkXZlIYdN1vZ2t7Z3dvv3RQPjw6PjmtnJ23TZprxn2WylR3Q2q4FIr7KFDybqY5TULJO+H4buF3JlwbkapHnGY8SOhQiVgwilYK/CyiyEmY5ioyg0rVrbtLkE3iFaQKBVqDylc/SlmecIVMUmN6npthMKMaBZN8Xu7nhmeUjemQ9yxVNOEmmC2PnpNrq0QkTrUthWSp/p6Y0cSYaRLazoTiyKx7C/E/r5djfBvMhMpy5IqtFsW5JJiSRQIkEpozlFNLKNPC3krYiGrK0OZUtiF46y9vknaj7rl176FRbdaKOEpwCVdQAw9uoAn30AIfGDzBM7zCmzNxXpx352PVuuUUMxfwB87nD6pfke4=</latexit><latexit sha1_base64="fCh6HOGbf9duTDzFuSO43tfFJgw=">AAAB9HicbVBNT8JAEJ36ifiFevSykZhwIi0XPZJ48YiJBRJoyHa7hQ3bbd2dkhDC7/DiQWO8+mO8+W9coAcFXzLJy3szmZkXZlIYdN1vZ2t7Z3dvv3RQPjw6PjmtnJ23TZprxn2WylR3Q2q4FIr7KFDybqY5TULJO+H4buF3JlwbkapHnGY8SOhQiVgwilYK/CyiyEmY5ioyg0rVrbtLkE3iFaQKBVqDylc/SlmecIVMUmN6npthMKMaBZN8Xu7nhmeUjemQ9yxVNOEmmC2PnpNrq0QkTrUthWSp/p6Y0cSYaRLazoTiyKx7C/E/r5djfBvMhMpy5IqtFsW5JJiSRQIkEpozlFNLKNPC3krYiGrK0OZUtiF46y9vknaj7rl176FRbdaKOEpwCVdQAw9uoAn30AIfGDzBM7zCmzNxXpx352PVuuUUMxfwB87nD6pfke4=</latexit><latexit sha1_base64="fCh6HOGbf9duTDzFuSO43tfFJgw=">AAAB9HicbVBNT8JAEJ36ifiFevSykZhwIi0XPZJ48YiJBRJoyHa7hQ3bbd2dkhDC7/DiQWO8+mO8+W9coAcFXzLJy3szmZkXZlIYdN1vZ2t7Z3dvv3RQPjw6PjmtnJ23TZprxn2WylR3Q2q4FIr7KFDybqY5TULJO+H4buF3JlwbkapHnGY8SOhQiVgwilYK/CyiyEmY5ioyg0rVrbtLkE3iFaQKBVqDylc/SlmecIVMUmN6npthMKMaBZN8Xu7nhmeUjemQ9yxVNOEmmC2PnpNrq0QkTrUthWSp/p6Y0cSYaRLazoTiyKx7C/E/r5djfBvMhMpy5IqtFsW5JJiSRQIkEpozlFNLKNPC3krYiGrK0OZUtiF46y9vknaj7rl176FRbdaKOEpwCVdQAw9uoAn30AIfGDzBM7zCmzNxXpx352PVuuUUMxfwB87nD6pfke4=</latexit><latexit sha1_base64="fCh6HOGbf9duTDzFuSO43tfFJgw=">AAAB9HicbVBNT8JAEJ36ifiFevSykZhwIi0XPZJ48YiJBRJoyHa7hQ3bbd2dkhDC7/DiQWO8+mO8+W9coAcFXzLJy3szmZkXZlIYdN1vZ2t7Z3dvv3RQPjw6PjmtnJ23TZprxn2WylR3Q2q4FIr7KFDybqY5TULJO+H4buF3JlwbkapHnGY8SOhQiVgwilYK/CyiyEmY5ioyg0rVrbtLkE3iFaQKBVqDylc/SlmecIVMUmN6npthMKMaBZN8Xu7nhmeUjemQ9yxVNOEmmC2PnpNrq0QkTrUthWSp/p6Y0cSYaRLazoTiyKx7C/E/r5djfBvMhMpy5IqtFsW5JJiSRQIkEpozlFNLKNPC3krYiGrK0OZUtiF46y9vknaj7rl176FRbdaKOEpwCVdQAw9uoAn30AIfGDzBM7zCmzNxXpx352PVuuUUMxfwB87nD6pfke4=</latexit>

o 0<latexit sha1_base64="ujoA4+E+/o6FzwTS2TLKj1lfggI=">AAAB9HicdVDLSsNAFJ34rPVVdelmaBFchUkMbd0V3LisYB/QhjKZTtqhk0ycmVRK6He4caGIWz/GnX/jpK2gogcuHM65l3vvCRLOlEbow1pb39jc2i7sFHf39g8OS0fHbSVSSWiLCC5kN8CKchbTlmaa024iKY4CTjvB5Cr3O1MqFRPxrZ4l1I/wKGYhI1gbyRd9TkONpRT3EA1KFWRf1quuV4XIRqjmuE5O3Jp34UHHKDkqYIXmoPTeHwqSRjTWhGOleg5KtJ9hqRnhdF7sp4ommEzwiPYMjXFElZ8tjp7DM6MMYSikqVjDhfp9IsORUrMoMJ0R1mP128vFv7xeqsO6n7E4STWNyXJRmHKoBcwTgEMmKdF8ZggmkplbIRljiYk2ORVNCF+fwv9J27UdZDs3XqVRXsVRAKegDM6BA2qgAa5BE7QAAXfgATyBZ2tqPVov1uuydc1azZyAH7DePgHfs5IP</latexit><latexit sha1_base64="ujoA4+E+/o6FzwTS2TLKj1lfggI=">AAAB9HicdVDLSsNAFJ34rPVVdelmaBFchUkMbd0V3LisYB/QhjKZTtqhk0ycmVRK6He4caGIWz/GnX/jpK2gogcuHM65l3vvCRLOlEbow1pb39jc2i7sFHf39g8OS0fHbSVSSWiLCC5kN8CKchbTlmaa024iKY4CTjvB5Cr3O1MqFRPxrZ4l1I/wKGYhI1gbyRd9TkONpRT3EA1KFWRf1quuV4XIRqjmuE5O3Jp34UHHKDkqYIXmoPTeHwqSRjTWhGOleg5KtJ9hqRnhdF7sp4ommEzwiPYMjXFElZ8tjp7DM6MMYSikqVjDhfp9IsORUrMoMJ0R1mP128vFv7xeqsO6n7E4STWNyXJRmHKoBcwTgEMmKdF8ZggmkplbIRljiYk2ORVNCF+fwv9J27UdZDs3XqVRXsVRAKegDM6BA2qgAa5BE7QAAXfgATyBZ2tqPVov1uuydc1azZyAH7DePgHfs5IP</latexit><latexit sha1_base64="ujoA4+E+/o6FzwTS2TLKj1lfggI=">AAAB9HicdVDLSsNAFJ34rPVVdelmaBFchUkMbd0V3LisYB/QhjKZTtqhk0ycmVRK6He4caGIWz/GnX/jpK2gogcuHM65l3vvCRLOlEbow1pb39jc2i7sFHf39g8OS0fHbSVSSWiLCC5kN8CKchbTlmaa024iKY4CTjvB5Cr3O1MqFRPxrZ4l1I/wKGYhI1gbyRd9TkONpRT3EA1KFWRf1quuV4XIRqjmuE5O3Jp34UHHKDkqYIXmoPTeHwqSRjTWhGOleg5KtJ9hqRnhdF7sp4ommEzwiPYMjXFElZ8tjp7DM6MMYSikqVjDhfp9IsORUrMoMJ0R1mP128vFv7xeqsO6n7E4STWNyXJRmHKoBcwTgEMmKdF8ZggmkplbIRljiYk2ORVNCF+fwv9J27UdZDs3XqVRXsVRAKegDM6BA2qgAa5BE7QAAXfgATyBZ2tqPVov1uuydc1azZyAH7DePgHfs5IP</latexit><latexit sha1_base64="ujoA4+E+/o6FzwTS2TLKj1lfggI=">AAAB9HicdVDLSsNAFJ34rPVVdelmaBFchUkMbd0V3LisYB/QhjKZTtqhk0ycmVRK6He4caGIWz/GnX/jpK2gogcuHM65l3vvCRLOlEbow1pb39jc2i7sFHf39g8OS0fHbSVSSWiLCC5kN8CKchbTlmaa024iKY4CTjvB5Cr3O1MqFRPxrZ4l1I/wKGYhI1gbyRd9TkONpRT3EA1KFWRf1quuV4XIRqjmuE5O3Jp34UHHKDkqYIXmoPTeHwqSRjTWhGOleg5KtJ9hqRnhdF7sp4ommEzwiPYMjXFElZ8tjp7DM6MMYSikqVjDhfp9IsORUrMoMJ0R1mP128vFv7xeqsO6n7E4STWNyXJRmHKoBcwTgEMmKdF8ZggmkplbIRljiYk2ORVNCF+fwv9J27UdZDs3XqVRXsVRAKegDM6BA2qgAa5BE7QAAXfgATyBZ2tqPVov1uuydc1azZyAH7DePgHfs5IP</latexit>

UBo 1<latexit sha1_base64="IPCAAOYnK8plyQD8KGX5h3mEmGY=">AAAB/nicdVBNSwMxEM36WetXVTx5CS2Cp7JbS1tvRS8eK9gPaEvJptk2mE2WZFYpS8G/4sWDIl79Hd78N2bbCir6YODx3gwz8/xIcAOu++EsLa+srq1nNrKbW9s7u7m9/ZZRsaasSZVQuuMTwwSXrAkcBOtEmpHQF6zt31ykfvuWacOVvIZJxPohGUkecErASoPcYfN8oHqCBUC0Vne4x2UAk0Gu4BbPapVSuYLdoutWvZKXklK1fFrGnlVSFNACjUHuvTdUNA6ZBCqIMV3PjaCfEA2cCjbN9mLDIkJvyIh1LZUkZKafzM6f4mOrDHGgtC0JeKZ+n0hIaMwk9G1nSGBsfnup+JfXjSGo9RMuoxiYpPNFQSwwKJxmgYdcMwpiYgmhmttbMR0TTSjYxLI2hK9P8f+kVSp6btG7Khfq+UUcGXSE8ugEeaiK6ugSNVATUZSgB/SEnp1759F5cV7nrUvOYuYA/YDz9gmvypXc</latexit><latexit sha1_base64="IPCAAOYnK8plyQD8KGX5h3mEmGY=">AAAB/nicdVBNSwMxEM36WetXVTx5CS2Cp7JbS1tvRS8eK9gPaEvJptk2mE2WZFYpS8G/4sWDIl79Hd78N2bbCir6YODx3gwz8/xIcAOu++EsLa+srq1nNrKbW9s7u7m9/ZZRsaasSZVQuuMTwwSXrAkcBOtEmpHQF6zt31ykfvuWacOVvIZJxPohGUkecErASoPcYfN8oHqCBUC0Vne4x2UAk0Gu4BbPapVSuYLdoutWvZKXklK1fFrGnlVSFNACjUHuvTdUNA6ZBCqIMV3PjaCfEA2cCjbN9mLDIkJvyIh1LZUkZKafzM6f4mOrDHGgtC0JeKZ+n0hIaMwk9G1nSGBsfnup+JfXjSGo9RMuoxiYpPNFQSwwKJxmgYdcMwpiYgmhmttbMR0TTSjYxLI2hK9P8f+kVSp6btG7Khfq+UUcGXSE8ugEeaiK6ugSNVATUZSgB/SEnp1759F5cV7nrUvOYuYA/YDz9gmvypXc</latexit><latexit sha1_base64="IPCAAOYnK8plyQD8KGX5h3mEmGY=">AAAB/nicdVBNSwMxEM36WetXVTx5CS2Cp7JbS1tvRS8eK9gPaEvJptk2mE2WZFYpS8G/4sWDIl79Hd78N2bbCir6YODx3gwz8/xIcAOu++EsLa+srq1nNrKbW9s7u7m9/ZZRsaasSZVQuuMTwwSXrAkcBOtEmpHQF6zt31ykfvuWacOVvIZJxPohGUkecErASoPcYfN8oHqCBUC0Vne4x2UAk0Gu4BbPapVSuYLdoutWvZKXklK1fFrGnlVSFNACjUHuvTdUNA6ZBCqIMV3PjaCfEA2cCjbN9mLDIkJvyIh1LZUkZKafzM6f4mOrDHGgtC0JeKZ+n0hIaMwk9G1nSGBsfnup+JfXjSGo9RMuoxiYpPNFQSwwKJxmgYdcMwpiYgmhmttbMR0TTSjYxLI2hK9P8f+kVSp6btG7Khfq+UUcGXSE8ugEeaiK6ugSNVATUZSgB/SEnp1759F5cV7nrUvOYuYA/YDz9gmvypXc</latexit><latexit sha1_base64="IPCAAOYnK8plyQD8KGX5h3mEmGY=">AAAB/nicdVBNSwMxEM36WetXVTx5CS2Cp7JbS1tvRS8eK9gPaEvJptk2mE2WZFYpS8G/4sWDIl79Hd78N2bbCir6YODx3gwz8/xIcAOu++EsLa+srq1nNrKbW9s7u7m9/ZZRsaasSZVQuuMTwwSXrAkcBOtEmpHQF6zt31ykfvuWacOVvIZJxPohGUkecErASoPcYfN8oHqCBUC0Vne4x2UAk0Gu4BbPapVSuYLdoutWvZKXklK1fFrGnlVSFNACjUHuvTdUNA6ZBCqIMV3PjaCfEA2cCjbN9mLDIkJvyIh1LZUkZKafzM6f4mOrDHGgtC0JeKZ+n0hIaMwk9G1nSGBsfnup+JfXjSGo9RMuoxiYpPNFQSwwKJxmgYdcMwpiYgmhmttbMR0TTSjYxLI2hK9P8f+kVSp6btG7Khfq+UUcGXSE8ugEeaiK6ugSNVATUZSgB/SEnp1759F5cV7nrUvOYuYA/YDz9gmvypXc</latexit>

LBo 1<latexit sha1_base64="IHnO3iEqQYVB4OGFYkCUDxMcNqI=">AAAB/3icdVA9SwNBEN3z2/gVFWxslgTBxnB3BqOdaGNhoWA+IBfC3mZPl+ztHrtzyhEt/Cs2ForY+jfs/DfuJRFU9MHA470ZZuaFieAGXPfDmZicmp6ZnZsvLCwuLa8UV9caRqWasjpVQulWSAwTXLI6cBCslWhG4lCwZtg/zv3mNdOGK3kBWcI6MbmUPOKUgJW6xY3To64KBIuAaK1u8E7AZQRZt1h2Kwf7e351D7sV1615vpcTv1bdrWLPKjnKaIyzbvE96CmaxkwCFcSYtucm0BkQDZwKdlcIUsMSQvvkkrUtlSRmpjMY3n+Ht6zSw5HStiTgofp9YkBiY7I4tJ0xgSvz28vFv7x2CtF+Z8BlkgKTdLQoSgUGhfMwcI9rRkFklhCqub0V0yuiCQUbWcGG8PUp/p80/IrnVrzzavmwNI5jDm2iEtpGHqqhQ3SCzlAdUXSLHtATenbunUfnxXkdtU4445l19APO2ycRnJYK</latexit><latexit sha1_base64="IHnO3iEqQYVB4OGFYkCUDxMcNqI=">AAAB/3icdVA9SwNBEN3z2/gVFWxslgTBxnB3BqOdaGNhoWA+IBfC3mZPl+ztHrtzyhEt/Cs2ForY+jfs/DfuJRFU9MHA470ZZuaFieAGXPfDmZicmp6ZnZsvLCwuLa8UV9caRqWasjpVQulWSAwTXLI6cBCslWhG4lCwZtg/zv3mNdOGK3kBWcI6MbmUPOKUgJW6xY3To64KBIuAaK1u8E7AZQRZt1h2Kwf7e351D7sV1615vpcTv1bdrWLPKjnKaIyzbvE96CmaxkwCFcSYtucm0BkQDZwKdlcIUsMSQvvkkrUtlSRmpjMY3n+Ht6zSw5HStiTgofp9YkBiY7I4tJ0xgSvz28vFv7x2CtF+Z8BlkgKTdLQoSgUGhfMwcI9rRkFklhCqub0V0yuiCQUbWcGG8PUp/p80/IrnVrzzavmwNI5jDm2iEtpGHqqhQ3SCzlAdUXSLHtATenbunUfnxXkdtU4445l19APO2ycRnJYK</latexit><latexit sha1_base64="IHnO3iEqQYVB4OGFYkCUDxMcNqI=">AAAB/3icdVA9SwNBEN3z2/gVFWxslgTBxnB3BqOdaGNhoWA+IBfC3mZPl+ztHrtzyhEt/Cs2ForY+jfs/DfuJRFU9MHA470ZZuaFieAGXPfDmZicmp6ZnZsvLCwuLa8UV9caRqWasjpVQulWSAwTXLI6cBCslWhG4lCwZtg/zv3mNdOGK3kBWcI6MbmUPOKUgJW6xY3To64KBIuAaK1u8E7AZQRZt1h2Kwf7e351D7sV1615vpcTv1bdrWLPKjnKaIyzbvE96CmaxkwCFcSYtucm0BkQDZwKdlcIUsMSQvvkkrUtlSRmpjMY3n+Ht6zSw5HStiTgofp9YkBiY7I4tJ0xgSvz28vFv7x2CtF+Z8BlkgKTdLQoSgUGhfMwcI9rRkFklhCqub0V0yuiCQUbWcGG8PUp/p80/IrnVrzzavmwNI5jDm2iEtpGHqqhQ3SCzlAdUXSLHtATenbunUfnxXkdtU4445l19APO2ycRnJYK</latexit><latexit sha1_base64="IHnO3iEqQYVB4OGFYkCUDxMcNqI=">AAAB/3icdVA9SwNBEN3z2/gVFWxslgTBxnB3BqOdaGNhoWA+IBfC3mZPl+ztHrtzyhEt/Cs2ForY+jfs/DfuJRFU9MHA470ZZuaFieAGXPfDmZicmp6ZnZsvLCwuLa8UV9caRqWasjpVQulWSAwTXLI6cBCslWhG4lCwZtg/zv3mNdOGK3kBWcI6MbmUPOKUgJW6xY3To64KBIuAaK1u8E7AZQRZt1h2Kwf7e351D7sV1615vpcTv1bdrWLPKjnKaIyzbvE96CmaxkwCFcSYtucm0BkQDZwKdlcIUsMSQvvkkrUtlSRmpjMY3n+Ht6zSw5HStiTgofp9YkBiY7I4tJ0xgSvz28vFv7x2CtF+Z8BlkgKTdLQoSgUGhfMwcI9rRkFklhCqub0V0yuiCQUbWcGG8PUp/p80/IrnVrzzavmwNI5jDm2iEtpGHqqhQ3SCzlAdUXSLHtATenbunUfnxXkdtU4445l19APO2ycRnJYK</latexit>

Update cut (5d)<latexit sha1_base64="c6ngN9L8FLSjxZuMoo4ccYC/niA=">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</latexit><latexit sha1_base64="c6ngN9L8FLSjxZuMoo4ccYC/niA=">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</latexit><latexit sha1_base64="c6ngN9L8FLSjxZuMoo4ccYC/niA=">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</latexit><latexit sha1_base64="c6ngN9L8FLSjxZuMoo4ccYC/niA=">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</latexit>

Solve Sod(vc,!

Dc )

<latexit sha1_base64="mEmEswofiI+t4k+/lMhRWr4UtQ0=">AAACB3icbVDLSgNBEJyNrxhfUY+CDCZCBAm7uegxoAePkZgHJJtldtJJhszuLDOzgbDk5sVf8eJBEa/+gjf/xsnjoIkFDUVVN91dfsSZ0rb9baXW1jc2t9LbmZ3dvf2D7OFRXYlYUqhRwYVs+kQBZyHUNNMcmpEEEvgcGv7wZuo3RiAVE+GDHkfgBqQfsh6jRBvJy55WBR8Bzle9bkcURh69xG0RQJ94tHN7kfeyObtoz4BXibMgObRAxct+tbuCxgGEmnKiVMuxI+0mRGpGOUwy7VhBROiQ9KFlaEgCUG4y+2OCz43SxT0hTYUaz9TfEwkJlBoHvukMiB6oZW8q/ue1Yt27dhMWRrGGkM4X9WKOtcDTUHCXSaCajw0hVDJzK6YDIgnVJrqMCcFZfnmV1EtFxy4696VcubCII41O0BkqIAddoTK6QxVUQxQ9omf0it6sJ+vFerc+5q0pazFzjP7A+vwBezmXrQ==</latexit><latexit sha1_base64="mEmEswofiI+t4k+/lMhRWr4UtQ0=">AAACB3icbVDLSgNBEJyNrxhfUY+CDCZCBAm7uegxoAePkZgHJJtldtJJhszuLDOzgbDk5sVf8eJBEa/+gjf/xsnjoIkFDUVVN91dfsSZ0rb9baXW1jc2t9LbmZ3dvf2D7OFRXYlYUqhRwYVs+kQBZyHUNNMcmpEEEvgcGv7wZuo3RiAVE+GDHkfgBqQfsh6jRBvJy55WBR8Bzle9bkcURh69xG0RQJ94tHN7kfeyObtoz4BXibMgObRAxct+tbuCxgGEmnKiVMuxI+0mRGpGOUwy7VhBROiQ9KFlaEgCUG4y+2OCz43SxT0hTYUaz9TfEwkJlBoHvukMiB6oZW8q/ue1Yt27dhMWRrGGkM4X9WKOtcDTUHCXSaCajw0hVDJzK6YDIgnVJrqMCcFZfnmV1EtFxy4696VcubCII41O0BkqIAddoTK6QxVUQxQ9omf0it6sJ+vFerc+5q0pazFzjP7A+vwBezmXrQ==</latexit><latexit sha1_base64="mEmEswofiI+t4k+/lMhRWr4UtQ0=">AAACB3icbVDLSgNBEJyNrxhfUY+CDCZCBAm7uegxoAePkZgHJJtldtJJhszuLDOzgbDk5sVf8eJBEa/+gjf/xsnjoIkFDUVVN91dfsSZ0rb9baXW1jc2t9LbmZ3dvf2D7OFRXYlYUqhRwYVs+kQBZyHUNNMcmpEEEvgcGv7wZuo3RiAVE+GDHkfgBqQfsh6jRBvJy55WBR8Bzle9bkcURh69xG0RQJ94tHN7kfeyObtoz4BXibMgObRAxct+tbuCxgGEmnKiVMuxI+0mRGpGOUwy7VhBROiQ9KFlaEgCUG4y+2OCz43SxT0hTYUaz9TfEwkJlBoHvukMiB6oZW8q/ue1Yt27dhMWRrGGkM4X9WKOtcDTUHCXSaCajw0hVDJzK6YDIgnVJrqMCcFZfnmV1EtFxy4696VcubCII41O0BkqIAddoTK6QxVUQxQ9omf0it6sJ+vFerc+5q0pazFzjP7A+vwBezmXrQ==</latexit><latexit sha1_base64="mEmEswofiI+t4k+/lMhRWr4UtQ0=">AAACB3icbVDLSgNBEJyNrxhfUY+CDCZCBAm7uegxoAePkZgHJJtldtJJhszuLDOzgbDk5sVf8eJBEa/+gjf/xsnjoIkFDUVVN91dfsSZ0rb9baXW1jc2t9LbmZ3dvf2D7OFRXYlYUqhRwYVs+kQBZyHUNNMcmpEEEvgcGv7wZuo3RiAVE+GDHkfgBqQfsh6jRBvJy55WBR8Bzle9bkcURh69xG0RQJ94tHN7kfeyObtoz4BXibMgObRAxct+tbuCxgGEmnKiVMuxI+0mRGpGOUwy7VhBROiQ9KFlaEgCUG4y+2OCz43SxT0hTYUaz9TfEwkJlBoHvukMiB6oZW8q/ue1Yt27dhMWRrGGkM4X9WKOtcDTUHCXSaCajw0hVDJzK6YDIgnVJrqMCcFZfnmV1EtFxy4696VcubCII41O0BkqIAddoTK6QxVUQxQ9omf0it6sJ+vFerc+5q0pazFzjP7A+vwBezmXrQ==</latexit>

