LINEARIZED AND DISTRIBUTED METHODS FOR POWER FLOW ANALYSIS AND
CONTROL IN SMART GRIDS AND MICROGRIDS
By
Niannian Cai

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Electrical Engineering - Doctor of Philosophy
2014

ABSTRACT
LINEARIZED AND DISTRIBUTED METHODS FOR POWER FLOW ANALYSIS AND
CONTROL IN SMART GRIDS AND MICROGRIDS
By
Niannian Cai
Optimization and control is a core part of Energy Management System (EMS), which receives
the data from supervisory control and data acquisition (SCADA) system, analyzes the data centrally and provides decision actions to the system operators. Unlike conventional power system,
smart grid and microgrid are more complex, dynamic and flexible, which requires a high level of
computational intelligence, speed and flexibility. This dissertation presents a linearized model to
analyze the optimization problem of smart grid, which can provide speed advantage as well as
enough accuracy. In the meantime, a distributed multi-agent based control system is proposed in
this dissertation for a flexible control of microgrids.
With the development of smart grid, power electronic control devices, such as Flexible AC
Transmission Systems (FACTS), are introduced into power system. They can help system operators adjust real and reactive power flows, provide voltage support or regulate voltage. The way to
determine the optimal size and location to install FACTS devices is a nonlinear optimization problem. Various nonlinear techniques have been proposed and developed to solve this optimization
model, such as descent methods, Newton’s methods, gradient projection methods, interior methods and so on. These nonlinear methods, to some extent, can provide accurate optimal solutions;
however, they are usually computationally expensive when dealing with large power systems with
tens of thousands of buses. And this computational speed sometimes cannot satisfy system operators’ requirements. Therefore, many industrial applications have utilized a DC optimal power flow
model which assumes a flat voltage magnitude over the system. This model can achieve the results
very fast, but it sacrifices accuracy and reactive power information.
To reach a better trade-off between accuracy and speed, in the first half of this dissertation, it
proposed a linearized power flow model for studying benefit of FACTS devices. This linear model

can achieve better accuracy than DC power flow model and maintain reactive power information
while the computational speed is not sacrificed.
In the meantime, the increasing penetration of renewable energy and its potential accommodation paradigm, microgrids, restrict traditional central control structure in terms of cost, flexibility and reliability. Distributed control is able to address these challenges in three aspects: more
economic efficiency by utilizing low-cost devices; more flexibility in terms of time-varying and
adaptive configurations or functions; and more robustness by continuing working in the presence
of single-point failure.
For the power balance control, this dissertation first proposed a distributed multi-agent system
without considering network losses and voltage regulation. In this proposed system, the information flows in parallel and results are obtained in a non-iterative way; therefore, this method achieves
superior performance in terms of speed without any convergence issues. In the case where power
losses are considerable and voltage regulation is expected, this dissertation proposed a distributed
multi-agent control for power balance based on Guess method. The proposed power flow algorithm fully makes use of communication time, and updates state information synchronously among
agents. Therefore it can also provide speed advantage over asynchronous methods.
For economic dispatch, this dissertation first proposed a distributed algorithm for microgrids
without considering network constraints. This proposed economic dispatch is merit for fast convergence and is applicable for on-line control. In order to consider the network constraints, this
dissertation presented a distributed multi-agent system which can consider the network constraints
with full or partial observation of system state information. The proof of convergence is also
presented in the dissertation.

Dedicated to:
My parents, Youxiang Cai and Junyun Yin
My wife Hui Fang,
And my brother Weiwei Cai

iv

ACKNOWLEDGMENTS

First of all, I would like to express my sincerely thank to my advisor Dr. Joydeep Mitra. I would
like to thank him not only for his enlightening guidance and continuous support for my graduate
studies, but also for all other non-academic merits that he teaches me in my daily life. He is one
of the nicest people that I have seen, and always stands besides me to give me advise whenever I
need. He is also very generous to tolerate and correct my countless funny mistakes. Without his
support, I can not make continuous process for my graduate study and finish my dissertation.
I am also very grateful to my committee members, Dr. Fang Z. Peng, Dr. Bingsen Wang, and
Dr. Jongeun Choi for their inspiring classes, valuable suggestions, and serving as my committee
members.
Many thanks go to my officemates and colleagues in ERISE lab, for their valuable discussions,
kindly help and priceless friendship, Ms. Nga Nguyen, Dr. Xufeng Xu, Mr. Mohammed Benidris,
Mr. Salem Elsaiahs, Mr. Samer Sulaeman, Ms. Yuting Tian, Mr. Valdama Johnson, Mr. Khaleel
O. Khadedah and Dr. Oluwafemi Ipinnimo.
I also would like to thank all my colleagues and friends in PELab, Dr. Shao Zhang, Dr. Shuitao
Yang and Mr. Yang Liu, for your collaboration in the transformer-less UPFC project. Your valuable
discussions and suggestions help me deeper understand the working principles of transfomer-less
UPFC. I also would like to thank Dr. Xi Lu, Dr. Qin Lei, Mr. Sisheng Liang, Mr. Xianhao Yu,
Dr. Shuai Jiang and all others friends in the PELab for their valuable discussions and helps in both
study and life.
I would like to express my special appreciation to my colleague, Mr. Yantao Song. Thank you
for your patient discussions and valuable suggestions about power electronic circuit and control. I
also want to thank you for your spiritual support and encouragement during my graduate study.
Last but not the least, I would like to thank my parents, my wife and my brother for their
unconditional care, support and understanding. The thanks that I would like to express for them is
beyond description.

v

TABLE OF CONTENTS

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

x

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Part I

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 The Need for New Optimization Techniques in Smart Grid . . . .
1.1.1 The Concept of Smart Grid . . . . . . . . . . . . . . . . .
1.1.2 Optimization Techniques in Traditional Grid . . . . . . . .
1.1.3 The Need for New Optimization Techniques in Smart Grid
1.2 The Need for New Control Techniques in Microgrid . . . . . . . .
1.2.1 The Concept of Microgrid . . . . . . . . . . . . . . . . .
1.2.2 Control Techniques in Traditional Grid . . . . . . . . . . .
1.2.3 The Need for New Control Techniques in Microgrids . . .
1.3 Scope of the Dissertation . . . . . . . . . . . . . . . . . . . . . .