Solve M0<latexit sha1_base64="a4zReeQPKHmVx2pJrbzkEtaTpfA=">AAAB8nicbVA9TwJBEN3DL8Qv1NJmI5hQkTsaLUlsbEwwCpgcJ9lb9mDD3u5ld46EXPgZNhYaY+uvsfPfuMAVCr5kkpf3ZjIzL0wEN+C6305hY3Nre6e4W9rbPzg8Kh+fdIxKNWVtqoTSjyExTHDJ2sBBsMdEMxKHgnXD8fXc706YNlzJB5gmLIjJUPKIUwJW8u+VmDBcvX1yq/1yxa27C+B14uWkgnK0+uWv3kDRNGYSqCDG+J6bQJARDZwKNiv1UsMSQsdkyHxLJYmZCbLFyTN8YZUBjpS2JQEv1N8TGYmNmcah7YwJjMyqNxf/8/wUoqsg4zJJgUm6XBSlAoPC8//xgGtGQUwtIVRzeyumI6IJBZtSyYbgrb68TjqNuufWvbtGpVnL4yiiM3SOashDl6iJblALtRFFCj2jV/TmgPPivDsfy9aCk8+coj9wPn8Aq9OQGA==</latexit><latexit sha1_base64="a4zReeQPKHmVx2pJrbzkEtaTpfA=">AAAB8nicbVA9TwJBEN3DL8Qv1NJmI5hQkTsaLUlsbEwwCpgcJ9lb9mDD3u5ld46EXPgZNhYaY+uvsfPfuMAVCr5kkpf3ZjIzL0wEN+C6305hY3Nre6e4W9rbPzg8Kh+fdIxKNWVtqoTSjyExTHDJ2sBBsMdEMxKHgnXD8fXc706YNlzJB5gmLIjJUPKIUwJW8u+VmDBcvX1yq/1yxa27C+B14uWkgnK0+uWv3kDRNGYSqCDG+J6bQJARDZwKNiv1UsMSQsdkyHxLJYmZCbLFyTN8YZUBjpS2JQEv1N8TGYmNmcah7YwJjMyqNxf/8/wUoqsg4zJJgUm6XBSlAoPC8//xgGtGQUwtIVRzeyumI6IJBZtSyYbgrb68TjqNuufWvbtGpVnL4yiiM3SOashDl6iJblALtRFFCj2jV/TmgPPivDsfy9aCk8+coj9wPn8Aq9OQGA==</latexit><latexit sha1_base64="a4zReeQPKHmVx2pJrbzkEtaTpfA=">AAAB8nicbVA9TwJBEN3DL8Qv1NJmI5hQkTsaLUlsbEwwCpgcJ9lb9mDD3u5ld46EXPgZNhYaY+uvsfPfuMAVCr5kkpf3ZjIzL0wEN+C6305hY3Nre6e4W9rbPzg8Kh+fdIxKNWVtqoTSjyExTHDJ2sBBsMdEMxKHgnXD8fXc706YNlzJB5gmLIjJUPKIUwJW8u+VmDBcvX1yq/1yxa27C+B14uWkgnK0+uWv3kDRNGYSqCDG+J6bQJARDZwKNiv1UsMSQsdkyHxLJYmZCbLFyTN8YZUBjpS2JQEv1N8TGYmNmcah7YwJjMyqNxf/8/wUoqsg4zJJgUm6XBSlAoPC8//xgGtGQUwtIVRzeyumI6IJBZtSyYbgrb68TjqNuufWvbtGpVnL4yiiM3SOashDl6iJblALtRFFCj2jV/TmgPPivDsfy9aCk8+coj9wPn8Aq9OQGA==</latexit><latexit sha1_base64="a4zReeQPKHmVx2pJrbzkEtaTpfA=">AAAB8nicbVA9TwJBEN3DL8Qv1NJmI5hQkTsaLUlsbEwwCpgcJ9lb9mDD3u5ld46EXPgZNhYaY+uvsfPfuMAVCr5kkpf3ZjIzL0wEN+C6305hY3Nre6e4W9rbPzg8Kh+fdIxKNWVtqoTSjyExTHDJ2sBBsMdEMxKHgnXD8fXc706YNlzJB5gmLIjJUPKIUwJW8u+VmDBcvX1yq/1yxa27C+B14uWkgnK0+uWv3kDRNGYSqCDG+J6bQJARDZwKNiv1UsMSQsdkyHxLJYmZCbLFyTN8YZUBjpS2JQEv1N8TGYmNmcah7YwJjMyqNxf/8/wUoqsg4zJJgUm6XBSlAoPC8//xgGtGQUwtIVRzeyumI6IJBZtSyYbgrb68TjqNuufWvbtGpVnL4yiiM3SOashDl6iJblALtRFFCj2jV/TmgPPivDsfy9aCk8+coj9wPn8Aq9OQGA==</latexit>

Solve S0t (!T )

<latexit sha1_base64="kudVdJ/W8oyCS/vheOSqFAL9Rjs=">AAACAHicbVA9TwJBEN3DL8SvUwsLm41ggg25o9GSxMYSw2cCB9lbBtiwd3vZ3SMhFxr/io2Fxtj6M+z8Ny5whYIvmeTlvZnMzPMjzpR2nG8rs7W9s7uX3c8dHB4dn9inZ00lYkmhQQUXsu0TBZyF0NBMc2hHEkjgc2j5k/uF35qCVEyEdT2LwAvIKGRDRok2Ut++qAk+BVyo9XXPKXZFACPSq98U+nbeKTlL4E3ipiSPUlT79ld3IGgcQKgpJ0p1XCfSXkKkZpTDPNeNFUSETsgIOoaGJADlJcsH5vjaKAM8FNJUqPFS/T2RkECpWeCbzoDosVr3FuJ/XifWwzsvYWEUawjpatEw5lgLvEgDD5gEqvnMEEIlM7diOiaSUG0yy5kQ3PWXN0mzXHKdkvtYzleKaRxZdImuUBG56BZV0AOqogaiaI6e0St6s56sF+vd+li1Zqx05hz9gfX5A5bylQI=</latexit><latexit sha1_base64="kudVdJ/W8oyCS/vheOSqFAL9Rjs=">AAACAHicbVA9TwJBEN3DL8SvUwsLm41ggg25o9GSxMYSw2cCB9lbBtiwd3vZ3SMhFxr/io2Fxtj6M+z8Ny5whYIvmeTlvZnMzPMjzpR2nG8rs7W9s7uX3c8dHB4dn9inZ00lYkmhQQUXsu0TBZyF0NBMc2hHEkjgc2j5k/uF35qCVEyEdT2LwAvIKGRDRok2Ut++qAk+BVyo9XXPKXZFACPSq98U+nbeKTlL4E3ipiSPUlT79ld3IGgcQKgpJ0p1XCfSXkKkZpTDPNeNFUSETsgIOoaGJADlJcsH5vjaKAM8FNJUqPFS/T2RkECpWeCbzoDosVr3FuJ/XifWwzsvYWEUawjpatEw5lgLvEgDD5gEqvnMEEIlM7diOiaSUG0yy5kQ3PWXN0mzXHKdkvtYzleKaRxZdImuUBG56BZV0AOqogaiaI6e0St6s56sF+vd+li1Zqx05hz9gfX5A5bylQI=</latexit><latexit sha1_base64="kudVdJ/W8oyCS/vheOSqFAL9Rjs=">AAACAHicbVA9TwJBEN3DL8SvUwsLm41ggg25o9GSxMYSw2cCB9lbBtiwd3vZ3SMhFxr/io2Fxtj6M+z8Ny5whYIvmeTlvZnMzPMjzpR2nG8rs7W9s7uX3c8dHB4dn9inZ00lYkmhQQUXsu0TBZyF0NBMc2hHEkjgc2j5k/uF35qCVEyEdT2LwAvIKGRDRok2Ut++qAk+BVyo9XXPKXZFACPSq98U+nbeKTlL4E3ipiSPUlT79ld3IGgcQKgpJ0p1XCfSXkKkZpTDPNeNFUSETsgIOoaGJADlJcsH5vjaKAM8FNJUqPFS/T2RkECpWeCbzoDosVr3FuJ/XifWwzsvYWEUawjpatEw5lgLvEgDD5gEqvnMEEIlM7diOiaSUG0yy5kQ3PWXN0mzXHKdkvtYzleKaRxZdImuUBG56BZV0AOqogaiaI6e0St6s56sF+vd+li1Zqx05hz9gfX5A5bylQI=</latexit><latexit sha1_base64="kudVdJ/W8oyCS/vheOSqFAL9Rjs=">AAACAHicbVA9TwJBEN3DL8SvUwsLm41ggg25o9GSxMYSw2cCB9lbBtiwd3vZ3SMhFxr/io2Fxtj6M+z8Ny5whYIvmeTlvZnMzPMjzpR2nG8rs7W9s7uX3c8dHB4dn9inZ00lYkmhQQUXsu0TBZyF0NBMc2hHEkjgc2j5k/uF35qCVEyEdT2LwAvIKGRDRok2Ut++qAk+BVyo9XXPKXZFACPSq98U+nbeKTlL4E3ipiSPUlT79ld3IGgcQKgpJ0p1XCfSXkKkZpTDPNeNFUSETsgIOoaGJADlJcsH5vjaKAM8FNJUqPFS/T2RkECpWeCbzoDosVr3FuJ/XifWwzsvYWEUawjpatEw5lgLvEgDD5gEqvnMEEIlM7diOiaSUG0yy5kQ3PWXN0mzXHKdkvtYzleKaRxZdImuUBG56BZV0AOqogaiaI6e0St6s56sF+vd+li1Zqx05hz9gfX5A5bylQI=</latexit>

Deliver x1 and x

2<latexit sha1_base64="Qub7oEYiHSJSUn4xsL9cTKJKgLk=">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</latexit><latexit sha1_base64="Qub7oEYiHSJSUn4xsL9cTKJKgLk=">AAACTHicdVDLSsQwFE3H9/gadekmOAgiMrR1cHQn6MKlgqPCtA5pmhnDpGlJbsWh9APduHDnV7hxoYhgOg9R0QuBk3PuPTc5QSK4Btt+skoTk1PTM7Nz5fmFxaXlysrqhY5TRVmTxiJWVwHRTHDJmsBBsKtEMRIFgl0GvaNCv7xlSvNYnkM/YX5EupJ3OCVgqHaFZt7ApKW6gZ/ZtYP9Pbe+t2PXbLvhuE4B3EZ9t557wO4gOzaLjF2OPYy9IBZhdpe3nettcx82EBkW4pfmXm/n7Up1bIzHxnhsjB3DFFVFozptVx69MKZpxCRQQbRuOXYCfkYUcCpYXvZSzRJCe6TLWgZKEjHtZ4N/5HjTMCHuxMocCXjAfp/ISKR1PwpMZ0TgRv/WCvIvrZVCZ9/PuExSYJIOF3VSgSHGRbI45IpREH0DCFXcvBXTG6IIBZN/2YQw/in+H1y4NceuOWf16uHGKI5ZtI420BZyUAMdohN0ipqIonv0jF7Rm/VgvVjv1sewtWSNZtbQjypNfwK9hrBN</latexit><latexit sha1_base64="Qub7oEYiHSJSUn4xsL9cTKJKgLk=">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</latexit><latexit sha1_base64="Qub7oEYiHSJSUn4xsL9cTKJKgLk=">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</latexit>

Deliver y and o o + 1<latexit sha1_base64="IM/XJETZhbYoZ8FlMlhHJb2taz8=">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</latexit><latexit sha1_base64="IM/XJETZhbYoZ8FlMlhHJb2taz8=">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</latexit><latexit sha1_base64="IM/XJETZhbYoZ8FlMlhHJb2taz8=">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</latexit><latexit sha1_base64="IM/XJETZhbYoZ8FlMlhHJb2taz8=">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</latexit>

Solve Mo<latexit sha1_base64="uMPnJEC0IkexU5yiF/cH45f220E=">AAAB8nicbVA9TwJBEN3DL8Qv1NJmI5hQkTsaLUlsbEwwCpgcJ9lb5mDD3u5ld4+EXPgZNhYaY+uvsfPfuMAVCr5kkpf3ZjIzL0w408Z1v53CxubW9k5xt7S3f3B4VD4+6WiZKgptKrlUjyHRwJmAtmGGw2OigMQhh244vp773QkozaR4MNMEgpgMBYsYJcZK/r3kE8DV2ydZ7Zcrbt1dAK8TLycVlKPVL3/1BpKmMQhDOdHa99zEBBlRhlEOs1Iv1ZAQOiZD8C0VJAYdZIuTZ/jCKgMcSWVLGLxQf09kJNZ6Goe2MyZmpFe9ufif56cmugoyJpLUgKDLRVHKsZF4/j8eMAXU8KklhCpmb8V0RBShxqZUsiF4qy+vk06j7rl1765RadbyOIroDJ2jGvLQJWqiG9RCbUSRRM/oFb05xnlx3p2PZWvByWdO0R84nz8LnZBX</latexit><latexit sha1_base64="uMPnJEC0IkexU5yiF/cH45f220E=">AAAB8nicbVA9TwJBEN3DL8Qv1NJmI5hQkTsaLUlsbEwwCpgcJ9lb5mDD3u5ld4+EXPgZNhYaY+uvsfPfuMAVCr5kkpf3ZjIzL0w408Z1v53CxubW9k5xt7S3f3B4VD4+6WiZKgptKrlUjyHRwJmAtmGGw2OigMQhh244vp773QkozaR4MNMEgpgMBYsYJcZK/r3kE8DV2ydZ7Zcrbt1dAK8TLycVlKPVL3/1BpKmMQhDOdHa99zEBBlRhlEOs1Iv1ZAQOiZD8C0VJAYdZIuTZ/jCKgMcSWVLGLxQf09kJNZ6Goe2MyZmpFe9ufif56cmugoyJpLUgKDLRVHKsZF4/j8eMAXU8KklhCpmb8V0RBShxqZUsiF4qy+vk06j7rl1765RadbyOIroDJ2jGvLQJWqiG9RCbUSRRM/oFb05xnlx3p2PZWvByWdO0R84nz8LnZBX</latexit><latexit sha1_base64="uMPnJEC0IkexU5yiF/cH45f220E=">AAAB8nicbVA9TwJBEN3DL8Qv1NJmI5hQkTsaLUlsbEwwCpgcJ9lb5mDD3u5ld4+EXPgZNhYaY+uvsfPfuMAVCr5kkpf3ZjIzL0w408Z1v53CxubW9k5xt7S3f3B4VD4+6WiZKgptKrlUjyHRwJmAtmGGw2OigMQhh244vp773QkozaR4MNMEgpgMBYsYJcZK/r3kE8DV2ydZ7Zcrbt1dAK8TLycVlKPVL3/1BpKmMQhDOdHa99zEBBlRhlEOs1Iv1ZAQOiZD8C0VJAYdZIuTZ/jCKgMcSWVLGLxQf09kJNZ6Goe2MyZmpFe9ufif56cmugoyJpLUgKDLRVHKsZF4/j8eMAXU8KklhCpmb8V0RBShxqZUsiF4qy+vk06j7rl1765RadbyOIroDJ2jGvLQJWqiG9RCbUSRRM/oFb05xnlx3p2PZWvByWdO0R84nz8LnZBX</latexit><latexit sha1_base64="uMPnJEC0IkexU5yiF/cH45f220E=">AAAB8nicbVA9TwJBEN3DL8Qv1NJmI5hQkTsaLUlsbEwwCpgcJ9lb5mDD3u5ld4+EXPgZNhYaY+uvsfPfuMAVCr5kkpf3ZjIzL0w408Z1v53CxubW9k5xt7S3f3B4VD4+6WiZKgptKrlUjyHRwJmAtmGGw2OigMQhh244vp773QkozaR4MNMEgpgMBYsYJcZK/r3kE8DV2ydZ7Zcrbt1dAK8TLycVlKPVL3/1BpKmMQhDOdHa99zEBBlRhlEOs1Iv1ZAQOiZD8C0VJAYdZIuTZ/jCKgMcSWVLGLxQf09kJNZ6Goe2MyZmpFe9ufif56cmugoyJpLUgKDLRVHKsZF4/j8eMAXU8KklhCpmb8V0RBShxqZUsiF4qy+vk06j7rl1765RadbyOIroDJ2jGvLQJWqiG9RCbUSRRM/oFb05xnlx3p2PZWvByWdO0R84nz8LnZBX</latexit>

Solve Sot (!T )

<latexit sha1_base64="1QdmaavMFRzB+ktReGqzxwauB1w=">AAACAHicbVC5TgMxEPWGK4QrQEFBY5EghSbaTQNlJBrKoJxSjpXXmU2s2OuV7Y0URWn4FRoKEKLlM+j4G5yjgIQnjfT03oxm5gUxZ9q47reT2tre2d1L72cODo+OT7KnZw0tE0WhTiWXqhUQDZxFUDfMcGjFCogIODSD0f3cb45BaSajmpnE0BVkELGQUWKs5GcvqpKPAeervunJQkcKGJBe7SbvZ3Nu0V0AbxJvRXJohYqf/er0JU0ERIZyonXbc2PTnRJlGOUwy3QSDTGhIzKAtqUREaC708UDM3xtlT4OpbIVGbxQf09MidB6IgLbKYgZ6nVvLv7ntRMT3nWnLIoTAxFdLgoTjo3E8zRwnymghk8sIVQxeyumQ6IINTazjA3BW395kzRKRc8teo+lXLmwiiONLtEVKiAP3aIyekAVVEcUzdAzekVvzpPz4rw7H8vWlLOaOUd/4Hz+APkjlUE=</latexit><latexit sha1_base64="1QdmaavMFRzB+ktReGqzxwauB1w=">AAACAHicbVC5TgMxEPWGK4QrQEFBY5EghSbaTQNlJBrKoJxSjpXXmU2s2OuV7Y0URWn4FRoKEKLlM+j4G5yjgIQnjfT03oxm5gUxZ9q47reT2tre2d1L72cODo+OT7KnZw0tE0WhTiWXqhUQDZxFUDfMcGjFCogIODSD0f3cb45BaSajmpnE0BVkELGQUWKs5GcvqpKPAeervunJQkcKGJBe7SbvZ3Nu0V0AbxJvRXJohYqf/er0JU0ERIZyonXbc2PTnRJlGOUwy3QSDTGhIzKAtqUREaC708UDM3xtlT4OpbIVGbxQf09MidB6IgLbKYgZ6nVvLv7ntRMT3nWnLIoTAxFdLgoTjo3E8zRwnymghk8sIVQxeyumQ6IINTazjA3BW395kzRKRc8teo+lXLmwiiONLtEVKiAP3aIyekAVVEcUzdAzekVvzpPz4rw7H8vWlLOaOUd/4Hz+APkjlUE=</latexit><latexit sha1_base64="1QdmaavMFRzB+ktReGqzxwauB1w=">AAACAHicbVC5TgMxEPWGK4QrQEFBY5EghSbaTQNlJBrKoJxSjpXXmU2s2OuV7Y0URWn4FRoKEKLlM+j4G5yjgIQnjfT03oxm5gUxZ9q47reT2tre2d1L72cODo+OT7KnZw0tE0WhTiWXqhUQDZxFUDfMcGjFCogIODSD0f3cb45BaSajmpnE0BVkELGQUWKs5GcvqpKPAeervunJQkcKGJBe7SbvZ3Nu0V0AbxJvRXJohYqf/er0JU0ERIZyonXbc2PTnRJlGOUwy3QSDTGhIzKAtqUREaC708UDM3xtlT4OpbIVGbxQf09MidB6IgLbKYgZ6nVvLv7ntRMT3nWnLIoTAxFdLgoTjo3E8zRwnymghk8sIVQxeyumQ6IINTazjA3BW395kzRKRc8teo+lXLmwiiONLtEVKiAP3aIyekAVVEcUzdAzekVvzpPz4rw7H8vWlLOaOUd/4Hz+APkjlUE=</latexit><latexit sha1_base64="1QdmaavMFRzB+ktReGqzxwauB1w=">AAACAHicbVC5TgMxEPWGK4QrQEFBY5EghSbaTQNlJBrKoJxSjpXXmU2s2OuV7Y0URWn4FRoKEKLlM+j4G5yjgIQnjfT03oxm5gUxZ9q47reT2tre2d1L72cODo+OT7KnZw0tE0WhTiWXqhUQDZxFUDfMcGjFCogIODSD0f3cb45BaSajmpnE0BVkELGQUWKs5GcvqpKPAeervunJQkcKGJBe7SbvZ3Nu0V0AbxJvRXJohYqf/er0JU0ERIZyonXbc2PTnRJlGOUwy3QSDTGhIzKAtqUREaC708UDM3xtlT4OpbIVGbxQf09MidB6IgLbKYgZ6nVvLv7ntRMT3nWnLIoTAxFdLgoTjo3E8zRwnymghk8sIVQxeyumQ6IINTazjA3BW395kzRKRc8teo+lXLmwiiONLtEVKiAP3aIyekAVVEcUzdAzekVvzpPz4rw7H8vWlLOaOUd/4Hz+APkjlUE=</latexit>

without 1 andcut (4d)

<latexit sha1_base64="gqzkpb3raZw+SREy1nziXl/ParI=">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</latexit><latexit sha1_base64="gqzkpb3raZw+SREy1nziXl/ParI=">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</latexit><latexit sha1_base64="gqzkpb3raZw+SREy1nziXl/ParI=">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</latexit><latexit sha1_base64="gqzkpb3raZw+SREy1nziXl/ParI=">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</latexit>

without 2 andcut (5d)

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Check if|UBoLBo|

|UBo| "?<latexit sha1_base64="lCy805AE+KkVzi8y2YkXER4Vn3Y=">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</latexit><latexit sha1_base64="lCy805AE+KkVzi8y2YkXER4Vn3Y=">AAACK3icbVC7TsMwFHV4E14FRhaLFqkLVdIFNhAsDAxFIgWpqSrHvWmtOnawnUpV2v9h4VcYYOAhVv4D9zHwOpKt43Pu1fU9UcqZNp735szNLywuLa+sumvrG5tbhe2dupaZohBQyaW6jYgGzgQEhhkOt6kCkkQcbqLe+di/6YPSTIprM0ihmZCOYDGjxFipVTgLKQgDiokOPu8C7WEWh6FbCmNFaD4Mzlry8NJew9H0MRyFHO7CPlGQasalKJ20CkWv4k2A/xJ/Ropohlqr8BS2Jc0SO5hyonXD91LTzIkyjHIYuWGmISW0RzrQsFSQBHQzn+w6wgdWaeNYKnuEwRP1e0dOEq0HSWQrE2K6+rc3Fv/zGpmJj5s5E2lmQNDpoDjj2Eg8Dg63mQJq+MASQhWzf8W0S2xKNj3t2hD83yv/JfVqxfcq/lW1eFqexbGC9tA+KiMfHaFTdIFqKEAU3aNH9IJenQfn2Xl3Pqalc86sZxf9gPP5BaSlp9w=</latexit><latexit sha1_base64="lCy805AE+KkVzi8y2YkXER4Vn3Y=">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</latexit><latexit sha1_base64="lCy805AE+KkVzi8y2YkXER4Vn3Y=">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</latexit>

o o + 1<latexit sha1_base64="gK/miazwLNjZlKXaeNiR5rKAb5I=">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</latexit><latexit sha1_base64="gK/miazwLNjZlKXaeNiR5rKAb5I=">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</latexit><latexit sha1_base64="gK/miazwLNjZlKXaeNiR5rKAb5I=">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</latexit><latexit sha1_base64="gK/miazwLNjZlKXaeNiR5rKAb5I=">AAACIXicdVDNS8MwHE3n15xfU49ewoYgKKOtY5u3gRePE9wHrGWkWbqFpU1JUmWU/Ste/Fe8eFBkN/GfMd1WUNEHgcd775f88ryIUalM88PIra1vbG7ltws7u3v7B8XDo47kscCkjTnjouchSRgNSVtRxUgvEgQFHiNdb3Kd+t17IiTl4Z2aRsQN0CikPsVIaWlQbCTO4pK+GHluYlauGjW7WrswK6ZZt2wrJXa9elmdcYcRXyEh+APk59ZsUCxnaZilYZaGllZSlMEKrUFx7gw5jgMSKsyQlH3LjJSbIKEoZmRWcGJJIoQnaET6moYoINJNFsvN4KlWhtDnQp9QwYX6fSJBgZTTwNPJAKmx/O2l4l9eP1Z+w01oGMWKhHj5kB8zqDhM64JDKghWbKoJwoLqXSEeI4Gw0qUWdAnZT+H/pGNXLLNi3VbLzdKqjjw4ASVwBixQB01wA1qgDTB4BM/gFbwZT8aL8W7Ml9GcsZo5Bj9gfH4BlTuf+w==</latexit>

Solve Sot (!T )

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Deliver x2 and y

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Deliver x1

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Update x1 and x

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Update cut (4d)<latexit sha1_base64="nkpcKwi3+vnhGSXPzQ6zbrZnQN0=">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</latexit><latexit sha1_base64="nkpcKwi3+vnhGSXPzQ6zbrZnQN0=">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</latexit><latexit sha1_base64="nkpcKwi3+vnhGSXPzQ6zbrZnQN0=">AAACK3icdVDLSsNAFJ34rPVVdelmaBEqSEliaeuu6MZlBdMW0lAmk2k7dDIJMxOhhP6PG3/FhS584Nb/cNIHqOiBgcO599x75/gxo1KZ5puxsrq2vrGZ28pv7+zu7RcODtsySgQmDo5YJLo+koRRThxFFSPdWBAU+ox0/PFVVu/cESFpxG/VJCZeiIacDihGSkv9wmXamw1xxdD3UrNy0ajZ1dqZWTHNumVbGbHr1fPqtMcjygPCFXTiACkCcaJguRqcTvuF0tIHlz649EFLKxlKYIFWv/DUCyKchHoYZkhK1zJj5aVIKIoZmeZ7iSQxwmM0JK6mHIVEeunszCk80UoAB5HQTx8zU787UhRKOQl93RkiNZK/a5n4V81N1KDhpZTHiSIczxcNEgZVBLPgYEAFwYpNNEFYUH0rxCMkEFY63rwOYflT+D9p2xXLrFg3dqlZXMSRA8egCMrAAnXQBNegBRyAwT14BC/g1Xgwno1342PeumIsPEfgB4zPL4sco4Q=</latexit><latexit sha1_base64="nkpcKwi3+vnhGSXPzQ6zbrZnQN0=">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</latexit>

LBo = St

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UBo = minUBo1, c>1 x

1

+ [↵ot + (o

t1)>x

1 + (ot2)

>x2]

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Figure 3.2. The GND algorithm workflow

third-stage can keep its own scenario set and return its own single cut. For multiple distribu-

tion systems in the third stage, they return multiple cuts to the second-stage problem based

on the single cut from respective scenario sets. From the third stage to the second stage, it

is promising to adopt the cut sharing [73] between each second-stage scenario to facilitate

the solution, which does not compromise exactness. Besides, The non-anticipativity in any

current stage is implicit since we only have one copy of wait-and-see variables in the previous

stage. For the convergence proof, we first give the following assumption.

Assumption 1. 1) The second-stage and third-stage problems are convex and have complete

recourse; 2) all recourse stages have finite support; 3) all the scenario sets are stage-wise in-

55

dependent.

These are all mild assumptions in the current electricity market in terms of system op-

erations. For the first two items, current industrial ED problems are mostly convex and

feasible (with load shedding allowed), and if we consider stochastic scenarios, they should be

finite in practice. For the third item, the transmission system should have an independent

and different scenario set with DSOs, as well as scenario sets between DSOs. Then based

on [61], consider the following proposition.

Proposition 1. Provided with Assumption 1, constraints (3-4c) and (3-5c) provide accurate

approximations for the second stage and the third stage, respectively.

Proof. From problem Sd to St, subproblem (3-7) is solved for each ωDc , yielding corre-

sponding optimal dual solutions. As problem Sd is convex and has complete recourse, the

Slater’s condition holds. Consider Q(x2,y, ωDc ) as the third-stage problem’s dual:

Q(x2,y, ωDc ) =u⊺(H2y + e)−

v(q⊺y +H⊺3x

∗2 + r2(ω

Dc )).

Based on the convexity of Sd, the subgradients provide:

Q(x2,y, ωDc ) ≥u⊺(H2y + e)−

v(q⊺y +H⊺3x

∗2 + r2(ω

Dc )).

Taking the expectation according to the distribution of uncertainty ωd, and using the sub-

gradient notations, we have:

Q(x2,y) ≥ αd +β⊺d1y +β

⊺d2x

∗2.

Thus, the affine function below for iteration o gives an exact outer linearization for problem

56

Sd and is returned to St:

F (y ∶ yo) = αod + (βod1)⊺y + (βod2)

⊺x∗2

When there are multiple Sd in the third stage, each of them returns the corresponding affine

cut to St. Thus, the optimality of the problem follows from the complete recourse of all

subsystems, and the approximation is exact if and only if Qo = maxoF (y ∶ yo)), i.e., con-

straint (3-5c). Proof is similar for the approximation of St in M , i.e., constraint (3-4c).

And here concludes the proof. ∎

Remark 1. As the approximation of constraints (3-4c) and (3-5c) is accurate for recourses,

based on Remark 4 and Remark 5 in [61], the cuts from subsequent stages formulate ac-

curate approximations only containing the subgradient information. Hence, provided with

Assumption 1, when the cuts provide exact approximations of later-stage problems, after a

sufficiently large number of forward and backward iterations, the algorithm will converge to

a solution within a sufficiently small gap from the global optimality with probability 1.

Remark 2. The assumption that the nodal voltage constraint is not binding is mild [71]

because the voltage deviations in distribution networks should be well controlled by voltage

regulators. It is unnecessary to consider the local voltage stability issue in a market clearing

problem from the ISO’s perspective since there is no such a market product. A Volt/VAR

optimization problem could be solved by local utilities to tackle this issue once the hierar-

chical market is cleared [74].

Remark 3. Note that in our case, M is a MIP. Hence, the final optimal solution satisfies

the potential MIP gap without loss of generality as the affine cuts are created from convex

subproblems. Practically, we admit that the GND’s algorithmic gap is hard to achieve zero,

and together with the MIP gap, the final solution might not be optimal. However, the the-

oretical global optimality remains if convexity holds for recourses.

Remark 4. The first three blocks in Figure 3.2 are for the initialization. However, if we

can find an excellent warm-start initial point, the algorithm will converge faster. Besides,

57

since the cut herein gives an affine approximation, cut sharing and cut selection can further

contribute to the acceleration.

Remark 5. The ISO’s master problem receives only the cut information consisting of sub-

gradients from the lower-level problems. With only the subgradients, one cannot recover the

full network information, which prevents the ISO from having the network information and

thus protects the confidentiality.

3.5. Numerical Experiments

To test the efficacy of the proposed T-D coordinated market operation and the solution

strategy, we carry out case studies on a modified Tran6+Dist7+Dist9 test system from [75]

and a Tran118+Dist30×5 from [46] for day-ahead UC & ED problems. We solve all of the

experiments by Gurobi on a Windows PC with quad-core Intel i7-6700 CPU and 8GB of

RAM.

3.5.1. Tran6+Dist7+Dist9 Test Case

We depict the system topology in Figure 3.3, which consists of one transmission system

(TS) and two radial ADNs. The original test system [75] did not consider the resistance

and reactive demand/generation limits. In this study, they are obtained via a fixed R/X

ratio (assume all distribution branches are with the same configuration and R/X ratio is 1:2)

and power factor (assumed to be V-I 0.95 lagging) using the existing data. We also have

four switches K1-K4 installed to enable the network reconfiguration of each ADN while the

switch can be reconfigured every 8 hours.

For uncertainties, one wind generator G3 and two PV units RG1 in ADN1 and RG2 in

ADN2 are added in the system, whose power outputs are uncertain and sampled from three

different historical datasets obtained from [76]. Note that for the renewables, we scale the

historical data to a distribution level, with the maximum output as 25MW. We also slightly

reduce the capacity of DG1 and DG2 in the original data. The hourly load profile is also

assumed to be stochastic that respects the sampled demand scenarios and is distributed

58

˜˜

˜˜

L1

L2

L3

L4

L5L

L6

L7

L8

L9

G1 G2

G3

DG1DG2

RG1

RG2

TB1

TB2 TB3

TB4

TB5 TB6

12

3

4

569

8

7

1

2

3

4

6 5

7

7.2%8.4%

11.2%

13.2%

40%

3%

5% 4.8%

3.8%3.4%

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Figure 3.3. Tran6+Dist7+Dist9 system topology.

according to the factor shown in Figure 3.3. The penalty cost for the load shedding is

$1,000/MW, in order to make the ADN decisions comparable with the TS.

First, 150, 200, and 200 scenarios for the renewable generators G3, RG1, and RG2

are sampled, and 180 scenarios for the demand are sampled using Monte-Carlo sampling

from the historical data, respectively. The number of joint renewable and demand scenarios

is very high and could jeopardize the computational efficiency. Then, we employ the Fast

Forward/Backward approach [34] in the GAMS SCENRED toolbox to reduce the numbers

of the joint renewable and load scenarios in the three grids. Figure 3.4 shows the sensitivity

analysis of scenario number reduction. We have a base scenario reduction ∆ that has 5, 10,

and 12 scenarios for the TS, ADN1, and ADN2, respectively, and we linearly increase the

number of the desired scenarios as shown in the x-axis of Figure 3.4. It can be seen that

using 10 × ∆ scenarios has a solution gap of 2.12% compared with using 50 × ∆ scenarios.

Hence, we argue that using 10×∆ scenarios achieves the tradeoff between the accuracy and

complexity. Note that the scenarios are i.i.d. wc for the two distribution systems are set

equal in this study, and the convergence criterion ε is 0.01%.

Figure 3.5 shows the algorithmic performance in this case. The GND algorithm con-

verges in the 31th iteration. The total load shedding upon convergence is zero. The costs

of the two ADNs start at extremely high levels in that the TS draws a large amount of

59

Figure 3.4. Sensitivity analysis on the scenario number

energy from the distribution sides to reduce its power outputs in the first iteration. This

phenomenon is because there is no restriction on the power exchange when the two ADNs

have to feed the TS even with high load shedding in the local area. However, the injections

from the TS to ADNs finally become incentivized as the operation cost of the conventional

units in the TS is much lower than the load shedding price in ADNs, and the DER units

in ADNs cannot fully support the local demands. Note that the solution reaches $36,449 in

iteration 17, and it is within a gap of 0.14%, as shown in Figure 3.5. Thus, if we have a

slightly higher tolerance on the optimality, we can greatly reduce the computation time.

Figure 3.6 provides the unit commitment decisions as well as the network reconfiguration

solutions of the master problem upon convergence. G1 is committed ON throughout the

time due to its low generation cost and large capacity. G2 is mainly used during the twilight

time since the PV generation is low at that time, but the energy consumption is high. ADN1

changes its topology once in this operation based on the optimized switch decisions, whereas

ADN2 remains the original topology. The topology change in ADN1 is mainly due to the

60

Figure 3.5. Algorithmic performance in the Tran6+Dist7+Dist9 case

capacity of lateral 3-7 at hour 16 cannot support the increasing demand of load L8 and L9,

and lateral 3-4 can relieve the congestion pressure by bifurcating the flow to two laterals,

i.e., 4-5 and 4-6, towards the demand nodes.

Besides, to illustrate the necessity of the T-D coordination, we carry out the other two

comparative experiments on isolated UC&ED and report the results in Table 3.1 in compar-

ison with the results of the T-D coordinated case. IUCED1 mode means that TS forecasts

the boundary power demand as all demands in the ADN, and IUCED2 mode forecasts

it as the maximum forecast demand minus the maximum generation outputs in the ADNs.

TDC−UCED stands for the proposed T-D coordinated UC&ED setting. It can be observed

from the table that in both isolated modes, there exists a notable power mismatch, even if

the IUCED2 mode can fairly approximate the coordination used by some current industry

61

Figure 3.6. Tran6+Dist7+Dist9 commitment and reconfiguration solutions

practice [46]. A small power mismatch, however, can still raise severe conditions for system

operators as it influences the system stability. We also notice that the received LMP in both

ADNs is decreasing, which is due to the relieved line congestion by the distribution network

reconfiguration considered in the T-D coordinated framework. This observation defends the

necessity of adopting the network reconfiguration-embedded T-D coordination in the future

market. We further conduct two additional analyses to illustrate the algorithmic perfor-

mance.

1). Exactness of the SOCP relaxation

While the branch flow model adopted in this work has been widely recognized as a nearly

exact relaxation of the actual AC power flow [66], we further demonstrate its exactness by

the SOCP gap, which is also a well-known index for evaluating the SOCP performance [77].

Table 3.2 tabulates the maximum SOCP gap values throughout scenarios in the two ADNs.

Generally, the smaller the gap is, the higher exactness the formulation achieves [77]. It

62

Table 3.1. Comparative Analysis on Isolation and Coordination.

Total Costs: $38,943.87

Power Mismatch 31.1% 37.8% Total Costs $4,324.82 $5,609.71

Received LMP $16.83/MWh $16.83/MWh Total Costs: $27,835.94

Power Mismatch 5.8% 8.4%

Total Costs $3,658.62 $5,119.84 Received LMP $15.47/MWh $15.47/MWh

Total Costs: $28,294.12

Power Mismatch 0% 0% Total Costs $3,316.20 $4,789.09

Received LMP $14.73/MWh $14.73/MWh

Table 3.2. Maximum SOCP Gaps of the two ADNs (p.u.)

Network ADN1 ADN2

SOCP gap 0.0064 0.0037

can be shown from Table 3.2 that our proposed model is accurate enough. Though this

approximation still cannot fully replace a complete AC power flow model, it gives an ex-

cellent starting point for such analyses run by ISOs with superior computational efficiency.

Figure 3.7 also shows that under normal system conditions, the nodal voltage constraints

are non-binding, which supports the strong duality of the SOCP-based subproblem.

2). Sensitivity analysis for initial values

To technically support Remark 4, we conduct a sensitivity analysis for initial values in

the algorithm. Note that we are particularly interested in the distribution reconfiguration

variables’ initial values since the unit commitment variables will render no infeasibility in

subsequent stages. We use five cases of initial values for the sensitivity analysis.

63

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ADN1

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ADN2

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Bus

Figure 3.7. Boxplot for voltage magnitudes of the two networks

• Case 1. Throughout the horizon, K1 = 1 and K2 = 0, K3 = 1 and K4 = 0.




• Case 5. The optimal configuration settings obtained by the simulation.

It is straightforward that Case 1 and Case 2 give a feasible set of initial values, while Case 3

and Case 4 give an infeasible one. Case 5 serves as a comparison with a perfect initial value.

Figure 3.8 depicts their performance. It is shown that all five cases converge to the optimal

solution, whereas Case 1 and Case 2 spend fewer iterations than Case 3 and Case 4, but

Case 5 achieves the fastest performance. The additional iterations spent in Case 3 and Case

4 are mostly feasibility-check iterations, where distribution systems return feasibility cuts.

64

This experiment shows that a feasible initial value leads to a better performance, whereas a

perfect initialization would further save the computational time.

3.5.2. Tran118+Dist34×5 Test Case

To test the scalability of the proposed method, we carry out experiments on a Tran118+Dist34×5

test case modified from [46], which consists of one IEEE 118-bus transmission system and

5 IEEE 34-bus distribution systems. A similar data modification such as R/X ratio and

power factor is conducted to make the original system suitable for our study. Moreover,

we also perform similar scenario generation and reduction for the additional ADNs (only 20

scenarios are considered in each ADN, and 10 scenarios are considered in the TS). The load

shedding cost is set as $1,000/kW in ADNs to keep comparability.

Three cases with different stopping criteria ε for the GND algorithm, namely 3%, 1%,

and 0.1%, are set up to test the algorithmic performance in a large system. Figure 3.9 shows

the comparative results. The boundary purchase here is defined by the sum of multiplica-

tions between boundary LMPs, which are the dual values of the boundary nodal balance

constraints in the second stage, and the power injections from the TS. We multiply the total

ADN cost and the purchase cost by 10 to make costs comparable. It can be observed that

the convergence speed greatly depends on the choice of the stopping criterion. And the

algorithm spends a great number of iterations to close the gap between 1% and 0.1%, which

is also a feature of optimality conditioned algorithms.

The CPU time for the case when ε = 1% is around 4.5 hours, which is acceptable consider-

ing such a stochastic, complex, and accurate modeling for both transmission and distribution

systems. However, directly running the corresponding deterministic equivalent SP problem

drains up the RAM and returns no solution. This observation defends the necessity of using

the proposed decomposition technique for acceleration. Note that with a more powerful sim-

ulation platform, such as high-performance CPU clusters, solving all the second-stage and

third-stage subproblems with each scenario realization could be implemented in parallel,

which can significantly reduce the computational time.

65

Figure 3.8. Sensitivity analysis on the initial values

3.6. Summary

To help the ISO with tackling the increasing system intermittency from the active dis-

tribution system and reduce the boundary power mismatch, we propose a T-D coordinated

market paradigm including UC and ED. This work is mainly from the ISO’s perspective

to optimize its daily operation considering the DSO’s performance. A generalized nested

decomposition method is tailored and efficiently decomposes the problem, which greatly fa-

cilitates the solution. The theoretical convergence proof of this algorithm is also articulated.

Numerical experiments corroborate the effectiveness of the proposed strategy and show the

necessity of the T-D coordination.

Future studies include exploring more effective cut selection from the convex relaxation

of subproblems, e.g., the Lagrangian dual cut, to reduce the computational time. Besides,

with the increasing DER installation in distribution systems, similar coordinations between

DSOs and microgrids await in-depth investigation.

66

Figure 3.9. Comparative analysis with different stopping criteria

67

Chapter 4

FUTURE ELECTRICITY MARKET DESIGNS AND GENERIC

SIMULATION PLATFORM

In this chapter, we comprehensively discuss the state-of-the-art progress of the electricity

market operations considering renewable resources, which is summarized in an overview. We

throw a draft design of the future market structure based on the renewable impacts and the

increasingly distributed framework. Then, we also present a general simulation platform for

researchers and industry partners to conduct state-of-the-art electricity market operations.

We implement this platform as a potential open-source simulation package for future users’

flexibility. The main contents of this chapter, thus, can be divided as follows.

• State-of-the-art market operations considering variable renewable energy are reviewed.

Several questions are raised against the development of the future electricity market.

• A draft design on the future market structure considering renewables and the dis-

tributed framework is discussed.

• Attempting to solve the revealed questions, a general electricity market operation sim-

ulation platform considering heterogeneous participants with novel market models is

presented.

4.1. Overview for State-of-the-art Market Operations with Renewables

Renewable energy has become one of the dominant renewable resources in current power

systems. To meet the growing requirement of the deregulated market structure, most market

operators in the world, e.g., the independent system operators (ISOs) in the U.S. and the

transmission system operators (TSOs) in Europe, have been conducting research in renew-

able integrations. With the accelerating penetration of renewable energy on both sides, how

68

to coordinate renewable generation with conventional generation technologies in short-term

market operations with reliability and robustness attracts research attention.

Short-term electricity market operations include energy markets, reserve markets, and

other ancillary service markets [78]. Unit commitment (UC) and economic dispatch (ED)

are two essential operation routines in the electricity market, which is also the main focus

of the research in this field [79]. UC and ED determine the process of daily generation feed-

ing system demands in multiple timescales. The reserve market ensures system reliability,

where the renewable participation can significantly alleviate the pressure of conventional

reserve providers [80]. Researchers are also interested in the PV’s capability providing other

ancillary services such as voltage regulation and frequency regulation, which are also still

open questions in the renewable-integrated market operation. With renewable uncertainty

considered in these operations, challenges for grid reliability and system dynamics arise.

While the ISOs keep using deterministic market operations to ensure tractability, nu-

merous research studies aiming at improving these market operations propose various ap-

proaches to tackle renewable uncertainties, such as stochastic programming (SP) and robust

optimization (RO). Additionally, before solving these problems, it is requisite to address the

renewable uncertainty modeling with effectiveness and accuracy, which is also under exten-

sive discussion with the development of data analytics and statistical learning algorithms.

It can be envisioned that in the future market operations, these uncertainty-based modeling

and optimization approaches can substantially contribute to the social welfare upon address-

ing the challenge of renewable intermittencies. However, before the real-world application

of these methods, their robustness and scalability need to be further verified by realistic

simulation and field demonstration. Recently, the Federal Energy Regulatory Commission

(FERC) releases an order that allows distributed energy resources (DERs) to participate

in the electricity wholesale market [81], the idea of the transactive energy and peer-to-peer

(P2P) market will gain more significant interest in utilities and ISOs, which is directly re-

lated to the household PV generation and retail market design.

In addition, long-term contractual arrangements for renewable integration can also influ-

69

ence their short-term operation cost. For example, mid-term or long-term power purchase

agreements (PPAs) have been widely used for solar power integration [82]. According to [83],

many European countries and the U.S. are actively promoting the feed-in-tariff mechanism

for solar energy to incentivize a long-term carbon-free electricity investment. In general, a

simplified and unified market structure that considers both short-term and long-term market

operations has proven to be an effective strategy to maximize the benefits of renewables [84].

Though these long-term measures have various impacts on renewable integration cost, this

review article will limit the range of discussion in the short-term electricity market.

4.1.1. Renewable Integration in the Electricity Market

As the installation of renewable resources grows sharply, market operators have developed

different rules for renewable integration in their daily operations. In this work, the discussion

on various forecasting methods, which have enjoyed thorough investigation in academia, is

not the main focus. Instead, the focus is on how these renewable generation forecasts can

be leveraged in realistic market operations. More specifically, renewable participation in

different types of electricity markets is delineated. These forecasting methods are briefly

discussed later.

4.1.1.1. Renewable Participation in the Energy Market

Solar units in most ISOs rarely bid into the markets. Instead, solar units are more

generally considered as negative demands and integrated to the demand forecast in daily

operations, especially for distributed PVs, which is reasonable in the transmission-level mar-

ket because of its emphasis on the bulk generation scheduling. For utility-scale solar units,

however, there have already been some active PPA contracts in some ISOs with high renew-

able penetration such as CAISO, and their bidding price keeps decreasing as the renewable

penetration grows. As the current installed capacity of solar energy is relatively incompara-

ble with the coal-fired and nuclear power plants, most of the generation portfolio still comes

from the conventional generation companies (GENCOs). However, the changes in market

70

design vary greatly. Some market rules and protocols are under substantial development in

places such as CAISO where the local solar penetration has already reached around 25.1%

to feed the daily demands in California and keeps increasing.