Part II

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1
2
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5
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Linearized Methods for the Optimization in Smart Grids . . . . 10

Chapter 2 Linearized Methods for UPFC Benefit Study . . . . . . . . .
2.1 Linearized AC Power Flow Model (Lossless Model) . . . . . . . .
2.2 Linearization of Transmission Line Constraint . . . . . . . . . . .
2.3 Power Flow Model of UPFC . . . . . . . . . . . . . . . . . . . .
2.4 Linearized Power Flow Model of UPFC . . . . . . . . . . . . . .
2.5 Model of UPFC Benefit Study . . . . . . . . . . . . . . . . . . .
2.5.1 Maximize Revenue for Constant Loading Conditions . . .
2.5.2 Maximize Revenue for Time-Varying Loading Conditions
2.6 Simulation Result . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 Maximize Revenue for Constant Loading Conditions . . .
2.6.2 Maximize Revenue for Time-Varying Loading Conditions
2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11
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41

Chapter 3 Optimal UPFC Placement and Parameter Selection to Minimize Generation
Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Problem Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Sensitivity Analysis and Branch Numbering . . . . . . . . . . . . . . . .
3.2.3 Configuration of the Algorithm . . . . . . . . . . . . . . . . . . . . . . .

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43
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48

vi

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3.3
3.4

3.2.3.1 Fitness Function .
3.2.3.2 Number of Genes .
3.2.3.3 Crossover . . . . .
3.2.3.4 Mutation . . . . .
3.2.3.5 Selection . . . . .
Case Study . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . .

Part III

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48
50
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53

Distributed Methods for the Control in Microgrid . . . . . . . . . 55

Chapter 4 Distributed Power Balance Control . . . . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
4.2 Parallel Method: Minimal Spanning Tree . . . . . . .
4.2.1 Stage 1: Discovery of Minimal Spanning Tree
4.2.2 Stage 2: Information Feedback Process . . .
4.2.3 Stage 3: Generation and Load Dispatch . . .
4.3 Multi-Agent System Implementation and Simulation
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . .

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56
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65

Chapter 5 Distributed Power Balance Control Considering Voltage Regulation and
Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Net Power Acquisition by Gauss Method . . . . . . . . . . . . . . . . . . . . . .
5.2.1 Distributed Construction of Ybus . . . . . . . . . . . . . . . . . . . . .
5.2.2 Agent Based Power Flow . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2.3 Net Power Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Multi-Agent Implementation Algorithm . . . . . . . . . . . . . . . . . . . . . .
5.4 Load and Generation Dispatch . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Simulation Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1 Five-Bus System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.1.1 Multi-agent Based Construction of Ybus . . . . . . . . . . . .
5.5.1.2 Multi-Agent Based Power Flow Calculation . . . . . . . . . . .
5.5.1.3 Netpower Acquisition . . . . . . . . . . . . . . . . . . . . . .
5.5.2 14-Bus System, 30-Bus System and 57-Bus System . . . . . . . . . . . .
5.5.2.1 14-Bus System . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2.2 30-Bus System . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2.3 57-Bus System . . . . . . . . . . . . . . . . . . . . . . . . . .
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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81

Chapter 6 Distributed Economic Dispatch
6.1 Introduction . . . . . . . . . . . . .
6.2 Economic Dispatch Model . . . . .
6.3 Proof of Convergence . . . . . . . .
6.4 Simulation Results . . . . . . . . .

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82
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92

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vii

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6.5

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

Chapter 7 Distributed Economic Dispatch Considering Network Constraints
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Proof of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4.1 Linearized Economic Dispatch Considering Network Constraints
7.4.2 Nonlinear Economic Dispatch Considering Network Constraints
7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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97
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108

Chapter 8 Multi-Agent Implementation and Multi-Level Control Architecture
8.1 Multi-Level Control Structure . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Multi-Agent Implementation . . . . . . . . . . . . . . . . . . . . . . . .
8.3 Simulation Platform of Lower Level . . . . . . . . . . . . . . . . . . . .
8.3.1 Master Power Electronic Interfaces . . . . . . . . . . . . . . . . .
8.3.2 Slave Power Electronic Interfaces . . . . . . . . . . . . . . . . .
8.4 Multi-Level Control Architecture . . . . . . . . . . . . . . . . . . . . . .
8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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110
110
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123

Part IV

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

Chapter 9 Contributions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . 126
9.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
9.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

viii

LIST OF TABLES

Table 2.1 Number of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Table 2.2 Number of Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Table 2.3 Number of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Table 2.4 Number of Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Table 2.5 Number of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Table 2.6 Number of Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
Table 2.7 Simulation Result for Constant Loading Conditions . . . . . . . . . . . . . . 34
Table 2.8 Simulation Result For Constant Loading Conditions with Deficient Reactive
Power Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Table 2.9 Load Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Table 2.10 Number of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Table 2.11 Number of Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Table 2.12 Simulation Result of Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Table 2.13 Simulation Result of Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Table 3.1 Simulation Optimal Placement and Parameter Selection of UPFC . . . . . . . 52
Table 3.2 Simulation Results of Subsequent Investment in the Case of 5-Year Utilization of UPFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Table 4.1 Parent Child Relationship Diagram . . . . . . . . . . . . . . . . . . . . . . . 60
Table 4.2 Simulation Results for Power Balance Control . . . . . . . . . . . . . . . . . 64
Table 5.1 Bus Input Data for Five-Bus System . . . . . . . . . . . . . . . . . . . . . . 76
Table 5.2 Line Input Data for Five-Bus System . . . . . . . . . . . . . . . . . . . . . . 76
Table 5.3 Local Yi Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Table 5.4 Results for the First Five Iteration . . . . . . . . . . . . . . . . . . . . . . . . 79
ix