Industry operations for the uncertainty-based renewable integration remain determinis-

tic. Currently, different ISOs have their own methods to take intermittent solar resources

into the consideration. Electric Reliability Council of Texas (ERCOT) uses the exact solar

forecast and performs adjustment if necessary, and during real-time operation, renewables

should follow the energy offer curves or be considered fixed as the telemetered output; Mid-

continent Independent System Operator (MISO) performs its planning and operation with

dispatchable solar units considering multiple levels of the forecast, i.e., peak, mean and min-

imum values; CAISO clears the market using the bid submitted by solar assets. Specifically,

for CAISO, a unique reserve called online PV solar reserve is procured based on 15% of

the hourly sum of all solar at-risk forecasts as a regular contingency reserve, for potential

loss of solar. On the contrary, the European electricity market has a different strategy from

the U.S. cases. The European Network of Transmission System Operators for Electricity

(ENTSO-E) has four submarkets in the market operation, i.e., the day-ahead spot market,

intraday market, balancing market, and imbalance settlement. The solar units, together

with other types of renewables, can bid in these submarkets with integration costs, but the

current solar integration in the E.U. is not incentivized because the designed cost scheme is

not competitive compared with the low carbon prices. To this end, researchers are actively

developing novel market structures and designs, especially the cost mechanism, to help ac-

commodate more renewables with reliable and robust operations.

For details about the operation, day-ahead schedulers not only use the forecasts provided

by the forecast vendors but use asset optimization software including the operating character-

istics of the installed renewable generators to make reliability-driven UC and ED decisions.

Then, real-time schedulers use the obligations from day-ahead schedulers and assessment

of the conditions throughout the operating day to adjust the decisions. It is convenient to

implement and monitor this process, which satisfies the requirement of interpretability and

71

high reliability in ISO’s operations.

Notably, based on the famous duck curve report from CAISO, it has also been indicated

that the operators use pre-dispatch and re-dispatch with high reserves to bolster the system

reliability with the renewable intermittency. For the daily market, ISOs perform a day-

ahead operation with a resolution of one hour for UC decisions, then a real-time dispatch

is performed with a resolution ranging from 5 minutes to 15 minutes, under the real-time

forecast and system requirements, together with contingency screening and corresponding

rescheduling if contingencies occur.

In addition, it is noteworthy that the market settlement for solar in the current U.S.

wholesale market remains identical with other generation types, which means that there is

no difference in terms of the final power deployment between renewables and conventional

generation. Taking ERCOT as an example, for the day-ahead operation, the settlement

price is exactly the day-ahead locational marginal price (LMP), whereas the real-time set-

tlement point price is the real-time LMP plus a UC reliability deployment price adder, i.e.,

the energy imbalance charge with a reserve price adder. Both the day-ahead price and the

real-time price apply to all resource types.

However, particularly for distribution systems, New York Independent System Operator

(NYISO) is now exploring a different settlement scheme of distributed PV in the market. It

is proposed that the under-generation and over-generation of DERs need to be taken into

consideration the settlement point price. From the research community, though, there have

been some papers regarding the renewable settlement such as the Gaussian residual bidding

proposed in and the stochastic bilevel programming-based settlement proposed in, solar set-

tlement in the market operation still requires further investigation.

As the intermittency issue of renewables is tackled deterministically by re-dispatch and

reserve procurements in the industry practice, the information of renewable generation pro-

files provided to ISOs should be as complete and accurate as possible. Especially for the solar

generation, as the distributed PV installations grow more and more popular, the coordination

between ISO and potential distribution-level markets is needed. While there have been some

72

works regarding the transmission and distribution coordination, it is well acknowledged that

it is important to enable the information exchange between transmission and distribution

markets with variable solar generation. Besides, with the transactive energy framework on

the demand side, small-scale local markets such as microgrids, are currently under research

and real-world experiment. These localized and decentralized markets possess considerable

potential to emerge in the future’s market hierarchy, requiring ISOs to find a better solution

for coordination between the wholesale market and retail markets.

On the methodology side, numerous studies aim at addressing the solar uncertainty di-

rectly in stochastic models, such as SP and RO, rather than the deterministic formulation.

While the technical details of these models are introduced in the subsequent section, the ad-

vantage of leveraging stochastic models is based on the reliable and robust solutions. With

the forecast or historical data of renewable resources at hand, one can model the solar un-

certainty in various ways and make risk-averse decisions by these models. In the summer

of 2019, CAISO provides its assessment on the summer load and resource for a short-term

UC using a scenario-based SP approach. ISO-New England (ISO-NE) and MISO also pay

attention to the application of RO in the UC and conduct several research projects. Though

these studies show the potential of leveraging stochastic models in the real-world market

operations, however, large-scale application of stochastic models in the electricity market is

not ready until issues such as the computational efficiency are fully addressed.

The main reason lies on the capacity difference between them. For instance, the entire

installation of wind capacity in the Pennsylvania-New Jersey-Maryland (PJM) area is nearly

14 times the solar installation. However, due to the rapidly increasing solar penetration, it

is much needed to model and capture solar variability and uncertainty in market operations.

Also, in the market optimization models, the uncertainty modeling for wind and solar is

dealt with similarly. For instance, the uncertainty set-based robust optimization is applied

to model both the solar uncertainty and wind uncertainty. We will detail the discussion on

the solar uncertainty modeling later. Meanwhile, some ISOs also follow the same forecast-

optimization-settlement procedure for wind and solar, like MISO and CAISO. It should be

73

noted that although the modeling techniques used for wind and solar are similar, the un-

derlying drivers for their output are inherently different: solar radiation vs. wind speed.

Besides, solar energy obeys the pattern of daily volatility, which differs from the seasonally

volatile feature of wind energy, posing a unique technical barrier for the development of solar

integration.

4.1.1.2. Renewable Participation in the Reserve Market

For the reserve provision from renewables, which has drawn extra attention for its in-

termittency issue, several research teams also gain some experiences from conducting pilot

projects. National Renewable Energy Laboratory (NREL) and Argonne National Labora-

tory (ANL) have already published reports and revealed the impact of renewables on the

power system reserve provision [43]. For a high renewable penetration scenario, balancing re-

serves should be increased and respond faster to maintain the system reliability and stability,

which increases the integration cost of renewable energy. To tackle this issue, North Ameri-

can Electric Reliability Corporation (NERC) recommends that PV plants must provide their

share of grid supports, and relevant research is performed for a grid-friendly PV plant, which

is reportedly capable of providing various ancillary services including the primary reserve in

the test. On the other hand, a significant economic opportunity for renewable systems is

revealed if renewables participate in secondary and tertiary controls of the reserve markets

as discussed in. Inspired by these findings, enabling renewables to provide such active power

reserves and other grid supports takes shape.

If we take solar energy as an example, in general, the research from this perspective

can be classified into two groups: the first is combining PV systems with auxiliary storage

systems to provide reserves; the second is making PV systems operate under the maximum

available capacity to provide reserves using the curtailed power. Particularly in, partial

shading conditions are considered, which is closely related to the realistic PV operating con-

ditions. In parallel with this research advancement, First Solar, NREL and CAISO started a

project in 2016 to test a 300 MW utility-scale PV power plant in California. The field tests

74

demonstrate that advanced controllers can be installed in PV systems to provide services

that range from spinning reserves, load following, ramping, frequency response, frequency

regulation, and voltage support, which paves a potential path that power electronics and

control strategies can help PV compete in the ancillary service market. Another demonstra-

tion project led by NREL, First Solar and ERCOT also comes to a similar conclusion. In

summary, the advancement of control strategies enables the provision of different levels of

reserves from PV systems for power system operations, when the value of this practice has

been substantiated so far.

In addition to the aforementioned technical research and demonstration projects, an-

other widely discussed issue is how the reserves provided by renewable units participate in

the markets. From this perspective, He et al. developed a general framework on the optimal

offering strategy for CSPs in a joint day-ahead energy, reserve and regulation market, which

adopts RO to address solar energy uncertainty and SP to address market price uncertainty.

Xu et al. extended this work by considering that an electric heater (EH) is additionally

integrated with CSPs, which further enhances the flexibility of CSPs in terms of the reserve

provision and bidding strategy in the market. The reserve provision from renewables is often

associated with the energy storage system (ESS), as proposed an annual bidding strategy

for industry-level PV plants with ESS participating in the secondary reserve market.

4.1.1.3. Potential Market Designs for the Solar Integration

Recent research on the renewable-integrated electricity market design focuses on how

to increase competitivity among market participants while maintaining system reliability.

Some featured development of new renewable integration mechanisms and market rules are

summarized as follows.

• Renewable bidding strategy. Renewable units are widely recognized as units with

a nearly zero marginal cost. As reported in [85], bidding zero for renewables in the

current LMP-based market environment may not help the investors recover the long-

75

term payback as the renewables are of high capital cost. L. Marshall et al. [86] proved

that, under the current market environment, zero-marginal-cost generators gain more

advantage when competing with other generators using collusive shadow prices, which

illustrates the inefficiency of the existing market mechanisms towards high renewable

penetration. Some other works focus on distribution market designs, including the

capacity-price game for renewable investors [87] and the average pricing market with

abundant renewables [88].

• Systems reliability. As distributed renewable units contribute largely to the total

solar installations, ISOs/RTOs concern that the uncertainties and fluctuations in the

downstream network may propagate back to the upstream market. Z. Li et al. [89]

provided a new distributed algorithm to cope with the transmission-distribution coor-

dination in economic dispatch without compromising each stakeholder’s confidentiality.

Then, P. Li et al. [48] tailored this method and applied it to a centralized distribu-

tionally robust optimization-based dispatch model, where the renewable uncertainty

is considered. The key factors to be considered in this type of market design include

the confidentiality of information exchanged between transmission and distribution

operators, and the uncertainty modeling of renewables on the distribution side.

• Transactive energy. Distributed market frameworks are under development to ac-

commodate the growing distributed energy resources, and it closely relates to the trans-

active market framework. Supported by the breakthrough of power electronic control

strategies (e.g., [90]), discussions over localized trading of solar power in an islanded

microgrid are presented [91]. A. Jafari et al. [92] proposed a new profit allocation

strategy for multi-energy microgrid management with energy storage-embedded dis-

tributed renewables. T. Khalili et al. [93] provided a novel multi-objective scheduling

framework for microgrid management with abundant renewables.

It is noteworthy that different market designs lead to different operation strategies, and

later sections in this chapter will delineate the market operation strategies focusing on how

to integrate the solar power reliably.

76

4.1.1.4. Impact of Future Market Designs

As discussed above, various market designs suggest transitioning the electricity market

structure from a conventional top-down hierarchy to a distributed bottom-up one due to

the proliferating renewable penetration on the distribution side. Particularly, the ancillary

service requirement and settlement will also be updated in the new market designs to cope

with new challenges introduced by the increasing renewables. However, such a transition

may impact future solar investment and the associated risk management.

Extensive works have been devoted to novel ancillary service procurement based upon

the new challenges raised by renewables. Apropos of the industrial practice, recently, P. Du

et al. [94] proposed a novel ancillary service market framework considering the increasing

renewables for ERCOT. Apart from the conventional regulation and primary frequency re-

serves, fast frequency response and contingency reserve can be added into the new market

framework with practical implementation protocols. They are to equalize the decreasing sys-

tem inertia and more severe frequency events from renewables. G. Zotti et al. [95] proposed

a framework called Ancillary Service 4.0, a top-down market structure, to appropriately

allocate the ancillary service in multi-scale system operations. Different scales of ancillary

service trading occur locally in the distribution systems or microgrids, monitored by a new

ancillary service operator in a distributed manner, whereas the central system operator col-

lects the cleared bids and offers. D. Jay et al. [96] proposed a Stackelberg game in the system

operation for the reactive power ancillary services, in which the ISO acts as a leader to co-

optimize and GENCOs act as followers to bid ancillary services. This asynchronous process

can serve as a reference to the centralized operation to system operators. To date, both the

centralized and decentralized approaches have been well investigated in terms of new designs

for ancillary services, which lays a solid foundation for the future market transformation.

As a result of the market structure transformation, renewable investment is also reconsid-

ered. Most of the studies, taking solar as an example, concerning the investment evaluation

are conducted for distributed energy resources, using performance metrics such as the En-

ergy Payback Time [97] and Baker-Finch parameterization [98]. The results show that a

77

single state-of-the-art PV panel is able to finish the investment payback in 7 to 12 months.

For utility-scale PV farm investment, though the payback time increases to years due to

higher investment cost, analyses show that solar is the most competitive renewable type

together with wind, considering various environmental factors [99]. It is also reported that

mergers and acquisitions could help with large-scale investments, which can also support risk

management. Concerning system operations, renewable energy has been practically proved

to be a cost-effective resource with long-term financial payoff, even in a full energy market

case [54]. However, many investors still hold a conservative attitude to the liquidity and

market power of solar energy, especially the utility-scale PV plants [100]. In fact, under

the current industry operation protocols, most of the bulk solar investments fail to pay back

without an appropriate feed-in-tariff, which is beyond the free market competition. Nonethe-

less, thanks to the future market structure transformation and advanced control strategies,

when renewables are free to provide all types of market services, it can be envisioned that

solar energy will gain significantly increased competitivity against other resources. Besides,

enabling multi-type ancillary service provisions could also reduce energy curtailment and

hence de-risk solar investment.

4.1.2. Renewable Uncertainty Modeling in Electricity Markets

As discussed in the previous section, the biggest challenge in the renewable-integrated

market is how to tackle the growing system supply intermittency. For industry practice, the

forecast values of renewable generation are directly put into the deterministic model, and the

supply intermittency is tackled via corrective control. The mainstream research, however,

suggests leveraging stochastic models and dynamic analytical methods on the uncertainties.

As the priority is modeling the uncertainty accurately and concisely, this section serves

as an introduction of the uncertainty modeling. In this chapter, the discussion on various

forecasting methods, which have enjoyed thorough investigation in academia, is not the main

focus. Instead, the focus is on how these renewable generation forecasts can be leveraged

in realistic market operations. More specifically, renewable participation in different types

78

of electricity markets is delineated. Note that these approaches are for the future solution

methodologies, which hold potentials to be applied in the renewable-integrated electricity

market studies. The interpretability of uncertainty modeling is also one of the industry’s

concerns. Typically, there are three ways of renewable uncertainty modeling, which are

detailed in the following subsections, respectively.

4.1.2.1. Sampling-based Modeling

The most widely used method in the renewable modeling, as well as other types of renew-

able models, is the sampling-based technique. As a basic idea of data analytics, the sampling

method draws a representative sample for the raw data profile by generating subsets cho-

sen from the overall population, each of which represents one realization of the variable

uncertainty with a probability. This method is welcomed by researchers for the excellent

interpretability and implemental convenience.

The approaches for the sampling technique vary according to the problem formulation

and data characteristics. For the sampling of a single-dimensional uncertain parameter in

the SP formulation, Monte-Carlo simulation [61] is the most popular method, which gener-

ates scenarios from the historical data profile concerning a specified probability distribution

function (PDF). When the uncertainty becomes multi-dimensional, a multi-stage scenario

tree [101] is the most common approach. The scenario tree constructs scenarios based on the

uncertainty dynamics in multiple stages, usually coinciding with the temporal correlations of

the underlying uncertainty, while the decisions are also made in each stage correspondingly.

The drawbacks of sampling-based techniques are apparent, as the accuracy of data real-

izations highly relies on the number of scenarios used in the model and the confidence of the

specified PDF. Many works adopt scenario reduction methods to balance the tradeoff be-

tween the computational complexity and model tractability. Typical works include the fast

forward/backward method [34], the Kolmogorov metric [102], and K-means clustering [103].

Interested readers can refer to [104] for more details about the taxonomy of scenario re-

duction. However, when there is a reduction of scenarios, there is always a reduction of

79

the solution quality. The tradeoff between accuracy and tractability for the sampling-based

method is the major obstacle before the real-world application. Besides, the choice of PDF

also affects this tradeoff, as the PDF’s fitting accuracy of the raw data controls the con-

fidence of the results. Particularly, Beta distribution [105] and kernel density estimated

non-Gaussian distribution [106] are the most popular metrics to draw daily scenarios for the

renewable uncertainty.

4.1.2.2. Boundary-based Modeling

To avoid the curse of dimensionality in the sampling-based approaches, the uncertainty

can be modeled within a confidence interval, which is often leveraged by RO techniques.

To formulate this boundary-based uncertainty set, extreme value identification and proba-

bilistic identification are the most popular approaches, of which the latter finds a specified

probability level of the look-ahead forecast output and hence combines multiple PDF values

to formulate the boundaries [79].

It is requisite that the boundary-based uncertainty set encloses finite extreme points

or extreme supports to guarantee the finite convergence of robust algorithms [22]. Thus,

the utilization of polyhedral sets [107] and ellipsoidal sets [108] for constructing the convex

boundary of the uncertainties leads the mainstream in the RO’s uncertainty modeling. Par-

ticularly, the ellipsoidal set is mainly considered for discrete or spatiotemporal correlated

scenarios [109]. Equations (4-1) and (4-1) introduce the general formulations of these sets,

Wpolyhedral = ω ∶ ωlower ≤ ω ≤ ωupper, (4-1)

Wellipsoidal = ω ∶ ∣∣ −√

Σ ⋅ ω∣∣2 ≤ Ω, (4-1)

where the polyhedral set is created by the value bounds of the uncertainties and the spa-

tiotemporal correlations are captured by the covariance matrix Σ within a 2-norm limit Ω

in the ellipsoidal set.

Apart from them, there are also other types of uncertainty sets which address the extreme

80

scenarios, such as the umbrella set [110] and chance-constrained (CC) reformulation [111].

However, for the renewable energy, using a polyhedral set to construct the robust uncertainty

enjoys the highest popularity, since the daily performance of renewable outputs can be easily

interpreted via box constraints with the maximum at noon and minimum at night.

Another kind of boundary-based uncertainty modeling tries to model the uncertainty by

constructing the ambiguity set instead of explicit box constraints. It uses a portion of the ob-

served data or temporal data to extract statistical properties from the moment information

or distribution divergence. And the ambiguity set does not require the exact and complete

data profile like SP or the overconfident boundary settings. The most typical formulation of

this moment-based boundary uncertainty set is shown in (4-3), which is created by moment

boundaries.

W =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

P (ω ∈ Ξ) = 1,

Fω ∶ (Eω − µ)⊺Σ−1(Eω − µ) ≤ γ1,

(E(ω − µ)(ω − µ)⊺ ≤ γ2Σ

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

, (4-3)

where Fω denotes the moment distribution, µ and Σ denote the mean and variance of the

empirical data, and hyperparameters γ1 and γ2 adjust the level of robustness. To be more

specific, the three constraints in (4-3) define the probability support, first and second order

moment boundaries, respectively.

Generally, this approach constructs the boundary with probability confidence from an

empirical dataset, which is naturally adaptive to the CC formulation [112]. When (4-3)

could conveniently evaluate the Kullback-Leibler divergence applied in [113], the ambiguity

set can also be constructed via the Wasserstein metric [114]. Both of them aim at evalu-

ating the distance between PDFs and only differ from the computational dimension. This

uncertainty modeling lays the foundation of the distributionally robust optimization, which

will be discussed in the next subsection.

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Table 4.1. Comparison of different learning-based approaches and their applications.

Applications Examples Features research Learning-based method

Economicapplications

Transmission markets

[115] Ensemble learning

[116] Recurrent neural net

[117] Autoregressive

[118] Generative adversarial net

Distribution markets


[120] Artificial neural net

[121] Radial basis function neural net

Engineeringapplications

Angle stability

[122] Decision tree

[123] Extreme learning machine

[124] Convolutional neural net

Voltage stability

[125] Extreme learning machine


[127] Convolutional neural net

4.1.2.3. Learning-based Modeling

Recent advance of data analytics, statistical pattern recognition, and machine learning

has enabled the learning-based modeling of renewable uncertainty. Besides the sampling-

based and boundary-based methods discussed above, the learning-based methods try to

extract or “learn” the statistical information of the existing data and then use the learned

model to represent the data patterns for forecasting and uncertainty modeling. Novel re-

newable forecasting methods have been used extensively to enhance the renewable-integrated

market, e.g., [115]. Generally, the improvement of the renewable forecast can dramatically

enhance the decision-making of the market operation. Scenario generation techniques have

also been proposed to capture renewable uncertainties in a stochastic optimization setting,

e.g., [118]. Table 4.1 gives a detailed comparison of relevant research articles focusing on

different applications.

It can be observed that for each category of applications, various learning-based methods

82

are adopted. For a more comprehensive review of recent advances in renewable forecast

techniques, readers can refer to [128]. It is shown that learning-based techniques can better

capture the dynamic nonlinearity and the spatiotemporal correlation of renewable generation

data. This advantage gives the learning-based methods a promising future. Besides, more

accurate renewable uncertainty modeling and forecast will also help reduce the investment

risk when performing portfolio analyses.

4.1.3. Future Solution Methodologies in the Renewable-Integrated Electricity Market

Associated with the modeling of solar uncertainty, the discussion on the solution method-

ologies, which highly depend on the type of market formulation and scenario generation for

the solar-integrated market, is also prevalent. Concerning the uncertainty-based market op-

erations, on the one hand, the solutions towards the transmission-level market, are mostly

related to model-based stochastic market formulations, such as SP and RO. These methods

have broad applications in UC, ED and market bidding/pricing problems, taking the cur-

rent research focus. For distribution- and customer-level market, on the other hand, which

raises recent research attention, the market formulation sometimes links with the model-

free dynamic game, while conventional stochastic models are found to be less applicable to

precisely describe the operation. From the application perspective, the upstream markets

are promising to solve the intermittency issues via these stochastic and robust models, as

the newly added stochastic analytical functionalities do not alter greatly the current market

rules. Besides, though the local solar-based market operations are mainly of the utility’s

interest, ISOs are also supporting the research conducted in this area to fortify the upstream

operations from the origin of the intermittencies. Table II tabulates the research literature

using these approaches (abbreviations are defined in the following) and the subsequent sub-

sections address the details of each. Note that in this section we extend the survey from

solar-integrated models to renewable-integrated models, as many approaches are claimed to

be general and scalable to all types of renewables.

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4.1.3.1. Stochastic Programming

SP theory has been under thorough investigation throughout the last decades and gained

considerable research interest in the market operation with uncertainties. Associated with

the sampling-based modeling techniques described in the previous section, SP leverages

the generated scenarios to implement reliability-assured stochastic models in a preventive

manner. Amongst the research in the market operation with uncertainties, the stage-wise

formulation receives widely adoption due to its natural fit to the market hierarchy. Taking

a solar uncertainty-based day-ahead UC (SUDAUC) problem as an example, a two-stage

formulation can be constructed as (4-4).

minIt,Pt

∑t

c⊺It +Eωg⊺Pt (4-4)

s.t. A ⋅ It ≤ J, ∀t,

B ⋅ It +C ⋅Pt ≤ K(ω), ∀t,

where the first stage determines the integer variables, It, including the start-up, shut-down,

and commitment decisions. The second stage finds the optimal dispatch Pt based on the

commitment decisions from the first stage. The first constraint in (4-4) is the first-stage

constraint, and the second constraint acts as a coupling constraint between two stages, in

which (ω) represents the uncertain parameter with the scenario index ω. The parameters

and constraints in this formulation are free to be assigned with other physical meanings,

such as production costs, reserve regulations and minimum ON/OFF time constraints. This

formulation befriends the operational hierarchy in the current wholesale market, and thus

gains vast applications in a large amount of literature. For the real-time operation problem,

(4-4) reduces to a single-stage SP model and thus is trivial to solve.