Table 5.5 Comparisons Between Power Balance Control Algorithms . . . . . . . . . . 81
Table 6.1 Generation Data for the Microgrid . . . . . . . . . . . . . . . . . . . . . . . 92
Table 7.1 Observed Area for Each Bus . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Table 7.2 Result Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
Table 7.3 Observed Area for Each Bus for IEEE 9-Bus System . . . . . . . . . . . . . 107
Table 7.4 Result Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
Table 8.1 Generation Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
Table 8.2 Data for PE Interfaces and Local Controllers . . . . . . . . . . . . . . . . . . 119
Table 8.3 Load Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

x

LIST OF FIGURES

Figure 2.1 Configuration of UPFC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Figure 2.2 A two node simple system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Figure 2.3 Transmission line constraint diagram. . . . . . . . . . . . . . . . . . . . . . 18
Figure 2.4 (a) Approximation of circular boundary. (b) Linear constraints for transmission line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Figure 2.5 Equivalent Circuit of UPFC . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Figure 2.6 Equivalent power injection model of UPFC . . . . . . . . . . . . . . . . . . 21
Figure 2.7 Vector diagram of UPFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Figure 2.8 Equivalent model of UPFC . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Figure 2.9 Feasible region of Vsp and Vsq . . . . . . . . . . . . . . . . . . . . . . . . 24
Figure 2.10 Linearization of Vsp and Vsq

. . . . . . . . . . . . . . . . . . . . . . . . . 25

Figure 2.11 UPFC investment versus capacity . . . . . . . . . . . . . . . . . . . . . . . 36
Figure 2.12 UPFC benefit versus capacity . . . . . . . . . . . . . . . . . . . . . . . . . 37
Figure 2.13 Production savings over equipment cost . . . . . . . . . . . . . . . . . . . . 37
Figure 3.1 (a) Bad solution space; (b) bad solution space; (c) good solution space . . . . 46
Figure 3.2 Flow chart of numbering branches . . . . . . . . . . . . . . . . . . . . . . . 49
Figure 3.3 Configuration of the algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 49
Figure 3.4 Crossover strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Figure 3.5 Mutation strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Figure 3.6 Average value of the population for the genetic algorithm with optimal branch
numbering and not optimal branch numbering. . . . . . . . . . . . . . . . . 53
Figure 4.1 Multi-agent system structure. . . . . . . . . . . . . . . . . . . . . . . . . . 58

xi

Figure 4.2 (a) Token transmission route. (b) Minimal spanning tree constructed. (c)
Information flow path for Stage 2. . . . . . . . . . . . . . . . . . . . . . . . 60
Figure 4.3 Simulation of multiple threads on a single-thread platform. . . . . . . . . . . 63
Figure 4.4 Two MAS structures: (a) completely parallel resulting in tightest coupling;
(b) completely sequential resulting in loosest coupling. . . . . . . . . . . . . 65
Figure 5.1 Information flow path for multi-agent system . . . . . . . . . . . . . . . . . 75
Figure 5.2 Tested five-bus system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Figure 6.1 Cost curve for agent m and agent n . . . . . . . . . . . . . . . . . . . . . . 88
Figure 6.2 Flow chart for economic dispatch . . . . . . . . . . . . . . . . . . . . . . . 91
Figure 6.3 Neighborhood relationship diagram . . . . . . . . . . . . . . . . . . . . . . 93
Figure 6.4 Generation dispatch result for five-agent system . . . . . . . . . . . . . . . . 93
Figure 6.5 Marginal cost for five-agent system . . . . . . . . . . . . . . . . . . . . . . 94
Figure 6.6 Generation cost for five-agent system . . . . . . . . . . . . . . . . . . . . . 94
Figure 6.7 Simulation results for economic dispatch . . . . . . . . . . . . . . . . . . . 95
Figure 7.1 Neighborhood relationship for IEEE 14 bus system . . . . . . . . . . . . . . 101
Figure 7.2 Generation cost curve for the first scenario of case 1 . . . . . . . . . . . . . 102
Figure 7.3 Generation cost curve for the second scenario of case 1 . . . . . . . . . . . . 103
Figure 7.4 Generation cost curve for the first scenario of case 2 . . . . . . . . . . . . . 103
Figure 7.5 Generation cost curve for the second scenario of case 2 . . . . . . . . . . . . 104
Figure 7.6 Generation cost curve for the first scenario of case 1 for AC models . . . . . 106
Figure 7.7 Generation cost curve for the second scenario of case 1 for AC models . . . 106
Figure 7.8 Generation cost curve for the first scenario of case 2 for AC models . . . . . 107
Figure 7.9 Generation cost curve for the second scenario of case 2 for AC models . . . 108
Figure 8.1 Proposed hierarchical control architecture. . . . . . . . . . . . . . . . . . . 111
Figure 8.2 Configuration of PE interface. . . . . . . . . . . . . . . . . . . . . . . . . . 114

xii

Figure 8.3 Control block for voltage-source power electronic interface. . . . . . . . . . 114
Figure 8.4 Control block for current-source power electronic interface. . . . . . . . . . 116
Figure 8.5 Microgrid configuration under study. . . . . . . . . . . . . . . . . . . . . . 118
Figure 8.6 Traces of messages for token transfer. . . . . . . . . . . . . . . . . . . . . . 119
Figure 8.7 Traces of messages for Information Feedback Process. . . . . . . . . . . . . 120
Figure 8.8 Traces of messages for Generation or Load Dispatch. . . . . . . . . . . . . . 121
Figure 8.9 (a) Convergence of marginal cost. (b) Dispatched values for the DGs. . . . . 121
Figure 8.10 (a) Convergence of marginal cost. (b) Dispatched values for the DGs. . . . . 122
Figure 8.11 (a) Convergence of marginal cost. (b) Dispatched values for the DGs. . . . . 122
Figure 8.12 (a) Real power output of DGs. (b) Reactive power output of DGs. (c) Real
power consumed by Load. (d) Reactive power consumed by Load. . . . . . . 123
Figure 8.13 (a) Output current transitions of DGs at 0.5s. (b) Output current transitions
of DGs at 1.0s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

xiii

Part I
Introduction

1

Chapter 1
Introduction
Optimization and control is a core part of Energy Management System (EMS), which receives the
data from supervisory control and data acquisition (SCADA) system, analyzes the data centrally
and provides decision actions to the system operators. However, with development of smart grid,
power electronic control devices, such as Flexible AC Transmission Systems (FACTS), are introducing into power system. Traditional DC optimization or AC optimization techniques turn out no
longer to satisfy accuracy or speed requirements. In the meantime, the increasing penetration of renewable energy and its potential accommodation paradigm, microgrids, restrict traditional central
control structure in terms of cost, flexibility and reliability. This dissertation utilizes linearization
and distributed techniques to develop innovative optimization and control methods for smart grid
and microgrid.

1.1
1.1.1

The Need for New Optimization Techniques in Smart Grid
The Concept of Smart Grid

According to [1], “a smart grid is a modernized electrical grid that uses information and communications technology to gather and act on information, such as information about the behaviors of
suppliers and consumers, in an automated fashion to improve the efficiency, reliability, economics,

2

and sustainability of the production and distribution of electricity”.
A smart grid is realized by implementing wide applications of a class of technology and devices, such as two-way communication technology, computer-based remote control, computedbased automation, wide area measurement systems, Phasor Measurement Units (PMU) and Flexible AC Transmission Systems (FACTS) devices, to bring electric delivery systems into modernization. According to EPRI [24], the roles of smart grid include engaging consumers, enhancing
efficiency, ensuring reliability and enabling renewable energy and electric transportation.

1.1.2

Optimization Techniques in Traditional Grid

Optimal power flow (OPF) problem, which came into focus dating back from 1960s, covers a
group of network related optimization problems in the field of power system planning, operation
or control. Examples of these problems can be: minimization of transmission loss, maximization
of market profit, minimization of number of altered control variables, minimization of transmission
line congestion, or minimization of pollution emission and operation cost et al. Mathematically, it
requires to solve a static constrained nonlinear optimization problem with one or more objective
functions and a set of equality and inequality constraints shown in equations (1.1)–(1.3). F (x, u)
is the objective function, which can be generation cost, load curtailment or transmission line congestion. g(x, u) and h(x, u) are equality and inequality constraints for the network states and
control variables. Its solution offers the optimal control variables that best satisfy objectives and
respect prespecified constraints. Numerous methods have been proposed to deal with OPF problem. These methods, based on their intrinsic characteristics, can be classified into two categories:
linear programming (LP) based methods and non-linear programming (NLP) based methods.

Minimize:

F (x, u)

(1.1)

Subject to:

g(x, u) = 0

(1.2)

h(x, u) ≤ 0

(1.3)

3

LP based methods are credited for their fast and reliable attributes, however, the requirement for
linearized model and accuracy consideration sometimes restrict their applications. A common LP
based method involves solving the OPF problem by DC power flow model [2–4]. In this method,
reactive power network equations are dropped; while all the voltage magnitudes are assumed to
be 1.0 per unit in the network. This method is popularly adopted in many commercial softwares
because of its speed and reliability. It can be used to calculate MW flows on transmission lines, but
it does not indicate any information about voltage magnitudes or MVar flows. Another common
LP based method solves full power flow model at an operating point and then linearizes the OPF
problem at that point. This method also benefits from high speed and reliability. However, it works
when the optimal point is near the operating point and it requires pre-acquisition of system operating point; while a lot of systems do not have this information. Refs. [5–7] provide a successive
LP based method for OPF problem. This method solves the non-linear network model by using
repetitive linear programmings to approach the optimal solution. It requires high computational
capability.
NLP based methods are favorable for high accuracy, but this merit is achieved by sacrificing
computational time and vast memory. Most NLP based methods existing to address OPF problem
include descent methods, Newton’s methods [8, 9], sequential quadratic programming [10, 11],
gradient projection methods [12, 13], augmented lagrangian methods [14, 15], interior point methods [16–18] and heuristic methods [19–21] et al. Any method mentioned above is only superior to
others for a certain format of nonlinear optimization problems. No single method is best at solving
all kinds of optimization problems.

1.1.3

The Need for New Optimization Techniques in Smart Grid

In smart grid, FACTS devices will be widely used to achieve flexible electricity transportation.
Traditional DC optimization models can not satisfy the optimization problem involving FACTS
devices because they do not provide reactive power and bus voltage magnitude information. For
some FACTS devices, such as STATCOM and UPFC, reactive power support and voltage regu4

lation is one of their dominant functions. Furthermore, DC optimization models sacrifice enough
accuracy in order to achieve high speed. Full AC optimization models can provide accurate and
full state information, however, the computation burden in terms of speed and complexity, sometimes, is forbidden. Besides, the AC models of a power system are usually highly nonlinear with
thousands of state and control variables; therefore, no AC optimization techniques can guarantee
to always achieve global optimum.
Suffering from the deficiencies of DC and AC optimization models, the development of smart
grid highly requires an optimization model that can achieve a better trade-off between speed and
accuracy. The new optimization model should be linear so that it can be solved by fast and reliable
LP method. It should also include complete state information with enough accuracy, so that it
can be used to analyze optimization problems involving reactive power and voltage control. This
comprehensive exam presents a generic framework for optimal power flow problems with FACTS
devices by using linearization methods to linearize power flow equations, transmission line constraints and FACTS devices. The proposed optimization model can be solved in one shot by LP
method; therefore, it has speed and reliability advantage as DC models. Additionally, this model
can provide voltage magnitude and reactive power flow information for the network, which traditional DC models can not offer. Furthermore, it also accomplishes more accurate results than DC
models with substantially less computational work than full AC power flow.