Another type of application on the stage-wise formulation is the temporal modeling of SP

with scenario trees. Retaking this SUDAUC problem as an example, (4-5) presents another

84

form.

minIt,ω ,Pt,ω

∑t

Eωc⊺It,ω + g⊺Pt,ω (4-5)

s.t. A ⋅ It,ω ≤ J, ∀t,∀ω,

B ⋅ It,ω +C ⋅Pt,ω ≤ K(ω), ∀t,∀ω,

(It,ω,Pt,ω) = (It,n,Pt,n), ∀t,∀ω,

where n denotes the root node of the scenario tree. This formulation shows a typical multi-

time-stage SP problem, which means the operation in each hour is solved individually with

one column in the scenario tree and updated hour by hour. The system modeling of this

temporal modeling is similar to the aforementioned hierarchical formulation, whereas the last

constraint in (4-5) represents a non-anticipativity constraint. This constraint guarantees the

same decisions obtained by scenarios from the same node in the tree and thus enforces the

wait-and-see nature of the next time stage when solving the problem in the current stage.

For solving the multi-stage SP problem, strategies can be classified into two groups:

stage-wise decomposition and scenario-wise decomposition. For the stage-wise decomposi-

tion, cutting-plane methods are the most typical techniques, which decompose the original

problem into several small problems with hierarchical or distributed stages. Hyperplanes are

used to relax the subproblems’ solutions and then delivered to the master problem, which cut

and thus shrink the optimality region of the original problem to accelerate the solution. The

most representative technique among the cutting-plane methods is Benders decomposition

(BD) or the L-shaped method, in which the hyperplanes are created via dual cuts. This

algorithm requires the subproblems to be convex, while it is trivial to group all nonconvex-

ities in the solar-integrated market operation such as the integer constraints to the master

problem. Besides, other cutting planes like the primal cut generation via the Dantzig-Wolfe

decomposition and recently updated stochastic decomposition also show their effectiveness

towards the multi-stage SP problem. It is noteworthy that, for the multi-stage SP of tem-

poral modeling, these stage-wise decomposition techniques are required to be nested [120],

85

while stochastic dynamic programming (SDP) is sometimes emphasized for its potency com-

pared with the nested decomposition.

Scenario-wise decomposition, on the other hand, tends to enhance the computability by

traversing scenarios instead of stages, which is related to augmented Lagrangian relaxation

approaches. Lagrangian relaxation is originally designed to relax the complicating con-

straints that involve nonconvexity and transition the original problem to the unconstrained

optimization. For its application in the multi-stage SP problem, the non-anticipativity con-

straint is therefore relaxed, by which the problem becomes single-scenario-based and solvable

in a distributed manner for each scenario’s realization. In most of the time when augmented

Lagrangian relaxation solves SP problems, a regularization term, which could be proximal

or entropic, is always required in the objective to penalize the optimality search, as shown

in (4-6) that formulates the version using the proximal regularization, as known as L2 regu-

larization in machine learning communities.

minIt,ω ,Pt,ω

c⊺It,ω + g⊺Pt,ω + λ⊺ω(It,ω − It) + α ⋅ ∣∣It,ω − It∣∣2 (4-6)

s.t. (It,ω,Pt,ω) ∈ F, ∀t,∀ω,

where λω denotes the Lagrangian multiplier of the non-anticipativity constraint, α denotes

the learning rate of the penalty, It denotes the non-anticipative decision, and F is the con-

straint space. The most popular augmented Lagrangian method is the augmented direction

method of multipliers (ADMM), which reduces to the method of progressive hedging (PH)

if all of the Lagrangian multipliers are updated in one iteration. However, it is notable

that, like other gradient-based methods, the convergence highly depends on the choice of

the learning rate α and is not guaranteed to be obtained within finite iterations. Thus, the

need for further investigating ADMM’s convergence arises in the research field of distributed

algorithms.

SP-based approaches benefit from exerting the exact historical data profiles to construct

the scenarios and thus yielding preventative solutions for future uncertainties. However,

86

the industrial application of SP-based methods to date is still limited, mainly owing to the

tremendous computational burden. As most of the research restrict themselves to small- or

medium-scale standard test systems, SP can be hard to scale in real-world systems. Nonethe-

less, as SP fathers nearly all of the subsequent uncertainty-based models, it is still beneficial

for both industry and academia to pay attention to the development of SP theory.

4.1.3.2. Robust Optimization

Though researchers have devoted considerable research efforts towards accelerating SP

solutions, the curse of dimensionality of SP caused by a plethora of scenarios prevents it

from practical implementations. In contrast, RO approaches leverage the boundary-based

modeling of uncertainty to capture the extreme scenario and hence greatly relieve the com-

putational burden. In this field, adaptive RO (ARO) and distributionally RO (DRO) have

great potential in future applications.

4.1.3.2.1. Adaptive robust optimization

The name “adaptive” stems from the fact that the first stage needs to adapt to the worst-

case scenario determined in the forward stages. Based on the polyhedral or ellipsoidal set

constructed for the uncertainty, ARO considers a boundary-based operation problem to find

the worst-case scenario, with which the robust and risk-averse solution is attained. Taking

the SUDAUC as an example, (4-7) formulates the problem with the same constraints in

(4-4):

minIt,Pt

∑t

c⊺It +maxu∈U

minPt

∑t

g⊺Pt (4-7)

s.t. A ⋅ It ≤ J, ∀t,

B ⋅ It +C ⋅Pt +D ⋅ u ≤ K(ω), ∀t,

where u denotes the uncertain parameter constrained in the uncertainty set U . To determine

the worst-case scenario, the subproblem needs to take a max-min form, and then the overall

87

problem finally yields a conservative solution averse to the worst case. This formulation

(4-7) lays the foundation of numerous ARO applications in the uncertainty-based market

operation.

For the solution strategy, decomposition techniques are also applied to accelerate. ARO

problems are not as difficult as SP problems with redundant scenarios to solve, and hence

the convex optimization techniques can efficiently tackle the problem. Among them, pri-

mal and dual decompositions still lead the mainstream, since they can conveniently solve

the problem to global optimality with tractable accelerations. As BD becomes a typical

representative of the dual decomposition, primal decompositions like column-and-constraint

generation (C&CG) shows better convergence performance. In contrast to adding dual cuts

in BD, the C&CG algorithm adds primal cuts and unifies the feasibility and optimality up-

dates. However, C&CG imposes more variables and constraints to the master problem in

each iteration than BD, which may lead to a longer solution time for each iteration. Thus,

the choice of primal and dual decompositions is based on the tradeoff between the conver-

gence speed and the computational time of one single iteration.

ARO-based models and strategies recently attract many industrial applications, as the

problem structure and decision procedure well fit the practice. When ARO has strong model

fidelity and ensures the system reliability, it has great potential to be generalized as a daily

operation procedure in the future solar-integrated market operation.

4.1.3.2.2. Distributionally robust optimization

When ARO constructs the uncertainty set by explicitly modeling the boundaries of values

as described in Section 4.1.2.2, DRO constructs the ambiguity set via the moment informa-

tion of the uncertain parameters. By leveraging the ambiguity set created by either moment

information or PDF’s distance information, DRO yields a less conservative solution than

ARO and better scalability than SP. As DRO has accumulated huge research interest in the

field of operations research (OR), its application in the uncertainty-based electricity market

operation recently dominates the research. The formulation of the DRO problem is identi-

88

cal with the ARO case, i.e., (4-7), but the tractability is phenomenally damaged, which is

mainly owing to the NP-hard nature from the dynamic moment ambiguity set. Even though

the ambiguity set is well-defined and convex, the problem could remain computationally

intractable.

To facilitate the solution, reformulation and decomposition techniques are proposed,

many of which remain in the stage of theoretical optimization derivation. As the prob-

lem structure is similar to the ARO problem, ARO’s solution strategies can also be applied

to solve DRO problems, especially when the ambiguity set is defined to be convex. In the OR

research field, on the other hand, literature focuses on using relaxation and reformulation to

convexify intricate ambiguity sets, which is rarely an issue in the electricity market case, as

the solar and other renewables possess trackable data patterns. However, the convexifying

relaxation of the ambiguity set still has future potential when the uncertainty is considered

to be multiplex.

Though DRO combines some of the advantages of SP and ARO, it is still hard to explain

to system operators that how these uncertainties are realized in the daily operation and how

it ensures the solution reliability, as they are modeled within a probability set. There also

lacks a metric to evaluate the final solution quality. As the research on the application of

DRO in the uncertainty-based market operation goes on, the real-world application of DRO

will slowly spread until its interpretability and evaluation process are well addressed.

4.1.3.3. Intelligent Model-free Approaches

Despite the dominance of using model-based scheduling techniques in the local energy

market, the dynamic game between prosumers, consumers and utilities is far from easy to

be modeled precisely. Heretofore, there is no silver bullet towards solving such a dynamic

and nonconvex problem to global optimality. Thus, intelligent model-free approaches with

learning-based forecast and scenario generation try to find a feasible and good enough so-

lution instead, which at least provides a practical reference and an evaluation metric for

decision makers. The most typical method among them is RL, which is to find an opti-

89

mal strategy for single or multiple agents to maximize rewards in an either model-based or

model-free environment. As the model-based RL techniques can be easily translated into

variants of conventional or approximate dynamic programming, here we mainly discuss the

model-free RL. Researchers have been looking into the application of this newly-revived in-

telligent method towards the complex and dynamic P2P game and other applications at

various market scales.

It is widely known that model-free RL techniques can be divided into two groups, i.e.,

value-based and policy-based methods. Value-based methods consider the problem as a

Markov decision process (MDP), in which the decision of the current state only depends on

the decision of the last state. The most common value-based approach is Q-learning together

with its deep NN-based version, Deep Q Network, which lies in the category of asynchronous

dynamic programming and heretofore is the most popular model-free approach adopted in

the PV-integrated local energy trading. This method updates one state in each decision

stage and eliminates the state-transition probabilities by assigning rewards and punishments

according to necessary conditions like the Bellman equation, as presented in (4-8).

Q(s, a) = R(s, a) + γ ∑s′∈S

p(s′∣s, a)π(a′∣s′)Q(s′, a′), (4-8)

where (s, a) and (s′, a′) denote the current and the next state and action pair respectively, π

denotes the agent behavior function, R denotes the reward value, p denotes the probability

and Q is the so-called Q value evaluating the quality of performing this action. From this

equation, it reflects a classical MDP where the current action quality is only associated with

the reward and possible opportunity costs from the previous stage. It also acts as an update

equation for the Q value as well as the revenue objective function.

The policy-based methods, on the other hand, evaluate the agents’ behaviors with pro-

gressive policies continuously instead of explicit and discrete reward values. In this case,

the value π(a′∣s′) in (8) becomes a probability distribution πθ(a′∣s′), where θ denotes the

dynamic policy. As the action is continuous and policy changes dynamically, the gradient

90

of the policy part in (4-8), i.e., Q minus R, needs to be computed to conduct state-space

evaluations. Though current research on the solar-integrated market operations seldom in-

vestigates this method, its effectiveness has already been corroborated for the joint bidding

and pricing design in the wholesale market and the strategic optimization in the domestic

energy scheduling, where the uncertain patterns are similar to the solar and other renew-

ables. As the value-based approaches consider discrete states that may not be applicable

in the real-time market environment, a model-free continuous solar-integrated market setup

cultivates considerable research potential from this standpoint.

4.1.4. Outlooks

As a salient participant in the current electricity market environment, solar energy poses

significant technical challenges such as increased uncertainties and intermittencies for system

operation and reliability. In this review, we mainly introduce the market operation with solar

energy integration in the energy, ancillary service and transactive markets, and comment on

the computational methodologies for enhancing the solar uncertainty modeling and operation

efficiency proposed in both the academic research and industry projects. In summary of

the discussions on the state-of-the-art solar-integrated market operations, future research

directions are revealed in terms of market paradigm design, solar uncertainty modeling and

solution strategies, which are specified as follows.

1. Design of the Market Paradigm. With the accelerating installation of renewable

resources in the distribution systems that lie at the lower level in the current electricity

market, the policy and paradigm of the market need to be updated. Most ISOs are

currently using the estimated and forecasted information from the utilities to perform

the deterministic market operation, whereas the increasing penetration of renewables

now threatens their accuracy. Additionally, how to coordinate the ongoing distribution-

level transactive energy market with the conventional market structure also calls for

answers. From this perspective, the future outlooks for both the industry and academia

are detailed in the following:

91

• Uncertainty-based policy-making and market applications. With the develop-

ment of uncertainty models and stochastic optimization techniques, transitioning

the current correction-based deterministic market operations to the risk-averse

stochastic operations meets the increasing demand of tackling renewable inter-

mittency and national development strategy.

• Development of unique renewable bidding strategies in energy and reserve mar-

kets. Renewable bidding and pricing rules are far from complete in the current

market operation. Besides, though the role of renewable energy in the electricity

market differs little from wind energy, it is requisite to design a more general and

unique market rule for renewable participation.

• Coordination with lower-level markets. As the utility-scale renewable installation

increases dramatically in distribution systems, market operators need to consider

the coordination between upstream and downstream markets, particularly the

transactive market.

2. Uncertainty Modeling and Market Operations. For the uncertainty modeling

of renewable energy, albeit resourceful literature has been investigating the stochastic

optimization throughout the years in the renewable-integrated market, the accuracy

and tractability of the current strategies need to be further reinforced. Hence, the next

research front converges to the improvement of the accuracy and interpretability of

renewable forecast and scenario generation methods, plus the data-driven optimization

for renewable-integrated market operations with fast and tractable solution strategies.

In summary, the research potentials are detailed as follows.

• Precise capture of the spatiotemporal correlation in the uncertainty modeling. Re-

cent studies on the scenario generation start considering this issue with short-team

correlations by using the learning-based uncertainty modeling, which is difficult to

be covered by sampling-based and boundary-based models. Besides, the medium

or long-term spatiotemporal correlation for renewable units in a large area and

the model fidelity of the learning-based methods deserve further research before

the real-world implementation.

92

• Data-driven models for the renewable-integrated market operation. The explo-

ration on data-driven methods earns a promising future from the recent advance-

ment of DRO on the distribution operation. However, the sophisticated inter-

pretability of the data-driven models prevents them from the large-scale applica-

tion and thereby needs to be investigated in future research.

• Algorithms with improved efficiency and accuracy. Current algorithms towards

stochastic models need to be upgraded to follow the trend of highly distributed and

decentralized market hierarchy. More accurate and more tractable solution strate-

gies are of capital importance. Specifically, for stochastic models with nonlinearly

correlated uncertainties, the performance of decomposition and distributed algo-

rithms like Bender’s Decomposition and Alternating Direction Method of Multi-

pliers calls for further enhancement. Apart from the distributed algorithms, the

advancement of the model-free intelligent algorithms such as policy-based rein-

forcement learning shows great potential for the data-driven market applications.

4.2. General Simulation Platform for Current and Future Short-term Electricity

Market Operations

4.2.1. Nomenclature

We provide a detailed notation list for this section.

1) Indices:

g / i Indices for generator / energy storage

t Index for time interval

2) Parameters:

93

RUg /RDg Ramp-up/down rate of unit g

RUi /RDi Ramp-up/down rate of ESS i

ResT PFR PFR response time

ResTAGC AGC response time

ResT SR SR response time

ResTNSR NSR response time

SUTimeg Startup time for unit g

∆fmax Maximum allowable frequency deviation

f0 Nominal frequency

DBg Governor deadband for unit g

Rig Equivalent droop curve for unit g

DCCg Droop curve coefficient of unit g

Pming Minimum power output of unit g

Pmaxg Maximum power output of unit g

CHmaxi Charging power limit of ESS i

DISmaxi Discharging power limit of ESS i

LRi Energy dissipation rate of ESS i

ηi Energy efficiency of ESS i

SOCmaxi Maximum SOC limit of ESS i

SOCmini Lowest SOC allowable for ESS i

3) Variables:

94

ug,t Unit commitment variable for unit g at interval t (Binary)

ci,t Charging indicator for ESS i at interval t (Binary)

di,t Discharging indicator for ESS i at interval t (Binary)

pg(i),t Active power of unit g (ESS i) at interval t

pfrupg(i),t

PFR provided by unit g (ESS i) at interval t

agcupg(i),t

AGC-up provided by unit g (ESS i) at interval t

agcdng(i),t

AGC-dn provided by unit g (ESS i) at interval t

srupg(i),t

SR provided by unit g (ESS i) at interval t

nsrupg,t NSR provided by unit g at interval t

disi,t Discharging power of ESS i at interval t

chi,t Charging power of ESS i at interval t

soci,t SOC of ESS i at interval t

4.2.2. Overview

Short-term electricity market operation has been summarized as a quintessential math-

ematical optimization with widely adopted yet evolving formulations. Typically, the inde-

pendent system operator (ISO) carries out and monitors the market operation in both the

day-ahead and real-time frames, including the unit commitment (UC) and economic dispatch

(ED) [129]. The operation involves multi-timescale interactions between different modules,

in which NP-hard models could be present, and thus it is not trivial to generalize in one

single mathematical program. Considering the evolving structure of modern electricity mar-

kets, developing a general simulation platform concerning novel system elements with easily

editable functions becomes an urgent task.

Though different ISOs might have different taxonomies and implementations on the mar-

95

ket operations, it is widely recognized that the day-ahead market and the real-time market

consist of the short-term market operation’s main body [130]. Particularly, day-ahead unit

commitment (DAUC) optimally determines the units’ commitment schedules on an hourly

basis, often serving as the day-ahead market clearing. Real-time unit commitment (RTUC)

re-commits fast-responsive units in a finer resolution with updated system information. Real-

time economic dispatch (RTED) clears the real-time market yielding the unit dispatch and

finalizes energy and reserve allocations. Under specific circumstances for market simulators,

day-ahead economic dispatch (DAED) follows by DAUC to retrieve shadow prices in the

day-ahead market. Hence, the coordination between these four multi-timescale operations

highlights the essence of the short-term electricity market operation.

Currently, quite a few software platforms have looked into the market operation and built

general tools to perform simulations. There is both open-source and commercial simulation

software. The advantages of commercial software like PLEXOS [131] and ENERLYTIX [132]

include that the operation is well-benchmarked by ISOs, and they are regulated and updated

by professional teams for customer cares. However, the commercial tools are not friendly

to academic researchers with limited accessibility, and they rarely consider state-of-the-art

models for future market implementations. Thus, researchers developed open-source plat-

forms such as FESTIV [133] and EGRET [134] trying to give a general solution for market

simulations to the research community. Nonetheless, with the rapid growth of renewable

technologies, current open-source tools are neither flexible for the multi-timescale operation

nor with state-of-the-art market models. Thus, we release a new open-source Python library,

MIDAS-S, abbreviated from Multi-timescale Integrated Dynamic And Scheduling - Schedul-

ing, to fill the gap between the existing open-source tools and practical market operations.

Implementing a complete multi-timescale market operation needs to take a large variety

of factors into accounts, such as the reserve product design and information exchange be-

tween different routines, which is the major feature that MIDAS-S nails down. For instance,

the unit startup and shutdown processes in many ISOs are uncontrollable. Still, they con-

tribute a moderate part of power during the transition state, which should be considered in

96

the power balance. Many existing academic works neglect this aspect, whereas G. Morales-

Espana et al. [135] have experimentally shown that the neglect could result in commitment

changes and a worse objective value. From the multi-timescale operation’s perspective, the

startup and shutdown processes become more difficult, which involve making real-time op-

erations aware of the MW levels of generators during startup/shutdown after UC yields the

commitment schedule. Similarly, another example would be that the energy storage system

(ESS) needs to perform energy arbitrage in the day-ahead market, as regulated by typical

ISOs [136]. In the real-time operations, ESSs should follow the day-ahead schedule or an

optimized dispatch signal.

As renewable technologies proliferate in the current power system, the market frame-

work should evolve with the optimized operation of sustainable resources and ESSs. Based

on enhanced forecast techniques, state-of-the-art market models have already considered

renewables in energy optimization. However, the ancillary service (AS) provision from re-

newables remains in the initial phase. Though smart inverters have been broadly deployed

in many regions’ renewable systems to support ASs [137], from the market operation’s side,

most ISOs in the US do not consider renewables providing ASs [138]. Nonetheless, the re-

search community has delved into the design regarding renewable-supported AS markets for

years. E. Ela. et al. [139] conducted pioneering works on integrating primary frequency

response (PFR) with other ancillary service products. B. Olek et al. [140] studied the steady

state and transient state of renewable providing ASs in a low-voltage network. More works

focus on the ESS performance on ASs. N. Cobos et al. [141] presented a robust scheduling

strategy for ESSs participating in the AS market. And N. Padmanabhan et al. [142] ana-

lyzed the ESS potential for providing spinning reserves via a production cost model. Above

all, our proposed tool enhances the recent development of renewable and ESS strategies for

AS provision, generalizes the choice of various models, and further validates the scheduling

results via the system dynamic simulation.

Under most scenarios in the current industry practice, the market operation is decoupled

with system dynamic analyses. With the integration of renewables and ESSs and their ca-

97

pability to provide reserves, executing dynamic simulation and posterior power flow analysis

becomes necessary to verify the market decisions. Very few works consider using dynamic

simulation to guide or correct the scheduling simulation. H. Yang et al. [143] proposed a fast

Fourier transformation-based reserve requirement determination used in the UC problem

with power flow check in PSS®E. In the proposed platform, PSS®E executes AC power

flow check for the RTED and performs dynamic simulation for renewable units such as AGC

modeling and inertia control simulation. The PSS®E results will guide the next-interval

RTED by updating the generator parameters, real-time power loss values, and the ESS dis-

patch.

In summary, we release an open-source Python library as MIDAS-S, which gives a flexi-

ble solution towards state-of-the-art multi-timescale market operations with validations from

dynamic simulations. We highlight the contributions of this work as follows.

• We propose a generalized flexible market operation consisting of day-ahead and real-

time scheduling that considers multi-timescale information exchange.

• MIDAS-S is the first open-source electricity market simulation tool that considers

multi-timescale operations with an interface for dynamic simulations.

• Multiple novel modeling features such as multi-timescale non-spinning reserve design

and AS-enabled ESS scheduling are present in the platform.

• The library is easily editable with flexible function utilities for different user-defined

modeling and executing alternatives.

We organize the remainder of this chapter as follows. Section 4.2.3 gives an overview

of the four operating modes in the platform upon user preferences. Section 4.2.4 discusses

highlights in the novel electricity market modeling, such as the multi-timescale AS model

designs and ESS operations. Section 4.2.5 presents the flexible features of the library with

user-friendly settings and the dynamic simulation interface. Section 4.2.6 provides simulation

examples from an 18-bus system and the WECC system. Section 4.2.7 concludes the chapter.

98

4.2.3. Working Mode Design for the Multi-timescale Operation

As a standard simulation tool for short-term market operations, MIDAS-S has four op-

eration modes that users can easily call and tune. We first give the following notes:

• For all modes, MIDAS-S could run for multiple days with adequate input data.

• According to user settings, all routines, i.e., DAUC, DAED, RTUC, and RTED, could

have adjustable look-ahead horizon and time resolution. For example, RTED can run

as a 2-hour look ahead with a 5-minute resolution.

• To retrieve day-ahead market clearing prices for some solvers, DAED is optional to be

executed right after any DAUC is executed.

Based on Figure 4.1, we detail the four modes as follows.

4.2.3.1. Mode 1: DAUC only

In some production cost modeling and commitment analysis scenarios, users would like

to only run DAUC with multiple days, for which Mode 1 is called. In this case, based on

the user-defined number of days, MIDAS-S will execute the DAUC routine with defined

look-ahead hours. The current DAUC uses the last-interval data yielded from the previous

DAUC as new inputs to update by default settings. In contrast, the very first DAUC will

need the user-input initial statuses.

4.2.3.2. Mode 2: RTED only

For a brief analysis of real-time market operations, users may only want to run RTED.

Mode 2 executes the RTED routine based on a virtual DAUC dataset provided by the user.