1.2
1.2.1

The Need for New Control Techniques in Microgrid
The Concept of Microgrid

A microgrid is a small-scale smart grid. It is a low voltage utility system formed by a cluster
of interconnected local small generations, including renewable (such as wind, PV, etc.) and conventional (micro-turbine, fuel cells, diesel generator) sources, intermediate energy storages and
loads [22]. As an electrical entity that facilitates high penetration of distributed generators, the
microgrid can be recognized and developed as a primary building block of smart grid.
5

A common expectation of a microgrid is that it should be capable of islanded (off-grid) operation. However, in this mode, several factors affect the ability of microgrids to achieve and maintain
stable operation. First, the unavailability of power support from the grid requires a global power
balance controller which is able to coordinate power generation and consumption in microgrids
and maintain system voltage and frequency. Further, if distributed generators (DGs) in a microgrid are required to be “plug and play”, no pre-determined economic dispatch would satisfy the
requirement of the microgrid; thus, an on-line economic dispatch controller is anticipated to reduce generation cost. Moreover, with substantial penetration of DGs, the controller of microgrid
is required to be flexible and adaptive; because it is difficult to estimate the number and the places
of DGs that are going to be connected. One of connecting nodes may supply power at one time but
absorb power at another time.

1.2.2

Control Techniques in Traditional Grid

Conventionally, power system monitoring, control and decision making are realized in a central
control structure. A control center utilizes a centralized supervisory control and data acquisition
(SCADA) system to remotely monitor the operation of the system spread over a large geographical
area, and an Energy Management System (EMS) to analyze the data, conduct necessary power flow
and optimal power flow computations and provide control actions to the operators.
This centralized control system plays a vital role in traditional power system, whose configuration is comparatively permanent and does not have high penetration of renewable energy and DGs.
However, as the movement of smart grid, especially the microgrid, increasing penetration of renewable energy and DGs is introduced into the system, which requires more flexible and scalable
control architecture. The central control architecture also suffers from high cost and low reliability. Because the centralized control center requires a costly super computing and data-storing
center, which can memorize astronomical amount of data obtained by the meters spread over the
system, analyze them in real time and compute for a possible control action. It is also vulnerable
to single-point failure. If the control center fails, the whole power system is under risk of collapse.
6

1.2.3

The Need for New Control Techniques in Microgrids

The configurations and operating modes of microgrids are allowed to dynamically change. DGs
may exhaust their fuel and shed; renewable energy may cut in or out with variable wind or solar
energy; storage units may continue switching between discharging and charging modes; different load resources may connect or disconnect. With the participation of plug-in hybrid/electric
vehicles, the extent of dynamic feature will increase considerably.
In the context of microgrids, conventional centralized structure restricts its application in terms
of cost, flexibility, and reliability. Distributed control is able to address these challenges in a
flexible and secure way by providing three merits: economic efficiency by integrating various
applications, utilizing lower-cost devices, and saving from the maintenance of the SCADA and
EMS system; flexibility in terms of time-varying and adaptive configurations or functions; and
robustness by continue working in the presence of single-point failure [23].
Multi-agent system (MAS) emerges as a potential solution that facilitates distributed control
for microgrids. An agent is an autonomous entity that can perceive and react to its environment
and communicate with others to achieve its local goal. MAS is a system consisting of two or
more agents. These agents collaborate and compete with each other to optimize its local goal, and
therefore overall to achieve a global goal. An agent has three characteristics: reactivity (reactive to
its environment), pro-activeness (exhibit goal-directed behavior), and social ability (communicate
with each other) [46]. Agents in microgrid are expected to implement tasks as sensory, communication and actuation [45].
Previously, MAS-based methods have been widely applied to various power system problems,
including microgrid control. Refs. [46, 47] have provided an excellent overview of such applications. Ref. [48] decomposed the optimal power flow problem and solved for the smaller subsystems
in parallel by using MAS. In [49, 50], by making use of autonomous and decentralized characteristics, MAS was introduced to solve electricity market problem. Refs. [39, 41, 51, 52] proposed
various methods to utilize MAS technique for power balance control and operation. Other MAS
applications include power system monitoring [53–55], protection [56], restoration [57–60] and
7

stability enhancement [61–63].
The distributed control techniques proposed in this dissertation are designed based on a decentralized multi-agent platform for the control and operation of microgrids. In this decentralized
architecture, all agents are hierarchically identical. There is no central or master agent, therefore,
it enhances the controller’s reliability and flexibility.

1.3

Scope of the Dissertation

This dissertation is organized as follows:
Chapter 2 utilizes linearization techniques to derive linearized optimization models for optimal
power flow problems with FACTS devices. It includes linearization of power flow equations,
linearization of transmission line constraints, and linearization of FACTS devices. It also presents
linearized models for revenue maximization problems of installing UPFC in both constant and
time-varying loading conditions.
Chapter 3 presents an efficient combined algorithm using genetic algorithm and linearized optimal power flow framework for the optimal location and parameter selection of UPFC. An optimal
branch numbering strategy is also proposed in this chapter to accelerate the convergence speed of
the algorithm. Simulation results showing the benefits of installing UPFC versus investments, as
well as diminishing returns of the subsequent investments are also presented in this chapter.
Chapter 4 proposes a distributed method for power balance control of microgrids without considering network losses and voltage regulation. The information flows in parallel and results are
obtained in non-iterative way; therefore, the algorithm can achieve superior performance in terms
of speed without any convergence issues.
Chapter 5 presents a distributed algorithm for power balance control of microgrids considering
network losses and voltage regulation based on Guess method. The distributed power flow algorithm fully makes use of communication time, and updates state information synchronously among
agents, which offers potential speed advantage.

8

Chapter 6 proposes a distributed economic dispatch algorithm for microgrids without considering network constraints. The Proof of convergence and proof of global optimization are presented
in this chapter. Simulation results show that the proposed algorithm is applicable for on-line economic dispatch.
Chapter 7 proposes a distributed algorithm for economic dispatch considering network constraints. Convergence of the algorithm is presented to ensure its stability. Simulation results show
that if the system states are globally observed, the proposed algorithm can achieve global optimization; if the system states are partially observed, the proposed algorithm can minimize total
generation cost, but it may be trapped by local optimum.
Chapter 8 presents the implementation of multi-agent system using Java Agent DEvelopment
(JADE) platform. It proposes a multi-level control architecture for the safe and economic operation
of microgrids. The upper MAS power balance layer accomplish exact power balance in three
sweeps, regardless of system size. The lower MAS based economic dispatch layer is able to
conduct economic dispatch on-line. Local control layer complies with the instructions from MAS
and implement control locally.
Chapter 9 discusses the conclusions and future work.

9

Part II
Linearized Methods for the Optimization in
Smart Grids

10

Chapter 2
Linearized Methods for UPFC Benefit
Study
Unified Power Flow Controller (UPFC) is one of the most widely used Flexible Alternating Current
Transmission Systems (FACTS) devices that can control real and reactive power flow through a
transmission line and regulate bus voltage simultaneously and independently. It promotes the grid
“smarter” by enabling flexible electric transportation. Utilization of UPFC can increase the transfer
capability of existing transmission lines and potentially reduce the demand for new transmission
facilities. From 2001 to 2008, Edison Electric Institute (EEI) invested nearly $57.5 billion in the
upgrade of its member company transmission infrastructure [26]. Future transmission expansion
can be deferred by developing UPFCs.
Vi

Vj

’
Vs Vi

TB

Ip+Iq
Shunt converter

Series converter

TE
VDC

Figure 2.1: Configuration of UPFC.

11

Zij

Figure 2.1 shows the configuration of UPFC. It is consisted of two back-to-back converters.
The series converter can generate a series voltage Vs in the range 0 ≤ Vs ≤ Vsmax with phase
angle ϕs in the range 0 ≤ ϕs ≤ 2π; therefore the terminal voltage Vi can be controlled for
the regulation of real and reactive power flow through the transmission line. The shunt converter
absorbs a real power current to compensate for the real power consumption of series converter in
order to maintain DC capacitor voltage, and injects a reactive power current to regulate bus voltage
Vi and provide reactive power support. Overall, UPFC has two main functions: real and reactive
power control of a transmission line, and bus voltage or shunt reactive power control.
Traditional UPFC benefit study either uses DC model or full AC model. DC model can computes the benefit and determines the optimum size of series converter; however, It does not provide
any reactive power and voltage magnitude information for the system; while reactive power flow
control and bus voltage regulation is an important function that UPFC can realize. It was therefore
considered important to extend the OPF framework to accommodate reactive power injections and
flows. On the other hand, series converter and parallel converter of UPFC are two independent
parts. How to determine the optimal capacity of each part for a given power system is also a key
problem for UPFC economic study. DC power flow study is capable of helping us determine the
optimal capacity of the series converter, but it cannot help us to decide the shunt size. Full AC
power flow optimization is a traditional solution to deal with both real and reactive power, but its
nonlinear characteristics and complexity requires a long time to find the optimal solution. Due
to this consideration, this dissertation proposed an innovative linearized ac power flow, which is
able to cope with both real and reactive power, but it also benefits from linear character and simple
implementation.

2.1

Linearized AC Power Flow Model (Lossless Model)

It is well known that the basic power flow equations for real and reactive power are presented as:

12

Pk = V k

Vm (Gkm cos δkm + Bkm sin δkm )

(2.1)

Vm (Gkm sin δkm − Bkm cos δkm )

(2.2)

m∈S
Qk = Vk
m∈S
where Pk and Qk are real and reactive power injected into node k; Vk and Vm are voltage magnitude at node k and node m; S is the set of all the buses in the system. δkm is the voltage angle
difference between node k and node m; Gkm and Bkm are the real and reactive part of (k,m)
element of bus admittance matrix (Ykm ).
In the steady state, power system is usually operated around the operating point; therefore bus
voltage magnitude is always around 1.0 p.u.

Vk = 1.0 + ∆Vk

(2.3)

If we ignore the small portion ∆Vk , the magnitude of the voltage at bus k can be approximated
by 1.0 p.u. However, it is important to note that this is only an approximation that enables the
linearization; it is not an assumption that the voltage magnitude equals 1.0 p.u.

Pk ≈

Vm (Gkm cos δkm + Bkm sin δkm )

(2.4)

Vm (Gkm sin δkm − Bkm cos δkm )

(2.5)

m∈S

Qk ≈
m∈S

If the phase angles between two adjacent buses are small enough, it is valid to make the approximation that sin δkm = δkm , cos δkm = 1. Then the equations above can be expressed
as:

Pk ≈

(Vm Gkm + Vm Bkm δkm )
m∈S

13

(2.6)

Qk ≈

(Vm Gkm δkm − Vm Bkm )

(2.7)

m∈S
Again,

Vm = 1.0 + ∆Vm

(2.8)

Take (2.8) into (2.6) and (2.7).