That is, users need to import a complete set of commitment results and all other input

parameters from the UC, such as the ESS arbitrage results. In real-time processes, this is an

ISO’s practice that the RTED considers all the UC results as input parameters [144]. Note

that Mode 2 executes RTED only, which can be regarded as a single unit dispatch system

with limited market functionalities. One should refer to Mode 3 or Mode 4 if real-time

99

RTEDDAUC DAUC DAUC DAUC

Day 1

is the specified day in the Input.xlsx.

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RTEDRTED RTED RTED RTED

Interval 1

is the specified interval in the Input.xlsx. The first interval (current interval) results in the RTED will be sent to dynamic simulations.

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DAUC RTEDRTED RTED RTED RTED

0:00 1:00 2:00

Day 0

is the specified interval in the Input.xlsx. After the DAUC is executed, the whole-day RTED will be executed before the next-day DAUC.Next-day DAUC uses current DAUC results as inputs.The first interval (current interval) results of the RTED will be sent to dynamic simulations.

23:00

DAUC RTEDRTED RTED RTED RTED

0:00 1:00 2:00 23:00

Day x<latexit sha1_base64="/SRrrqno98F43FYtXqDHrqi0QnU=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7AckoWw2m3bpZjfsTsQS+jO8eFDEq7/Gm//GbZuDtj4YeLw3w8y8KBPcgOt+O5WNza3tnepubW//4PCofnzSMyrXlHWpEkoPImKY4JJ1gYNgg0wzkkaC9aPJ7dzvPzJtuJIPMM1YmJKR5AmnBKzkB5ESccCheJoN6w236S6A14lXkgYq0RnWv4JY0TxlEqggxviem0FYEA2cCjarBblhGaETMmK+pZKkzITF4uQZvrBKjBOlbUnAC/X3REFSY6ZpZDtTAmOz6s3F/zw/h+QmLLjMcmCSLhclucCg8Px/HHPNKIipJYRqbm/FdEw0oWBTqtkQvNWX10mv1fTcpnd/1Wi3yjiq6Aydo0vkoWvURneog7qIIoWe0St6c8B5cd6dj2VrxSlnTtEfOJ8/vuORfg==</latexit><latexit sha1_base64="/SRrrqno98F43FYtXqDHrqi0QnU=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7AckoWw2m3bpZjfsTsQS+jO8eFDEq7/Gm//GbZuDtj4YeLw3w8y8KBPcgOt+O5WNza3tnepubW//4PCofnzSMyrXlHWpEkoPImKY4JJ1gYNgg0wzkkaC9aPJ7dzvPzJtuJIPMM1YmJKR5AmnBKzkB5ESccCheJoN6w236S6A14lXkgYq0RnWv4JY0TxlEqggxviem0FYEA2cCjarBblhGaETMmK+pZKkzITF4uQZvrBKjBOlbUnAC/X3REFSY6ZpZDtTAmOz6s3F/zw/h+QmLLjMcmCSLhclucCg8Px/HHPNKIipJYRqbm/FdEw0oWBTqtkQvNWX10mv1fTcpnd/1Wi3yjiq6Aydo0vkoWvURneog7qIIoWe0St6c8B5cd6dj2VrxSlnTtEfOJ8/vuORfg==</latexit><latexit sha1_base64="/SRrrqno98F43FYtXqDHrqi0QnU=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7AckoWw2m3bpZjfsTsQS+jO8eFDEq7/Gm//GbZuDtj4YeLw3w8y8KBPcgOt+O5WNza3tnepubW//4PCofnzSMyrXlHWpEkoPImKY4JJ1gYNgg0wzkkaC9aPJ7dzvPzJtuJIPMM1YmJKR5AmnBKzkB5ESccCheJoN6w236S6A14lXkgYq0RnWv4JY0TxlEqggxviem0FYEA2cCjarBblhGaETMmK+pZKkzITF4uQZvrBKjBOlbUnAC/X3REFSY6ZpZDtTAmOz6s3F/zw/h+QmLLjMcmCSLhclucCg8Px/HHPNKIipJYRqbm/FdEw0oWBTqtkQvNWX10mv1fTcpnd/1Wi3yjiq6Aydo0vkoWvURneog7qIIoWe0St6c8B5cd6dj2VrxSlnTtEfOJ8/vuORfg==</latexit><latexit sha1_base64="/SRrrqno98F43FYtXqDHrqi0QnU=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0mKoMeCF48V7AckoWw2m3bpZjfsTsQS+jO8eFDEq7/Gm//GbZuDtj4YeLw3w8y8KBPcgOt+O5WNza3tnepubW//4PCofnzSMyrXlHWpEkoPImKY4JJ1gYNgg0wzkkaC9aPJ7dzvPzJtuJIPMM1YmJKR5AmnBKzkB5ESccCheJoN6w236S6A14lXkgYq0RnWv4JY0TxlEqggxviem0FYEA2cCjarBblhGaETMmK+pZKkzITF4uQZvrBKjBOlbUnAC/X3REFSY6ZpZDtTAmOz6s3F/zw/h+QmLLjMcmCSLhclucCg8Px/HHPNKIipJYRqbm/FdEw0oWBTqtkQvNWX10mv1fTcpnd/1Wi3yjiq6Aydo0vkoWvURneog7qIIoWe0St6c8B5cd6dj2VrxSlnTtEfOJ8/vuORfg==</latexit>


DAUC RTUC

RTED

RTUC

RTED

RTUCRTUC

RTED

RTUCRTUC

RTED

RTUC

RTED

Run Hour 0 RTUC for Hour 1 toHour 3

Run Hour 1RTUC for Hour 2 toHour 4

Run Hour 2 RTUC for Hour 3 toHour 5

11 RTED incurrent hour


0:00 0:10

0:05

1:00

0:55

1:10

1:051:55

2:05

2:00

At the beginning of every hour, run a look-ahead RTED with 12 intervals (every 5 minutes).After the DAUC is executed, the whole-day RTUC + RTED will be executed before the next-day DAUC.The last interval results of the last interval RTED in one hour will be sent to the RTUC for the initial status of the hour-ahead RTUC.The RTUC commitment results will be used for the next-hour RTED.Next-day DAUC uses current DAUC results as inputs.The first-interval RTED in the next day uses the last-interval RTED result in the current day as inputs.The first interval (current interval) results of the RTED will be sent to dynamic simulations.

23:00

DAUC RTUC

RTED

RTUC

RTED

RTUCRTUC

RTED

RTUCRTUC

RTED

RTUC

RTED11 RTED incurrent hour


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0:55

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Last RTED

Last RTED

Day 0

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Last RTED

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(a). Mode 1: DAUC Only

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(d). Mode 4: DAUC + RTUC + RTED

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(c). Mode 3: DAUC + RTED

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(b). Mode 2: RTED Only

Figure 4.1. Operation modes of MIDAS-S.

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market operation details are required.

4.2.3.3. Mode 3: DAUC + RTED

Mode 3 serves as a simplified version of Mode 4, which runs DAUC first and then runs

RTED accordingly. As shown in Fig. 1 (c), after the day’s DAUC is executed, all RTEDs for

this day will be executed based on the day’s DAUC results. When it comes to the next day,

the DAUC uses the previous day’s DAUC results as initial inputs. This is because practically

in many ISOs, one day’s DAUC is executed on the last day, such as in ERCOT [145], when

the previous day’s RTEDs have not been finished yet. However, the day’s first RTED will use

not only the day’s DAUC commitments but the last RTED dispatch results of the previous

day as inputs. It makes the RTED instances consecutive.

4.2.3.4. Mode 4: DAUC + RTUC + RTED

Mode 4 is the complete execution of both the day-ahead and real-time market operation

process, which is also the most complicated. RTUC serves as a recommitment process for

fast-responsive units according to the varying real-time market environment. To better

introduce, Fig. 1 (d) uses one time setting as an example. We set the time resolutions of

DAUC, RTUC, and RTED as 1 hour, 1 hour, and 5 minutes, respectively, whereas their

simulation horizons are 26 hours, 3 hours, and 2 hours. That is, in each hour, 12 RTEDs are

executed.

The design philosophy is that the current hour’s RTUC runs for intervals starting from

the next hour. For example, the RTUC run at Hour 1 is for Hour 2-3-4 and is executed

right after the first RTED instance at Hour 1 is executed. We detail the reasoning of this

design as follows. For example, if the RTUC starting from 03:00 schedules a generator with a

startup time of 20 minutes to start up at 03:00, its commitment effective time is 03:00. The

generator will then need to start up from 02:40 in real-time due to its startup time. Hence,

the RTED starting from 02:40 should be aware of the unit’s startup, which means the RTUC

starting from 03:00 should have been executed at least before 02:40. Since the purpose of

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RTUC is to reschedule fast-responsive units in real-time, we deem 1 hour of the startup

time by default for fast-responsive units. Hence, the RTUC executes one hour before the

operating time as default in MIDAS-S. In the same example, Hour 1’s RTUC’s initial inputs

should be the initial inputs of Hour 2, which comes from the results of Hour 1’s RTED.

Based on the previous design, when the user sets each routine’s time frames in this mode,

the time coupling needs to be strict. For instance, since Hour 1’s RTUC runs for Hour 2-3-4,

it requires Hour 1’s dispatch result as initial inputs, which should come from the first RTED

at Hour 1. Hence, the first RTED at Hour 1 should have at least a 1-hour time horizon.

Similarly, since the RTED horizon is 2 hours in this example, the last RTED runs for 01:55-

03:55. Hence, the RTUC horizon at Hour 1 should have at least 3 hours time horizon to

cover for Hour 2-3-4. It is recommended that first-time users adopt the default setting, as

this example shows.

4.2.4. Novelties on Electricity Market Models

This section discusses MIDAS-S model novelties concerning evolving operations of re-

newables and ESS participations in both the energy and AS markets. We highlight our AS

designs and ESS operations as follows.

4.2.4.1. Novel AS Designs

In MIDAS-S, we accommodate four kinds of AS products in the market, i.e., PFR, reg-

ulation reserves (often refers to automatic generation control, AGC), spinning reserve (SR),

and non-spinning reserve (NSR). Though different ISOs may have different taxonomies on

the AS market, their general functionalities remain similar. Note that these reserves are

related to the frequency response, and we do not consider voltage response procurement as

there is no such relative product in the current U.S. market [146]. Note that in MIDAS-S,

we only consider the reserve capacity procurement.

a). PFR models.

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Though there are very few ISOs that procure PFR in the market, with the rapid growth

of renewable technologies, PFR will become more and more important thanks to its fast-

responsive feature. Hence, for the future market design, in MIDAS-S, we accommodate the

PFR model considering thermal units’ governor control and response time, which results in

the following steady-state model.

0 ≤ pfrupg,t ≤∆fmax −DBg

Rig, (4-9)

Rig =DCCg ⋅ f0Pmaxg

, (4-10)

where (4-9) gives the physical limits of governor’s PFR provision, (4-10) defines the gov-

ernor droop curve. Ref. [139] provides more information regarding the modeling of generator

governors. Note that for renewable units, by employing the virtual governor, we could still

model their PFR provisions via this formulation, which we also conduct analyses in the sys-

tem dynamic simulation. Other PFR formulations considering system inertia and power loss

such as [147] are also available in MIDAS-S.

b). AGC and SR models.

Regulation reserves take a critical role in the ISOs’ reserve categories. They are often

expressed as how much power the unit can adjust due to frequency events, which means they

relate to the units’ ramping rate. It is similar for the SR, as another type of the upward

reserves. Hence, we present the following steady-state model for regulation reserves and SR.

0 ≤ agcupg,t ≤ ResTAGC ⋅RUg, (4-11)

0 ≤ agcdng,t ≤ ResTAGC ⋅RDg, (4-12)

0 ≤ srupg,t ≤ ResTSR ⋅RUg, (4-13)

where constraint (4-11), (4-12), and (4-13) regulate the AGC-up, AGC-down, and SR, re-

spectively, by the response time and ramp rates.

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c). NSR models.

NSR is also a vital reserve product in the current electricity market, but very few works

have considered its complete modeling in the literature. It is non-trivial to include NSR in

the multi-timescale scheduling framework since the response time of NSR is typically 15∼30

minutes, which trespasses multiple intervals of the real-time operation. Though many ISOs

perform the NSR scheduling in both day-ahead and real-time, real-time NSR scheduling

typically occurs in the RTUC while RTEDs fix NSR values only to adjust units’ headroom

and ramp room [144]. Hence, we design the following model in DAUC and RTUC.

0 ≤ nsrupg,t ≤ (1 − ug,t) ⋅ [Pming,t +RUg ⋅ (ResT

NSR − SUTimeg)] + ug,t ⋅RUg ⋅ResTNSR, (4-14)

where constraint (4-14) means that when the unit is offline, during the NSR response time,

the maximum NSR it can provide is the minimum power output, using the startup time,

plus the power level that it can ramp to using the rest of time. This implies that the unit is

not eligible to provide NSR if its startup time is higher than the NSR response time. If the

unit is online, it could also provide NSR restricted by the operation ramp rate.

The NSR scheduling is separately scheduled in both DAUC and RTUC, whereas for

RTED, the NSR values are fixed to reflect the RTUC schedules in Mode 4 or DAUC sched-

ules in Mode 3. Also, note that all renewables and ESSs are not eligible to provide NSR,

which is also a common practice [144].

d). Unit headroom and ramp room.

The unit headroom and ramp room constraints for each generator are as follows.

104

Idle<latexit sha1_base64="fSrLzLPZZ24JVO5mANYijxAcEic=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbFU0lE0GPBi94q2A9oQ9lsJu3S3U3Y3Qil9C948aCIV/+QN/+NSZuDtj4YeLw3w8y8IBHcWNf9dkpr6xubW+Xtys7u3v5B9fCobeJUM2yxWMS6G1CDgitsWW4FdhONVAYCO8H4Nvc7T6gNj9WjnSToSzpUPOKM2ly6DwUOqjW37s5BVolXkBoUaA6qX/0wZqlEZZmgxvQ8N7H+lGrLmcBZpZ8aTCgb0yH2MqqoRONP57fOyFmmhCSKdVbKkrn6e2JKpTETGWSdktqRWfZy8T+vl9roxp9ylaQWFVssilJBbEzyx0nINTIrJhmhTPPsVsJGVFNms3gqWQje8surpH1Z99y693BVa5wXcZThBE7hAjy4hgbcQRNawGAEz/AKb450Xpx352PRWnKKmWP4A+fzB9vEjgQ=</latexit><latexit sha1_base64="fSrLzLPZZ24JVO5mANYijxAcEic=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbFU0lE0GPBi94q2A9oQ9lsJu3S3U3Y3Qil9C948aCIV/+QN/+NSZuDtj4YeLw3w8y8IBHcWNf9dkpr6xubW+Xtys7u3v5B9fCobeJUM2yxWMS6G1CDgitsWW4FdhONVAYCO8H4Nvc7T6gNj9WjnSToSzpUPOKM2ly6DwUOqjW37s5BVolXkBoUaA6qX/0wZqlEZZmgxvQ8N7H+lGrLmcBZpZ8aTCgb0yH2MqqoRONP57fOyFmmhCSKdVbKkrn6e2JKpTETGWSdktqRWfZy8T+vl9roxp9ylaQWFVssilJBbEzyx0nINTIrJhmhTPPsVsJGVFNms3gqWQje8surpH1Z99y693BVa5wXcZThBE7hAjy4hgbcQRNawGAEz/AKb450Xpx352PRWnKKmWP4A+fzB9vEjgQ=</latexit><latexit sha1_base64="fSrLzLPZZ24JVO5mANYijxAcEic=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbFU0lE0GPBi94q2A9oQ9lsJu3S3U3Y3Qil9C948aCIV/+QN/+NSZuDtj4YeLw3w8y8IBHcWNf9dkpr6xubW+Xtys7u3v5B9fCobeJUM2yxWMS6G1CDgitsWW4FdhONVAYCO8H4Nvc7T6gNj9WjnSToSzpUPOKM2ly6DwUOqjW37s5BVolXkBoUaA6qX/0wZqlEZZmgxvQ8N7H+lGrLmcBZpZ8aTCgb0yH2MqqoRONP57fOyFmmhCSKdVbKkrn6e2JKpTETGWSdktqRWfZy8T+vl9roxp9ylaQWFVssilJBbEzyx0nINTIrJhmhTPPsVsJGVFNms3gqWQje8surpH1Z99y693BVa5wXcZThBE7hAjy4hgbcQRNawGAEz/AKb450Xpx352PRWnKKmWP4A+fzB9vEjgQ=</latexit><latexit sha1_base64="fSrLzLPZZ24JVO5mANYijxAcEic=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbFU0lE0GPBi94q2A9oQ9lsJu3S3U3Y3Qil9C948aCIV/+QN/+NSZuDtj4YeLw3w8y8IBHcWNf9dkpr6xubW+Xtys7u3v5B9fCobeJUM2yxWMS6G1CDgitsWW4FdhONVAYCO8H4Nvc7T6gNj9WjnSToSzpUPOKM2ly6DwUOqjW37s5BVolXkBoUaA6qX/0wZqlEZZmgxvQ8N7H+lGrLmcBZpZ8aTCgb0yH2MqqoRONP57fOyFmmhCSKdVbKkrn6e2JKpTETGWSdktqRWfZy8T+vl9roxp9ylaQWFVssilJBbEzyx0nINTIrJhmhTPPsVsJGVFNms3gqWQje8surpH1Z99y693BVa5wXcZThBE7hAjy4hgbcQRNawGAEz/AKb450Xpx352PRWnKKmWP4A+fzB9vEjgQ=</latexit>


1) Discharge 2) Idle 3) Charge



1) Discharge 2) Idle 3) Charge

(a) PESS Operation (b) BESS Operation

DISmaxi

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DISmaxi


DISmaxi


DISmaxi


DISmaxi


DISmaxi


CHmaxi

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CHmaxi


CHmaxi


CHmaxi


CHmaxi


CHmaxi


Discharge<latexit sha1_base64="lCKUo0nitpV1sbRP921KL/3kW9I=">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</latexit><latexit sha1_base64="lCKUo0nitpV1sbRP921KL/3kW9I=">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</latexit><latexit sha1_base64="lCKUo0nitpV1sbRP921KL/3kW9I=">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</latexit><latexit sha1_base64="lCKUo0nitpV1sbRP921KL/3kW9I=">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</latexit>

Discharge<latexit sha1_base64="lCKUo0nitpV1sbRP921KL/3kW9I=">AAACJHicdVDLSgMxFM34tr6qLt0Ei6AgZWYsVnEj6MKlglWhHUomvVNDMw+SO2IZ5mPc+CtuXPjAhRu/xUxtQUUPBA7n3JPcHD+RQqNtv1tj4xOTU9Mzs6W5+YXFpfLyyoWOU8WhwWMZqyufaZAiggYKlHCVKGChL+HS7x0V/uUNKC3i6Bz7CXgh60YiEJyhkdrlg6w1uKSpur6X2dX9vV23trttV2277rhOQdx6baeWtxBu0Q+yY6H5NVNdyPN2uTIK0FGAjgLUMUqBChnitF1+aXVinoYQIZdM66ZjJ+hlTKHgEvJSK9WQMN5jXWgaGrEQtJcN9svphlE6NIiVORHSgfo9kbFQ637om8mQ4bX+7RXiX14zxWDPy0SUpAgR/3ooSCXFmBaN0Y5QwFH2DWFcCbMrLRpgHE2vJVPC6Kf0f3LhVh276py5lcOtYR0zZI2sk03ikDo5JCfklDQIJ3fkgTyRZ+veerRerbev0TFrmFklP2B9fAIUSaH9</latexit><latexit sha1_base64="lCKUo0nitpV1sbRP921KL/3kW9I=">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</latexit><latexit sha1_base64="lCKUo0nitpV1sbRP921KL/3kW9I=">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</latexit><latexit sha1_base64="lCKUo0nitpV1sbRP921KL/3kW9I=">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</latexit>

Charge<latexit sha1_base64="WzYY03QBfw+BGc2iBv+D8WAWcuM=">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</latexit><latexit sha1_base64="WzYY03QBfw+BGc2iBv+D8WAWcuM=">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</latexit><latexit sha1_base64="WzYY03QBfw+BGc2iBv+D8WAWcuM=">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</latexit><latexit sha1_base64="WzYY03QBfw+BGc2iBv+D8WAWcuM=">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</latexit>

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Idle<latexit sha1_base64="FeqvreppSWqHIrt+NidiNjANPX4=">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</latexit><latexit sha1_base64="FeqvreppSWqHIrt+NidiNjANPX4=">AAACHXicdVBJSwMxGM241rqNevQSLIKCDBk7xfZW8KK3CnaBdiiZTKYNzSwkGaEM80e8+Fe8eFDEgxfx35huoKIPAo/3vW/J8xLOpELo01haXlldWy9sFDe3tnd2zb39loxTQWiTxDwWHQ9LyllEm4opTjuJoDj0OG17o8tJvX1HhWRxdKvGCXVDPIhYwAhWWuqbTtabDumKgedmyELlmlNDZ8iqlKvIqWqCnFqlZuc9L+Z+du1zmud9s7RwwoUTLpzQttAUJTBHo2++9/yYpCGNFOFYyq6NEuVmWChG9MBiL5U0wWSEB7SraYRDKt1selgOj7XiwyAW+kUKTtXvHRkOpRyHnnaGWA3l79pE/KvWTVVQdTMWJamiEZktClIOVQwnUUGfCUoUH2uCiWD6VkiGWGCidKBFHcLip/B/0jq3bGTZN06pfjqPowAOwRE4ATa4AHVwBRqgCQi4B4/gGbwYD8aT8Wq8zaxLxrznAPyA8fEFQnKe3w==</latexit><latexit sha1_base64="FeqvreppSWqHIrt+NidiNjANPX4=">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</latexit><latexit sha1_base64="FeqvreppSWqHIrt+NidiNjANPX4=">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</latexit>

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pfrupi,t

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agcupi,t

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agcdni,t

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srupi,t


pfrupi,t


agcupi,t


srupi,t


agcdni,t


pfrupi,t


agcupi,t


srupi,t


agcdni,t


pfrupi,t


agcupi,t


srupi,t


agcdni,t


pfrupi,t


agcupi,t


srupi,t


agcdni,t


pfrupi,t


agcupi,t


srupi,t


agcdni,t

Figure 4.2. Two schemes for ESSs providing ASs.

Pming ≤ pg,t + pfr

upg,t + agc

upg,t + sr

upg,t + nsr

upg,t ≤ P

maxg , (4-15)

Pming ≤ pg,t − agc

dng,t ≤ P

maxg , (4-16)

pg,t − pg,t−1 + pfrupg,t + agc

upg,t + sr

upg,t + nsr

upg,t ≤ RUg, (4-17)

pg,t − pg,t−1 − agcdng,t ≤ −RDg. (4-18)

In MIDAS-S, we let all reserve products share the same headroom and ramp room. For

some reserves, due to the response time difference, weighting factors might be applied to

adjust their contributions of headroom, which can be user-defined in MIDAS-S. Also, note

that some ISOs do not consider ASs in the ramping constraint, e.g., MISO [144], but we still

provide this flexibility to include (4c) and (4d) in the model.

4.2.4.2. MIDAS-S ESS Designs

In the MIDAS-S default setting, we model the ESS degradation cost as a linear cost

function in the objective function, and use a three-status implementation of ESSs, i.e.,

105

charging, discharging, and idle. The operation constraints of ESSs are as follows.

pi,t = disi,t − chi,t, (4-19)

ci,t + di,t ≤ 1, (4-20)

pi,t − pi,t−1 + agcupi,t + pfr

upi,t + sr

upi,t ≤ RUi, (4-21)

pi,t − pi,t−1 − agcdni,t ≥ −RDi, (4-22)

chi,t ≤ ci,t ⋅CHmaxi , (4-23)

disi,t ≤ di,t ⋅DISmaxi . (4-24)

soci,t − soci,t−1 ⋅ (1 −LRi) = chi,t ⋅ ηi − disi,t/ηi, (4-25)

soci,t + agcdni,t ⋅ResT

AGC ≤ SOCmaxi , (4-26)

soci,t − agcupi,t ⋅ResT

AGC − pfrupi,t ⋅ResTPFR − srupi,t ⋅ResT

SR ≥ SOCmini . (4-27)

They are vanilla ESS operational constraints, where we enforce the idle status of ESSs in

(4-20), and in (4-20)-(4-21) and (4-26)-(4-27), we enforce the similar ramp room constraints

as in the generation part. The state-of-charge (SOC) limits consider AS capacities.