Pk ≈

(Vm Gkm + Bkm δkm + ∆Vm Bkm δkm )

(2.9)

(Gkm δkm + ∆Vm Gkm δkm − Vm Bkm )

(2.10)

m∈S

Qk ≈
m∈S

Note that both ∆Vm and δkm are small numbers around zero. Therefore, it is reasonable to
eliminate the terms that contain a product of ∆Vm and δkm . Hence:

Pk ≈

(Vm Gkm + Bkm δkm )

(2.11)

(Gkm δkm − Vm Bkm )

(2.12)

m∈S

Qk ≈
m∈S

Let us focus on equation (2.11) first, it can be further written as:

Pk =
m∈S
Since

Bkm δk −

Vm Gkm +
m∈S





Bkm =




n=k

Bkm δm

(2.13)

m∈S

bkn + bkk for m = k
−bkm

for m = k

where bkk is the total shunt susceptance at bus k. bkm is the susceptance of branch connecting
14

node k and node m. Hence,

Bkm δk = [−bk1 − ... + (
m∈S

bkm + bkk ) − ... − bkN ]δk
m=k

(2.14)

= bkk δk
Finally, linearized real power equation can be obtained by rearranging equation (2.13):

Vm Gkm −

Pk =
m∈S

Bkm δm − (Bkk − bkk )δk

(2.15)

m=k

In the same manner, the linearized reactive power can be expressed as:

Gkm δm − (Gkk − gkk )δk −

Qk = −

Vm Bkm

(2.16)

m∈S

m=k

In matrix form, the linearized AC power flow can be described as:








 P   −B

=
−G
Q
where





B =




(B11 − b11 )
B21

(2.17)

...

B1N

(B22 − b22 ) ...

B2N

...

BN 1

BN 2
G

G  δ 


−B
V

B12

...





 11

 G21
G=

 ...


...

... (BN N − bN N )

G12

...

G22

...

...

...

...



G1N


G2N 
...

GN 1 GN 2 ... GN N

15



















G =




(G11 − g11 )

G12

...

G1N

(G22 − g22 ) ...

G21
...

...

GN 1

GN 2



B

 11

 B21
B=

 ...


...

...









... (GN N − gN N )

B12

...

B22

...

...

G2N



...



B1N


B2N 
...






BN 1 BN 2 ... BN N

2.2

Linearization of Transmission Line Constraint

In traditional DC optimal power flow calculation, reactive power flow is always not considered.
Power flow constraints of transmission lines are simply assumed to be real power flow constraints,
which can be stated as equation (2.18). This simplification will introduce considerable error in the
optimal power flow calculation, since reactive power flow sometimes takes up a non-neglectable
portion in the power flow of transmission line.

max |
|Pij | ≤ |Sij

(2.18)

In the linearized power flow proposed in section 2.1, it is accessible to obtain voltage magnitude and reactive power state information. Consequently, it provides a prerequisite to linearize
transmission line constraint considering reactive power. First consider a simple system shown in
Fig. 2.2.

P12+jQ12

V1∠θ1

Z=R+jX

V2 ∠θ 2

Figure 2.2: A two node simple system.
16

Power flow in this transmission line can be calculated as:

P12 =

X
R
(V12 − V1 V2 cos δ12 ) +
V1 V2 sin δ12
Z2
Z2

(2.19)

Q12 =

R
X
(V12 − V1 V2 cos δ12 ) −
V1 V2 sin δ12
Z2
Z2

(2.20)

Practically, the difference between δ1 and δ2 is not significant, it is reasonable to make the
approximation that sin δ12 ≈ δ12 , cos δ12 ≈ 1. Considering in the transmission system, R

X.

Therefore, it is also acceptable to approximate that R ≈ 0, X ≈ Z. When the system is operated
near operating point, then we have V1 ≈ 1.0 p.u., V2 ≈ 1.0 p.u.. Consequently, equations (2.19)
and (2.20) can be simplified as:
δ − δ2
P12 = 1
X

(2.21)

V − V2
Q12 = 1
X

(2.22)

Originally, expression of transmission line power flow constraint is:

|S12 | =

2 + Q2 ≤ |S max |
P12
12
12

(2.23)

Equation (2.23) indicates that eligible real and reactive power flow without violating transmismax . The circle shown in Fig. 2.3
sion line constraint is bounded by a circle, whose radius is S12
depicts the boundary restricting real and reactive power flow of transmission line. Obviously the
constraint is not linear.
To linearize the constraint, a piecewise approximation of the boundary can be utilized. In
Fig. 2.4 (a), a total number of twelve straight lines are adopted to approach the circular boundary.
However, considering constraints of bus voltage magnitude, six of the constraints in which reactive
power value is larger than real power flow value can be neglected.
17

Q12
S12max

θ
P12

Figure 2.3: Transmission line constraint diagram.

Q12

Q12

30°

30°

P12

(a)

P12

(b)

Figure 2.4: (a) Approximation of circular boundary. (b) Linear constraints for transmission line

18

Then the number of linear constraints for a transmission line is reduced to six as shown in Fig.
2.4 (b). Notice that the two constraints perpendicular to P axis are the old constraints for traditional
DC power flow, mathematically stated as:

max ≤ P ≤ S max
−S12
12
12

(2.24)

The other four constraints perpendicular to diameters with angles 30◦ or −30◦ shown in Fig.
2.4 (b) can be geometrically formulated as:

max ≤ P − 0.58Q ≤ 1.15S max
−1.15S12
12
12
12

(2.25)

max ≤ P + 0.58Q ≤ 1.15S max
−1.15S12
12
12
12

(2.26)

Equations (2.24–2.26) are integrated together to construct linear constraints of transmission
lines for linearized optimal power flow. In this paper, 6 linearized constraints are used. However,
generally, the number of constraints can be other number depending on the individual situations.
If high accuracy is expected, a larger number of constraints can be utilized to approach the circle
constraint. If less computation is preferred, fewer number of constraints can be used. The degree
of the diameters which is perpendicular to the constraints is also a variable that can be changed
based on preference.