For the ESSs’ headroom modeling, we consider a more refined formulation concerning

various types of ESSs. Due to their power/energy ramping time, we treat the pumped energy

storage system (PESS) and battery energy storage system (BESS) differently in MIDAS-S.

Figure 4.2 depicts the operation logic of different flexibilities of ESS types. On the one hand,

some ESS types, such as PESSs, require more time to change status in real-time. Hence,

performing reserve scheduling for this type should only consider the reserve limit from the

same direction. On the other hand, fast-responsive ESSs, such as BESSs, can quickly switch

status, which allows it to allocate more reserves by using its opposite direction. Based on

106

Figure 4.2, we give an exact formulation for these two flexibility schemes of ESSs as follows.

PESS ∶=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

pfrupi,t + agcupi,t + sr

upi,t ≤ (1 − ci,t) ⋅DISmax

i + chi,t − disi,t,

agcdni,t ≤ (1 − di,t) ⋅CHmaxi + disi,t − chi,t.

(4-28)

BESS ∶=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

pfrupi,t + agcupi,t + sr

upi,t ≤DIS

maxi + chi,t − disi,t,

agcdni,t ≤ CHmaxi + disi,t − chi,t.

(4-29)

It is straightforward to see that constraint (4-29) excludes all binary decision variables in

(4-28) to achieve the logic. It is flexible in MIDAS-S to claim one ESS as PESS-type or

BESS-type. Also, note that the ESS charging and discharging dispatch together with their

binary indicators are termed as parameters in the RTED since the ESSs should participate

in the energy arbitrage conducted in the day-ahead market operation. Only the reserve

optimization and the SOC equations are included in RTED. To better reflect the real-time

operation of ESSs, a linear interpolation of ESS dispatch based on DAUC results is performed

in RTED. Note that all the formulations described in this section refer to [?], which is a

prior work for the MIDAS framework. Then, we implement them in a generalized way in

the MIDAS-S library.

4.2.4.3. Other Formulations

We omit other general formulations of UC and ED since they have been widely discussed

and benchmarked in the literature, where we refer to [148] for interested readers. MIDAS-S

provides a very flexible model library that can be modified and enhanced by users. Different

modeling alternatives can be selected and tested in MIDAS-S.

4.2.5. Modularized Executables for User Flexibility

For different user preferences, modularized executables in each routine can be called

and modified based upon various simulation goals. For high-level usages, users only need

to modify the input data (by default in the form of MS Excel and CSV formats, as in

107

Listing 4.1. DAUC Model Example

1 UCmdl = pe.AbstractModel ()

2 Params.DAUC_read_params(UCmdl)

3 Vars_base.load_base_vars(UCmdl)

45 # Define variables

6 getattr(Vars_gen_power , "DAUC_Piecewise_power")(UCmdl)

7 getattr(Vars_ess_power , "Idle_2var_power_soc")(UCmdl)

89 # Unit constraints

10 getattr(Constr_gen_ancillary_service , "DAUC_AS")(UCmdl)

11 getattr(Constr_ess_ancillary_service , "DAUC_AS")(UCmdl)

1213 # System constraints

14 getattr(Constr_power_balance , "Nodal_power_balance")(UCmdl)

15 getattr(Constr_sys_reserve , "AS_Coupled")(UCmdl)

1617 # Objective function

18 getattr(Objective_function , "DAUC_objective_basic")(UCmdl)

Listing 4.2. Generation AS Constraint Implementation

1 def _Define_PFRup(md):

23 def pfr_pos_cap_rule(md , gi , hi):

4 return md.gen_pfr_pos[gi ,hi] <= (md.Df_Max - md.Gen_DB[gi]) /

md.Gen_Ri[gi]

56 md.gen_pfr_pos_cap = pe.Constraint(md.DPGEN_SET , md.INTER_SET , rule

= pfr_pos_cap_rule)

78 @add_model_attr(component_name ,

9 requires = ’params_loader ’: None ,

10 ’base_var_loader ’: None ,

11 ’power_var_loader ’: None ,

12 )

13 def DAUC_AS(md):

1415 _Define_PFRup(md)

108

FESTIV [133]) to change the problem setting and working mode, and the results will be

automatically documented in both MS Excel and Python binary formats. However, users

might want to modify the embedded optimization model internally. As MIDAS-S is built

upon the Python optimization package Pyomo, descriptive implementation is allowable. The

modularized executables that MIDAS-S offers could also help users easily customize their

models and test operational variants. We discuss the scheduling part and dynamic part

separately on this matter.

4.2.5.1. Scheduling Simulation Part

Taking the DAUC routine as an example, Listing 4.1 provides a simplified sketch of

DAUCModel.py. Imports and function returns are omitted for brevity. After calling this func-

tion, the object UCmdl contains all the DAUC model information and is sent to the mode co-

ordinator to execute. If the user would like to modify the model, taking the generator AS con-

straint for an example (Line 10), user could open its callback, i.e., Constr gen ancillary service.py,

as shown in Listing 4.2, define a new function with a new name similar to DAUC AS, and mod-

ify the object call getattr in DAUCModel.py accordingly. The add model attr() method is

the utility protecting buffer for this flexible implementation.

The function Define PFRup in Listing 2 implements constraint (1a), which is very in-

tuitive that all the parameters, variables, and constraints definitions are descriptive. Note

that upon changing the model, users should follow the calling sequence as in Listing 1, i.e.,

parameters → variables → constraints → objective function. This is a typical modeling se-

quence of any optimization model in Pyomo. If a prerequisite argument is missing, a warning

will be raised.

Another advantage of the modularized executables is that users could call only a part of

the MIDAS-S functions for utility usage. Flexible callable functions are stored in the utility

library and are independent of each other. For example, suppose the user only wants to

derive the power transfer distribution factors for the DC power flow. In that case, it can

109

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Price dataand constraints

<latexit sha1_base64="UCyqUzx+HcWtM0mufxXChKJWGWA=">AAACB3icbVDLSsNAFJ3UV62vqEtBBovQRSlJQXRZcOOygn1AG8rNZNoOnUzCzEQpoTs3/oobF4q49Rfc+TdO0yy09cDA4Zx7uXOOH3OmtON8W4W19Y3NreJ2aWd3b//APjxqqyiRhLZIxCPZ9UFRzgRtaaY57caSQuhz2vEn13O/c0+lYpG409OYeiGMBBsyAtpIA/u02a7iByaCKgYRYB5BgIeRpASUZmI0sMtOzcmAV4mbkzLK0RzYX/0gIklIhSYclOq5Tqy9FKRmhNNZqZ8oGgOZwIj2DBUQUuWlWY4ZPjdKdt48oXGm/t5IIVRqGvpmMgQ9VsveXPzP6yV6eOWlTMSJpoIsDg0TjnWE56XggJnEmk8NASKZ+SsmY5BAtKmuZEpwlyOvkna95l7UnNt6uVHJ6yiiE3SGKshFl6iBblATtRBBj+gZvaI368l6sd6tj8Vowcp3jtEfWJ8/cY6YVw==</latexit>

PV, wind, and load forecasting

<latexit sha1_base64="b2gD4J2m8wtO6qRT/i0AV872XpA=">AAACAHicbVDLSgMxFM3UVx1foy5cuAkWoasyUxBdFqrgsmpf0BlKJpNpQzPJkGSEMnTjr7hxoYhbP8Odf2P6WGjrCYHDOfcm954wZVRp1/22CmvrG5tbxW17Z3dv/8A5PGorkUlMWlgwIbshUoRRTlqaaka6qSQoCRnphKP61O88Eqmo4E09TkmQoAGnMcVIG6nvnPhcUB4RrqFvzn3T9+2H+s113ym5FXcGuEq8BSmBBRp958uPBM4S8xJmSKme56Y6yJHUFDMysf1MkRThERqQnqEcJUQF+WyBCTw3SgRjIc01k8zU3x05SpQaJ6GpTJAeqmVvKv7n9TIdXwU55WmmCcfzj+KMQS3gNA0YUUmwZmNDEJbUzArxEEmEtcnMNiF4yyuvkna14l1U3LtqqVZexFEEp+AMlIEHLkEN3IIGaAEMJuAZvII368l6sd6tj3lpwVr0HIM/sD5/AE44lNM=</latexit>

RTSCED

<latexit sha1_base64="tfaY9ttyj4+QMwj0RiCmy+K9jKw=">AAACAHicbVC7TsMwFHXKq4RXgIGBxaJC6lQllRCMlbowFmjaSk1UOY7bWnXsyHaQqqgLv8LCAEKsfAYbf4PbZoCWY1k6Oude+94TpYwq7brfVmljc2t7p7xr7+0fHB45xycdJTKJiY8FE7IXIUUY5cTXVDPSSyVBScRIN5o05373kUhFBW/raUrCBI04HVKMtJEGzlnABeUx4RoG5ty3g8B+aPrNgVNxa+4CcJ14BamAAq2B8xXEAmeJeQkzpFTfc1Md5khqihmZ2UGmSIrwBI1I31COEqLCfLHADF4aJYZDIc01kyzU3x05SpSaJpGpTJAeq1VvLv7n9TM9vAlzytNME46XHw0zBrWA8zRgTCXBmk0NQVhSMyvEYyQR1iYz24Tgra68Tjr1mndVc+/qlUa1iKMMzsEFqAIPXIMGuAUt4AMMZuAZvII368l6sd6tj2VpySp6TsEfWJ8/ZQSU4g==</latexit>

RTSCUC

<latexit sha1_base64="Kl95cX1BkUbBPO6W1wFg1t5Q++Y=">AAACAHicbVC7TsMwFHXKq4RXgIGBxaJC6lQllRCMRWVgLIK0lZqochynterYke0gVVUXfoWFAYRY+Qw2/ga3zQAtx7J0dM699r0nyhhV2nW/rdLa+sbmVnnb3tnd2z9wDo/aSuQSEx8LJmQ3QoowyomvqWakm0mC0oiRTjRqzvzOI5GKCv6gxxkJUzTgNKEYaSP1nZOAC8pjwjUMzLm5DgL7vuk3+07FrblzwFXiFaQCCrT6zlcQC5yn5iXMkFI9z810OEFSU8zI1A5yRTKER2hAeoZylBIVTuYLTOG5UWKYCGmumWSu/u6YoFSpcRqZyhTpoVr2ZuJ/Xi/XyVU4oTzLNeF48VGSM6gFnKUBYyoJ1mxsCMKSmlkhHiKJsDaZ2SYEb3nlVdKu17yLmntXrzSqRRxlcArOQBV44BI0wC1oAR9gMAXP4BW8WU/Wi/VufSxKS1bRcwz+wPr8ATILlME=</latexit>

DASCUC

<latexit sha1_base64="J+M90l4HgHkE0NE2TMbANUxCq5w=">AAACG3icbVA9SwNBEN2LXzF+nVraLAYhVbgLiJYBGyuJYD4gF8Le3lyyZG/32N1TQsj/sPGv2FgoYiVY+G/cJFdo4oOBx3szzMwLU8608bxvp7C2vrG5Vdwu7ezu7R+4h0ctLTNFoUkll6oTEg2cCWgaZjh0UgUkCTm0w9HVzG/fg9JMijszTqGXkIFgMaPEWKnv1gIhmYhAGHwDD0FQSogagcGpklFGDbYKERFWGQccgWYDoftu2at6c+BV4uekjHI0+u5nEEmaJXYJ5UTrru+lpjchyjDKYVoKMg0poSMygK6lgiSge5P5b1N8ZpUIx1LZskfO1d8TE5JoPU5C25kQM9TL3kz8z+tmJr7sTZhIMwOCLhbFGcdG4llQOGIKqOFjSwhVzN6K6ZAoQo2Ns2RD8JdfXiWtWtU/r3q3tXK9ksdRRCfoFFWQjy5QHV2jBmoiih7RM3pFb86T8+K8Ox+L1oKTzxyjP3C+fgBnKaDt</latexit>

Newmarket productand rule designs

<latexit sha1_base64="rfAqyReJfRlzY1gCWaVo//ZfVh8=">AAACGHicbVBNS8NAEN3Urxq/qh69LBahp5oURI8FETxWsK3QhLLZTNulm03Y3RRC6M/w4l/x4kERr735b9y0PWjrg2Uf780wMy9IOFPacb6t0sbm1vZOedfe2z84PKocn3RUnEoKbRrzWD4FRAFnAtqaaQ5PiQQSBRy6wfi28LsTkIrF4lFnCfgRGQo2YJRoI/Url56ImQhBaHwnQA4z7Hk2EWHxSVAgJ1BQRUcQphxUv1J16s4ceJ24S1JFS7T6lZkXxjSNzATKiVI910m0nxOpGeUwtb1UQULomAyhZ6ggESg/nx82xRdGCfEgluaZDefq746cREplUWAqI6JHatUrxP+8XqoHN37ORJJqEHQxaJByrGNcpIRDJoFqnhlCqGRmV0xHRBKqTZa2CcFdPXmddBp196ruPDSqzdoyjjI6Q+eohlx0jZroHrVQG1H0jF7RO/qwXqw369P6WpSWrGXPKfoDa/YDXhafQw==</latexit>

Energyandreserveschedules

<latexit sha1_base64="lDaOO6O7PhBAk9F3veWjL3l0OZE=">AAACC3icbVBNS8NAEN34WetX1KOX0CL0VJOC6LEgFE+2gmkLTSib7aZdutmE3YlQQu9e/CtePCji1T/gzX/jts1BWx8sPN6bmZ15QcKZAtv+NtbWNza3tgs7xd29/YND8+i4reJUEuqSmMeyG2BFORPUBQacdhNJcRRw2gnG1zO/80ClYrG4h0lC/QgPBQsZwaClvlnyRMzEgAqwXMHA84rN2/Nmo6GJAgypoqpvlu2qPYe1SpyclFGOVt/88gYxSSM9lHCsVM+xE/AzLIERTqdFT09NMBnjIe1pKnBElZ/Nb5laZ1oZWGEs9dNLzdXfHRmOlJpEga6MMIzUsjcT//N6KYRXfsZEkgIVZPFRmHILYmsWjDVgkhLgE00wkUzvapERlpiAjq+oQ3CWT14l7VrVuajad7VyvZLHUUCnqIQqyEGXqI5uUAu5iKBH9Ixe0ZvxZLwY78bHonTNyHtO0B8Ynz+XEpoJ</latexit>

UnitON/OFFstatuses

<latexit sha1_base64="lDaOO6O7PhBAk9F3veWjL3l0OZE=">AAACC3icbVBNS8NAEN34WetX1KOX0CL0VJOC6LEgFE+2gmkLTSib7aZdutmE3YlQQu9e/CtePCji1T/gzX/jts1BWx8sPN6bmZ15QcKZAtv+NtbWNza3tgs7xd29/YND8+i4reJUEuqSmMeyG2BFORPUBQacdhNJcRRw2gnG1zO/80ClYrG4h0lC/QgPBQsZwaClvlnyRMzEgAqwXMHA84rN2/Nmo6GJAgypoqpvlu2qPYe1SpyclFGOVt/88gYxSSM9lHCsVM+xE/AzLIERTqdFT09NMBnjIe1pKnBElZ/Nb5laZ1oZWGEs9dNLzdXfHRmOlJpEga6MMIzUsjcT//N6KYRXfsZEkgIVZPFRmHILYmsWjDVgkhLgE00wkUzvapERlpiAjq+oQ3CWT14l7VrVuajad7VyvZLHUUCnqIQqyEGXqI5uUAu5iKBH9Ixe0ZvxZLwY78bHonTNyHtO0B8Ynz+XEpoJ</latexit>

UnitON/OFFstatuses

<latexit sha1_base64="WmqZXZKwW2YAbOVykZtPkQLe6IM=">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</latexit>

Timeseries PV,wind, and Loaddata (minute-level)

<latexit sha1_base64="y39MRmZjYI2XC2yBFtWbguMBPKM=">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</latexit>

Timeseries PV,wind, and Loaddata (second-level)

P Y T H O N

<latexit sha1_base64="l0KwfSGdzn8pcOlzPINWFh/wtYs=">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</latexit>

pg,t, ploss

and real-timereserves

<latexit sha1_base64="Lu5TLWK/g2bjabCZseAWg4cLPtQ=">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</latexit>

update at theend of everyRTED interval

P S S / E

<latexit sha1_base64="SEdEJcHzP7r83C6QvAKSvMnOHSc=">AAACN3icbVDLSgMxFM34tr5GXboJFsGFlpmC6FIpghulgtXCzFDupGkNZpIhyQhl6F+58Tfc6caFIm79A9N2BG29EDiccx8nJ04508bznp2p6ZnZufmFxdLS8srqmru+ca1lpghtEMmlasagKWeCNgwznDZTRSGJOb2J72oD/eaeKs2kuDK9lEYJdAXrMALGUi33Ig+HSwLVjaPcq3jD2psA/VBIJtpUGHyeccP2wV4Jw9JJ7RTXgJOMg5Gq33LLPxN4EvgFKKOi6i33KWxLkiV2MeGgdeB7qYlyUIYRTvulMNM0BXIHXRpYKCChOsqHlvt4xzJt3JHKPmtsyP6eyCHRupfEtjMBc6vHtQH5nxZkpnMU5UykmaGCjA51Mo6NxIMQcZspSgzvWQBEMesVk1tQQIyNumRD8Me/PAmuqxX/oOJdVsvHu0UcC2gLbaNd5KNDdIzOUB01EEEP6AW9oXfn0Xl1PpzPUeuUU8xsoj/lfH0Doc2oMA==</latexit>

Multi-areaACE Calculator

<latexit sha1_base64="JgOjh0CwV4U+KmG6WJpXxKLF9mU=">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</latexit>

Dynamic AGCDispatcher

<latexit sha1_base64="rs2p2lhPh9lYtkdr/iThe/2syEM=">AAAB/nicbVDLSgNBEJz1GeMrKp68DAYhp7Arih4DevAYwTwgG8PspJMMmZ1dZnrFsAT8FS8eFPHqd3jzb5wke9DEgoaiqpvuriCWwqDrfjtLyyura+u5jfzm1vbObmFvv26iRHOo8UhGuhkwA1IoqKFACc1YAwsDCY1geDXxGw+gjYjUHY5iaIesr0RPcIZW6hQO/WuQyGh8n/oIj5iigPG4Uyi6ZXcKuki8jBRJhmqn8OV3I56EoJBLZkzLc2Nsp0yj4BLGeT8xEDM+ZH1oWapYCKadTs8f0xOrdGkv0rYU0qn6eyJloTGjMLCdIcOBmfcm4n9eK8HeZTsVKk4QFJ8t6iWSYkQnWdCu0MBRjixhXAt7K+UDphlHm1jehuDNv7xI6qdl77zs3p4VK6Usjhw5IsekRDxyQSrkhlRJjXCSkmfySt6cJ+fFeXc+Zq1LTjZzQP7A+fwBwz+V7g==</latexit>

ptie<latexit sha1_base64="Wwi/M9jKjSFhreHnvlA7jPW8dmM=">AAAB73icbVBNS8NAEN3Ur1q/qh69LBahp5KIoseCHjxWsB/QhrLZTtqlm03cnQgl9E948aCIV/+ON/+N2zYHbX0w8Hhvhpl5QSKFQdf9dgpr6xubW8Xt0s7u3v5B+fCoZeJUc2jyWMa6EzADUihookAJnUQDiwIJ7WB8M/PbT6CNiNUDThLwIzZUIhScoZU6vVuQyGjYL1fcmjsHXSVeTiokR6Nf/uoNYp5GoJBLZkzXcxP0M6ZRcAnTUi81kDA+ZkPoWqpYBMbP5vdO6ZlVBjSMtS2FdK7+nshYZMwkCmxnxHBklr2Z+J/XTTG89jOhkhRB8cWiMJUUYzp7ng6EBo5yYgnjWthbKR8xzTjaiEo2BG/55VXSOq95lzX3/qJSr+ZxFMkJOSVV4pErUid3pEGahBNJnskreXMenRfn3flYtBacfOaY/IHz+QNzvY+A</latexit>

f

<latexit sha1_base64="owf0xSPcefkPKktmnl7xFES89Rk=">AAAB8HicbVBNSwMxEM3Wr1q/qh69BIvQU9ktiB4rPehFqGBrpV1KNjvbhmaTJckKZemv8OJBEa/+HG/+G9N2D9r6YODx3gwz84KEM21c99sprK1vbG4Vt0s7u3v7B+XDo46WqaLQppJL1Q2IBs4EtA0zHLqJAhIHHB6CcXPmPzyB0kyKezNJwI/JULCIUWKs9Hh13cS3MgQ+KFfcmjsHXiVeTiooR2tQ/uqHkqYxCEM50brnuYnxM6IMoxympX6qISF0TIbQs1SQGLSfzQ+e4jOrhDiSypYweK7+nshIrPUkDmxnTMxIL3sz8T+vl5ro0s+YSFIDgi4WRSnHRuLZ9zhkCqjhE0sIVczeiumIKEKNzahkQ/CWX14lnXrNO6+5d/VKo5rHUUQn6BRVkYcuUAPdoBZqI4pi9Ixe0ZujnBfn3flYtBacfOYY/YHz+QOwJY+Y</latexit>

AGC Model

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preserve

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Extended-termed System Dynamic Model

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PFRControl

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AGC

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VoltageControl

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InertiaControl

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SRControl

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Freq./Volt.Control

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pAGC4sec

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Update everyRTED interval

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Update everyRTED interval

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Power Flow Analysis

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Multi-timescale Scheduling Model

Figure 4.3. Scheduling-dynamic information exchange.

be achieved by just running input.utils.cal ptdf(), where input is the object generated

from the general input data sheet. These useful utility functions are all implemented in a

vectorized fashion with fast and memory-saving features.

For DAUC and RTUC, we only generate the Pyomo abstract model one time since the

mixed-integer programming problem is time-consuming to save when the system size grows.

Then, for subsequent UC problems, we keep updating the model parameters in a rolling

fashion. For DAED and RTED, We generate the model for DAED and RTED every time,

as RTED does not impose a huge computational burden.

4.2.5.2. Dynamic Simulation Part

The dynamic simulation is by default executed in PSS®E, but other simulation platforms

could also interface with MIDAS-S on this matter. After each RTED run, the scheduling part

will save the RTED dispatch result in both CSV and Python binary formats. Afterward, the

results will be sent to PSS®E for dynamic analyses, while the RTED routine will be halted,

110

waiting for the updated information is sent from the PSS®E, serving as new initial inputs

for the next RTED. The information exchange process between the scheduling and dynamic

simulation parts is detailed in Fig. 3. We omit the detailed discussion over the dynamic

simulation part, which will be presented in another paper together with more comprehensive

numerical experiments.

4.2.6. Simulation Examples

This section verifies the proposed platform by conducting market analyses on an 18-

bus system with 5 generators and 1 ESS and a reduced WECC 243-bus system with 197

generators for scalability analyses. These two analyses are documented in the example

gallery of MIDAS-S, where the network and unit data can also be found. Listing 3 shows the

code executing four work modes for the 18-bus system with built-in summary and plotting

functions. The dynamic simulation is embedded for both test systems (denoted as closed-

loop simulation). All the subsequent simulations are performed in Intel Python 3.6.3 with

MIDAS-S 0.6.6, NumPy 1.19.2, Pandas 1.1.2, and Pyomo 5.7 on a 2.9 GHz quad-core Intel

Core i7 CPU with 16GB memory. We employ Gurobi 9.1 as the optimizer. Also, note that

all figures in this section are generated from the built-in plotting functions.

4.2.6.1. 18-bus Test System

?? depicts the 18-bus system topology. Since DAUC and RTED are included in Mode 3

and 4, we focus on the economic assessment of them. Two consecutive days are considered

in the simulation, while the second day’s timeseries data is a copy of the first day adding

Gaussian noise.

a). Mode 4 General Assessment.

In Mode 4, DAUC, RTUC, and RTED routines are executed in the multi-timescale coor-

dinative fashion. Figure 4.5 depicts the dispatch share between generators in DAUC, RTUC,

and RTED for the two days, and Table I tabulates the cost summary in Day 2. G1 and G3

111

Listing 4.3. Sample code for the 18-bus system run

1 import midass as ms

23 results_mode1 = ms.run_mode1(solver_name = "gurobi",

4 prefix_dir = r"midass/data /18 _bus/",

5 verbose = True)

6 results_mode4 = ms.run_mode4(solver_name = "gurobi",

7 prefix_dir = r"midass/data /18 _bus/",

8 psse_folder = r"PSSE/",

9 case_name = r"Case1",

10 run_name = r"Case1_24hr",

11 close_loop_flag = False ,

12 ess_interpolate_method = True ,

13 ess_dispatchable = False ,

14 verbose = True)

1516 Fig_Plot = ms.FigurePlot("DAUC", results_mode4)

are inflexible units with a startup time longer than 1 hour, whereas G2 and G4 are flexible

units with a startup time of less than 1 hour. Thus, G2 and G4 are re-committable in RTUC.

We consider all four AS types in this test case, while the AS requirements are set as different

portions of the load.

The overall dispatch pattern is intuitive. During the day when PV output is abundant,

the ESS is charging using the excess PV power. When the time goes to the night, as the

peak load occurs, the ESS is discharging to relieve the dispatch burden, during which the

fast-start gas unit G2 is brought up for the demand increase. Since we assume RTUC uses

the same timeseries profiles as in DAUC, there are not many differences between Figure 4.5

(a) and (b), while in Day 2, for Hour 7-10, RTUC shows a different dispatch profile with

DAUC. When the load and renewable profiles are the same, this dispatch difference could

be owing to the reserve relocation in RTUC when the simulation horizon for RTUC is only

3 hours. And RTED will strictly follow the RTUC commitments.

112

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Nuclear<latexit sha1_base64="gHMbEVbZIubbZTVVoGDah1KMbtw=">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</latexit><latexit sha1_base64="gHMbEVbZIubbZTVVoGDah1KMbtw=">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</latexit><latexit sha1_base64="gHMbEVbZIubbZTVVoGDah1KMbtw=">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</latexit><latexit sha1_base64="gHMbEVbZIubbZTVVoGDah1KMbtw=">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</latexit>

Hydro<latexit sha1_base64="d0CUnXMe0qNpPf53viThKLth+EI=">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</latexit><latexit sha1_base64="d0CUnXMe0qNpPf53viThKLth+EI=">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</latexit><latexit sha1_base64="d0CUnXMe0qNpPf53viThKLth+EI=">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</latexit><latexit sha1_base64="d0CUnXMe0qNpPf53viThKLth+EI=">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</latexit>

ESS<latexit sha1_base64="dovIA4K2k4RbYRfGb6/D/bJa1Tc=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0l6sceCCB4rtR/QhrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RS2tnd294r7pYPDo+OT8ulZR8epYthmsYhVL6AaBZfYNtwI7CUKaRQI7AbT24XffUKleSwfzSxBP6JjyUPOqLFS667VGpYrbtVdgmwSLycVyNEclr8Go5ilEUrDBNW677mJ8TOqDGcC56VBqjGhbErH2LdU0gi1ny1PnZMrq4xIGCtb0pCl+nsio5HWsyiwnRE1E73uLcT/vH5qwrqfcZmkBiVbLQpTQUxMFn+TEVfIjJhZQpni9lbCJlRRZmw6JRuCt/7yJunUqp5b9R5qlYabx1GEC7iEa/DgBhpwD01oA4MxPMMrvDnCeXHenY9Va8HJZ87hD5zPH9kxjW8=</latexit><latexit sha1_base64="dovIA4K2k4RbYRfGb6/D/bJa1Tc=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0l6sceCCB4rtR/QhrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RS2tnd294r7pYPDo+OT8ulZR8epYthmsYhVL6AaBZfYNtwI7CUKaRQI7AbT24XffUKleSwfzSxBP6JjyUPOqLFS667VGpYrbtVdgmwSLycVyNEclr8Go5ilEUrDBNW677mJ8TOqDGcC56VBqjGhbErH2LdU0gi1ny1PnZMrq4xIGCtb0pCl+nsio5HWsyiwnRE1E73uLcT/vH5qwrqfcZmkBiVbLQpTQUxMFn+TEVfIjJhZQpni9lbCJlRRZmw6JRuCt/7yJunUqp5b9R5qlYabx1GEC7iEa/DgBhpwD01oA4MxPMMrvDnCeXHenY9Va8HJZ87hD5zPH9kxjW8=</latexit><latexit sha1_base64="dovIA4K2k4RbYRfGb6/D/bJa1Tc=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0l6sceCCB4rtR/QhrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RS2tnd294r7pYPDo+OT8ulZR8epYthmsYhVL6AaBZfYNtwI7CUKaRQI7AbT24XffUKleSwfzSxBP6JjyUPOqLFS667VGpYrbtVdgmwSLycVyNEclr8Go5ilEUrDBNW677mJ8TOqDGcC56VBqjGhbErH2LdU0gi1ny1PnZMrq4xIGCtb0pCl+nsio5HWsyiwnRE1E73uLcT/vH5qwrqfcZmkBiVbLQpTQUxMFn+TEVfIjJhZQpni9lbCJlRRZmw6JRuCt/7yJunUqp5b9R5qlYabx1GEC7iEa/DgBhpwD01oA4MxPMMrvDnCeXHenY9Va8HJZ87hD5zPH9kxjW8=</latexit><latexit sha1_base64="dovIA4K2k4RbYRfGb6/D/bJa1Tc=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0l6sceCCB4rtR/QhrLZTtqlm03Y3Qgl9Cd48aCIV3+RN/+N2zYHbX0w8Hhvhpl5QSK4Nq777RS2tnd294r7pYPDo+OT8ulZR8epYthmsYhVL6AaBZfYNtwI7CUKaRQI7AbT24XffUKleSwfzSxBP6JjyUPOqLFS667VGpYrbtVdgmwSLycVyNEclr8Go5ilEUrDBNW677mJ8TOqDGcC56VBqjGhbErH2LdU0gi1ny1PnZMrq4xIGCtb0pCl+nsio5HWsyiwnRE1E73uLcT/vH5qwrqfcZmkBiVbLQpTQUxMFn+TEVfIjJhZQpni9lbCJlRRZmw6JRuCt/7yJunUqp5b9R5qlYabx1GEC7iEa/DgBhpwD01oA4MxPMMrvDnCeXHenY9Va8HJZ87hD5zPH9kxjW8=</latexit>

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23Hour

0

500

1000

1500

2000

2500

3000

Act

ive

Pow

er(M

W)

PV

Load

(a) (b)Figure 4.4. 18-bus diagram.

Table 4.2. Economic Assessment of Different Routines in Day 2

DAUC RTUC RTED

Gen. Cost ($) 247,697 247,654 240,243

ESS Cost ($) 44 44 37

LS Penalty Cost ($) 0 0 22

Ramp Penalty Cost ($) 0 0 73,543

TX Penalty Cost ($) 0 0 0

SOC Penalty Cost ($) 0 0 0

PV Curtailment (MWh) 27 27 59

Gen.: Generator, LS: Load shedding.

For the unit startup/shutdown trajectory modeling, it is straightforward to observe that,

in (c) and (f) during the startup and shutdown processes of G2, RTED models the linear

trajectories explicitly. Especially in (f), as G2’s startup time is 20 minutes, when the RTUC

shown in (e) determines G2’s startup at 18:00, RTED traverses four intervals in (f) for the

113

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RTED

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Day 1

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Day 2

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(b)<latexit sha1_base64="1Sw5w5L4WGO4t3SvlfgUX4pI7cs=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahXkpSED0WvOitgm2FppTNdtMu3WzC7kQsoX/DiwdFvPpnvPlv3LY5aOuDgcd7M8zMCxIpDLrut1NYW9/Y3Cpul3Z29/YPyodHbROnmvEWi2WsHwJquBSKt1Cg5A+J5jQKJO8E4+uZ33nk2ohY3eMk4b2IDpUIBaNoJd9H/oRBmFXZ+bRfrrg1dw6ySrycVCBHs1/+8gcxSyOukElqTNdzE+xlVKNgkk9Lfmp4QtmYDnnXUkUjbnrZ/OYpObPKgISxtqWQzNXfExmNjJlEge2MKI7MsjcT//O6KYZXvUyoJEWu2GJRmEqCMZkFQAZCc4ZyYgllWthbCRtRTRnamEo2BG/55VXSrte8i5p7V680bvM4inACp1AFDy6hATfQhBYwSOAZXuHNSZ0X5935WLQWnHzmGP7A+fwB1j6RkA==</latexit>

(c)

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(d)<latexit sha1_base64="FthFuRLqtktQPpF+8OaFxvJLtnU=">AAAB83icbVBNS8NAEN3Ur1q/qh69LBahXkpSED0WvOitgm2FppTNdtIu3WzC7kQsoX/DiwdFvPpnvPlv3LY5aOuDgcd7M8zMCxIpDLrut1NYW9/Y3Cpul3Z29/YPyodHbROnmkOLxzLWDwEzIIWCFgqU8JBoYFEgoROMr2d+5xG0EbG6x0kCvYgNlQgFZ2gl30d4wiDMqnA+7Zcrbs2dg64SLycVkqPZL3/5g5inESjkkhnT9dwEexnTKLiEaclPDSSMj9kQupYqFoHpZfObp/TMKgMaxtqWQjpXf09kLDJmEgW2M2I4MsveTPzP66YYXvUyoZIUQfHFojCVFGM6C4AOhAaOcmIJ41rYWykfMc042phKNgRv+eVV0q7XvIuae1evNG7zOIrkhJySKvHIJWmQG9IkLcJJQp7JK3lzUufFeXc+Fq0FJ585Jn/gfP4A2UqRkg==</latexit>

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(f)

Figure 4.5. 18-bus system’s Mode 4 dispatch results.

startup process. Then, once the startup process is completed at 18:00, thanks to the high

operation ramp rate of G2, it ramps sharply to a high MW output for the peak load time.

The shutdown process is also explicitly shown in this test case. It is noteworthy that the

startup and shutdown process design discussed in subsection 4.2.3 contributes to the explicit

modeling as shown in (c) and (f). Users could also choose other startup/shutdown modeling

approaches as MIDAS-S can flexibly switch.

We could also observe the real-time ESS performance that strictly follows the DAUC and

RTUC values. When the ESS performs the energy arbitrage in the day-ahead market coor-

dinating with the PV power, real-time ESS dispatch values are interpolated but restricted to

the SOC limits. However, the AS schedules for ESSs are still dispatchable in the real-time

process in case any sudden event occurs.

114

In MIDAS-S, we generally put slack variables with customized penalty costs for most

operational constraints, including the power balance, ramping constraints, power flow equa-

tions, SOC operations, and AS requirements, which is also an ISO’s practice [144]. Accord-

ing to Table I, we could see that MIDAS-S conducts the market operation under moderate

penalty values. We set very high penalty values, e.g., $100,000/MW for both load shedding

and ramp violation. In practical operations, the penalty values are unavoidable in the op-

timization. Then, they will be removed after the market clearing is complete by a pricing

rerun with violated constraint relaxation [149]. For PV curtailment, since we allow PV to

provide ASs in MIDAS-S, the curtailment can be used to provide upward reserves. However,

as we install an ESS in the system, the ESS takes the major responsibility to provide ASs

due to its flexible feature, and PV is prone to use its full headroom to charge the ESS during

the day. We could also see that the three modes’ unit production costs are very close, which

proves the consistency of the multi-timescale operation.

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PESS<latexit sha1_base64="befXQOvzoW2GBHFTqv8lQi5cYvc=">AAAB9HicbVBNS8NAEN34WetX1aOXYBE8laQgeiyKoLdK7Qe0oWy2k3bpZhN3J8US+ju8eFDEqz/Gm//GbZuDtj4YeLw3w8w8PxZco+N8Wyura+sbm7mt/PbO7t5+4eCwoaNEMaizSESq5VMNgkuoI0cBrVgBDX0BTX94PfWbI1CaR/IBxzF4Ie1LHnBG0UheB+EJ/SC9uqnVJt1C0Sk5M9jLxM1IkWSodgtfnV7EkhAkMkG1brtOjF5KFXImYJLvJBpiyoa0D21DJQ1Be+ns6Il9apSeHUTKlER7pv6eSGmo9Tj0TWdIcaAXvan4n9dOMLj0Ui7jBEGy+aIgETZG9jQBu8cVMBRjQyhT3NxqswFVlKHJKW9CcBdfXiaNcsk9Lzn35WLlLosjR47JCTkjLrkgFXJLqqROGHkkz+SVvFkj68V6tz7mrStWNnNE/sD6/AG3eJIT</latexit>

BESS

Figure 4.6. ESS Schedules in the 18-bus system.

115

b). Mode 4 ESS Performance with ASs.

We further validate the ESS implementation of the two operation schemes for different

flexibilities on the AS provision. Figure 4.6 provides the comparison between setting the

ESS as PESS-type and BESS-type. Note that we only show the DAUC results here since

ESSs perform the energy arbitrage in the day-ahead market. It is clear the ESS in the BESS

case covers a higher portion of ASs than the ESS in the PESS case as it can provide ASs by

exceeding its opposite limits. As described in subsection 4.2.4 B, the BESS type has better

flexibility of switching conditions and thus gains more capability to provide ASs. Figure 4.6

validates the exactness of the proposed ESS operating schemes. We also observe that the

total cost of using the BESS type is lower than the case using the PESS type because when

the AS burden is relieved by ESSs, generating units could leverage more power to perform

the economic dispatch. Concerning the future dominance of ESS installations, system oper-

ators need to consider the ESS flexibilities in market operations.

c). Comparison between Four Modes.

To further validate the performance, we run Mode 1, 2, 3, and 4 on the 18-bus test

system, and Table 4.3 gives the cost comparison. It is clear that for all modes, DAUC yields

the same results. And Mode 2’s RTEDs have the same outcomes as Mode 4’s RTEDs since

we feed in the previous results obtained by DAUC and RTUC in Mode 4. However, when

comparing Mode 3 and Mode 4, we could find that Mode 4’s RTEDs come to better results

with lower production costs and penalty values. In Mode 3, the ramping penalty is high since

Table 4.3. Cost Comparison between Four Modes in Day 2

DAUC ($) Mode 1 Mode 2 Mode 3 Mode 4

Prod. Cost 247,742 N/A 247,742 247,742

Penalty Cost 0 N/A 0 0

RTED ($) Mode 1 Mode 2 Mode 3 Mode 4

Prod. Cost N/A 240,298 242,893 240,298

Penalty Cost N/A 0 528,327 73,602

Prod: Production.

116

Table 4.4. Computational Time for Test Systems

(Unit: sec) Mode 1 Mode 2 Mode 3 Mode 4

18-bus W 1.32 493 496 542

18-bus S 0.84 127 128 151

WECC W 287 − − −

WECC S 193 − − −

W: Complete wall time; S: Solver time.

the ESS value inheritance is directly applied to the RTED and no generator is rescheduled

in RTUC, which shrinks the ramp room of RTED’s generators. This proves the importance

of using RTUC in the short-term market operations.

4.2.6.2. Scalability Analysis

To test the scalability of MIDAS-S, we further run tests on the WECC 243-bus system

and Table 4.4 tabulates the computational time for the two systems. For Mode 2, 3, and 4,

the time is for one-day simulations. Note that we report both the solver execution time and

the complete wall time including the time used for parsing models and information exchange.

We could see that when the system size grows, the computational time also increases. But it

is still within the acceptable range of execution time considering the workstation capability.

4.2.7. Summary

We propose and validate a Python-based open-source simulation library, MIDAS-S, for

the short-term electricity market operations. The library is highly flexible for parameter

tuning, result I/O, and model modifications based on different user preferences. We en-

hance the current electricity market operation model by new AS modeling and refined ESS

operations, considering the recent advancement of renewable technologies. The MIDAS-S

library also accommodates a unique and novel structure of DAUC-RTUC-RTED, validated

via simulation-based economic assessment. The generalized implementation of complicated

time-coupled market operations could help the open-source community understand com-

117

mercial electricity market frameworks. The ready-to-use interface between the scheduling

and dynamic simulation paves a more accurate path for market analyses and provides new

insights for future market operations.

118

Chapter 5

CONCLUDING REMARKS

Till now, we have conducted research on both the long-term and short-term market

operations, considering the massive penetration of renewable energy. By leveraging the

uncertainty-based optimization techniques, the decision-makings in electricity markets can be

largely enhanced. The proposed stochastic optimization-based market framework provides a

promising candidate solution towards the challenge that the proliferating renewables impose.

We also investigate a combined simulation platform for future’s potential electricity market

designs, with emphasis on renewable impacts on the electricity market. Below is a detailed

summary of what we have achieved in the finished works.

• Long-term Operation: We proposed a novel modeling framework and decomposi-

tionbased solution strategy combining SP and RO to deal with multiplex uncertainties

in coordinated mid- and long-term power system planning. After solving a multi-year

generation and transmission planning problem from an independent system operator

(ISO)’s perspective, we analyzed the economic values of renewable energy comparing

with other generation technologies in terms of the long-term investment payback. Con-

sidering the N-k contingencies and resource uncertainties, the hybrid stochastic and

robust model was efficiently solved by a combined algorithm encoding the column-

andconstraint generation and L-shaped method. We showed that the optimal invest-

ment portfolio in mid-term planning is the combined generation mix that includes

both renewables and conventional units. In a longer-term planning study, the renew-

able companies show their growing cost-effectiveness due to the low operation cost.

Note that the study we performed did not consider the energy storage and governmen-

tal and community incentives, strengthening the renewables’ competitiveness. Hence,

it is envisaged that a 100% renewable power system becomes feasible and economic in

the future.

119

• Short-term Operation: We proposed a three-stage unit commitment model for the

market operation of transmission and distribution coordination under the uncertainties

of renewable generation and demand variations. Since the central operator will require

more information about the increasingly active distribution networks, the T-D coordi-

nation becomes a promising solution. The novel T-D market paradigm we proposed

quantitatively analyzed the typical market operation under a high distribution-level

DER penetration. We showed that when considering the T-D coordination, the T and

D boundary mismatches can be favorably reduced to zero, which saves hundreds of

millions of dollars as operation costs. Besides, eliminating the active power mismatch

could potentially help mitigate the frequency stability issue in both systems. To ef-

ficiently achieve this goal, our devised generalized decomposition technique greatly

facilitates the solution of the coordinated market operation while considering complex

system components such as the network reconfiguration.

• Industry-level market simulation tool: Heretofore, there lacks a general market

operation simulation software that can integrate with the system dynamic simulation.

Market operators keep a strong interest in capable software that can help simulate

a realistic market environment considering both steady-state market products and

transient-state system reliability. We design and implement such an open-source soft-

ware based on Python programming language that could interface with commercial or

free solvers and professional power system simulation software such as PSS®E. Within

this open-source tool, various users from either industry or academia could easily get

access to the core of the electricity market operations adopted in a general way for

most scenarios in the world’s electricity market. Novel research ideas and traditional

engineering requirements synergize well within this platform to make a better future

for the progress of the electricity market.

120

Appendix A

LINEAR RELAXATION TECHNIQUE

The product q of one binary variable z and one continuous variable x can be relaxed as

follows. The relaxation is tight due to the convexity of the McCormick Envelope.

q ≤ x +M(1 − z), (A-1)

q ≥ x −M(1 − z), (A-2)

q ≤Mz, (A-3)

q ≥ −Mz. (A-4)

121

Appendix B

FEASIBILITY GUARANTEE OF THE GND ALGORITHM

We provide two alternatives to guarantee the feasibility of the proposed GND algorithm.

Upon infeasibility, a feasibility-check subproblem for the third-stage problem will need to

be solved and return a feasibility cut to the second stage. Note that the second stage will

not incur any infeasibility since the transmission ED problem has no integer and the load

shedding variables enforce the feasibility of operational constraints. Hence, we only formulate

the third-stage feasibility-check problem, as shown in (B-1).

∀c ∶ minjmn,h,c

H

∑h

∑mn

j+mn,h,c + j−mn,h,c, (B-1)

subject to

Constraint (3-3a)-(3-3f), (3-3l)-(3-3r), (B-1a)

xDmn,h,c ≥ 0, ∀h, (B-1b)

xDmn,h,c = 0, ∀n ∈ ADT ,∀h, (B-1c)

xDmn,h,c + xDnm,h,c = 1 + j+mn,h,c − j

−mn,h,c,

∀n ∈ FD/FDQ ,∀h, (B-1d)

xDmn,h,c + xDnm,h,c = z

∗q,h,c + j

+mn,h,c − j

−mn,h,c,

∀n ∈ FDQ ,∀h, (B-1e)

∑n∶(m,n)∈FD

xmn,h,c = 1, ∀m ∈ AD/ADT ,∀h, (B-1f)

j+mn,h,c, j−mn,h,c ≥ 0, ∀(m,n) ∈ FD,∀h, (B-1g)

If we write in a compact form and for each scenario, we attain (B-2).

∀c,∀ωDc ∶ Sd(wc, ωDc ) = min

jj, (B-2)

122

subject to

∥H2y∗ +K2z + e∥2 ≤ q⊺y∗ + p⊺z + k⊺j +H3x

∗2 + r2(ω

Dc ), (B-2a)

where j+mn,h,c and j−mn,h,c are positive slack variables imposed on the distribution reconfig-

uration constraints, j is their vector form. After we solve (B-2), the feasibility cut can be

formulated as in (B-3).

u⊺(H2y∗ + e) − v(q⊺y∗ + k⊺j +H3x

∗2 + r2(ω

Dc )) ≤ 0, (B-3)

Note that any infeasible set of distribution reconfigurations will incur infeasibility in all

scenarios of this distribution network. Hence, upon encountering an infeasible subproblem

instance, a feasibility cut will be generated and all other scenarios will be terminated be-

cause (B-3) contains enough feasibility information. The feasibility cut will be added into the

second-stage problem. Afterwards, the second-stage problem will be solved and add a cut

containing both feasibility and optimality information to the first stage. Then the iteration

continues from the first stage. This procedure is called Fast Forward and Fast Back [72],

which also coincides with (B-3). One of this method’s benefits is that distribution networks

can return their feasibility cut individually without compromising other distribution systems’

optimality.

Adding feasibility cuts, however, naturally increases the computational burden as the

algorithm would require more iterations to converge. As discussed in [150], we can simply

replace the constraints in problem (3-3) with the ones in problem (B-1) and add a huge

penalty for these slack variables in the objective function, which makes the third-stage prob-

lem always feasible. This manuever could avoid adding any feasibility cut while maintaining

feasibility. Again, as shown in (B-1), not all operational constraints should be relaxed but

only the ones with integer variables as they are the potential sources of infeasibility.

Note that, though without a concrete proof, evidence has been found that enforcing a

feasible subproblem appears to be more computationally efficient than adding feasibility

123

cuts [150] and corresponds to the ISO’s practice [151]. However, adding cuts is an exact and

canonical way. Nevertheless, both methods are able to eliminate the infeasibility issue to

different extents.

124

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