2.3

Power Flow Model of UPFC

UPFC steady state equivalent circuit is shown in Figure 2.5 [25]. The shunt converter is installed
on the bus side so that it can provide/absorb reactive power into/from the grid and participate in
the voltage regulation if necessary. The other converter is installed in series with the transmission
line to provide a series voltage Vs , so that voltage Vi and the power flow passing through this
transmission line can be regulated. Iq is the reactive power portion of the current of the shunt
19

converter. It is always perpendicular to voltage vector Vi , expressed in equation (2.27). Ip is the
real power portion of the shunt current. Its function is to absorb or inject real power from/into
grid, so as to keep capacitor voltage Vdc constant, expressed in equation (2.28). It is in parallel
with vector Vi , described in (2.29). There are three independent control variables for UPFC, series
voltage magnitude Vs , series voltage phase ϕs and magnitude of shunt reactive current Iq . They
satisfy 0 ≤ Vs ≤ Vsmax , 0 ≤ ϕs ≤ 2π, 0 ≤ Iq ≤ Iqmax . Vsmax and Iqmax are two variables
that determine UPFC capacity.

Vi
P'ij
Q'ij

Grid

j

i
I'ij
V'i

Vs
−+

Zij

Vj
P'ji
Q'ji

I''ij

Ip Iq

Figure 2.5: Equivalent Circuit of UPFC

Iq ⊥ Vi

Ip =

Re[Vs Iij∗ ]
Vi

Ip // Vi

(2.27)

(2.28)

(2.29)

The series converter and shunt converter of UPFC can be modeled as a voltage source and a
current source respectively [25]. It is easy to see that the optimal generation and power flow values
inside the Grid in Fig. 2.6 will be equivalent to those inside the Grid in Fig. 2.5, as long as the
power flowing out of bus i and the power flowing into bus j are the same in Fig. 2.5 and Fig. 2.6.
20

Therefore, the UPFC model in Fig. 2.6 is equivalent to the power injection model of Fig. 2.5 when
the power injection values are equal to the values shown in Fig. 2.6.

Grid

Vi
Pij−P'ij
Qij−Q'ij

Vj

i
Pij
Qij

j
Pji
Qji

Zij

Pji−P'ji
Qji−Q'ji

Figure 2.6: Equivalent power injection model of UPFC
The vector diagram for UPFC is shown in Figure 2.7.

V'i Vs Vsmax
φs
Vi

Iq Ip

Ip I'ij
Iq
I''ij

Vj

Iqmax
Figure 2.7: Vector diagram of UPFC
Power flow from bus i to bus j can be computed as:

Sij = Pij + jQij = Vi (jVi B/2 + Iq + Ip + Iij )∗
Power flow from bus j to bus i can be computed as:

21

(2.30)

Sji = Pji + jQji = Vj (jVj B/2 − Iij )∗

(2.31)

Therefore, real and reactive power flow through the transmission line after installing UPFC can
be formulated as:

Pij =(Vi2 + Vs2 )Gij + 2Vi Vs Gij cos(ϕs − δij ) − Vj Vs (Gij cos ϕs + Bij sinϕs )

(2.32)

− Vi Vj (Gij cos δij + Bij sin δij )

Qij = − Vi Iq − Vi2 (Bij + B/2) − Vi Vs [Gij sin(ϕs − δij ) + Bij cos(ϕs − δij )]

(2.33)

− Vi Vj (Gij sin δij − Bij cos δij )

Pji = Vj2 Gij − Vj Vs (Gij cos ϕs − Bij sin ϕs ) − Vi Vj (Gij cos δij − Bij sin δij )

Qji = − Vj2 (Gij + B/2) − Vj Vs (Gij sin ϕs − Bij cos ϕs ) + Vi Vj (Gij sin δij

(2.34)

(2.35)

+ Bij cos δij )
The injected power at two buses can be computed by:

Si = Vi (−Ip − Iq − Vs /Zij )∗

(2.36)

Sj = Vj (Vs /Zij )∗

(2.37)

Hence, the equivalent injected active and reactive power are:

22

Pi = −Vs2 Gij − 2Vi Vs Gij cos(ϕs − δij ) + Vj Vs (Gij cos ϕs + Bij sinϕs )

(2.38)

Qi = Vi Iq + Vi Vs [Gij sin(ϕs − δij ) + Bij cos(ϕs − δij )]

(2.39)

Pj = Vj Vs (Gij cos ϕs − Bij sin ϕs )

(2.40)

Qj = −Vj Vs (Gij sin ϕs + Bij cos ϕs )

(2.41)

The effect of UPFC is equivalent to two injected sources at bus i and bus j respectively, shown
in Figure 2.8.

jBij

i

j
Pj+jQj

Pi+jQi
Vi

Vj
Figure 2.8: Equivalent model of UPFC

2.4

Linearized Power Flow Model of UPFC

For the transmission system, |Gij | is a very small number. It is widely acceptable to approximate
Gij ≈ 0. The angle difference between two adjacent buses is also very small. Hence, ϕs − δij ≈
ϕs , sin δij ≈ δij and cos δij ≈ 1. If the system is operated around operating point. Then Vi ≈ 1.0,
Vj ≈ 1.0.
Then the equation (2.38-2.41) can be approximated as:

23

Pi = Vs Bij sinϕs

(2.42)

Qi = Iq + Vs Bij cos ϕs

(2.43)

Pj = −Vs Bij sin ϕs

(2.44)

Qj = −Vs Bij cos ϕs

(2.45)

2 +V2 = V2 ≤ V2
Let Vs sin ϕs = Vsp , Vs cos ϕs = Vsq . Vsp , Vsq satisfy Vsp
smax . Hence,
s
sq
Vsp , Vsq can only lie in the cycle with radius Vsmax , shown in Figure 2.9.

Vsq
0

φs

Vsmax
Vsp

Figure 2.9: Feasible region of Vsp and Vsq
The cycle constraint on Vsp , Vsq can be linearized by six linear constraints, shown in Figure
2.10.
Mathematically, the constraints for Vsp , Vsq can be described as:

−Vsmax ≤ Vsq ≤ Vsmax

24

(2.46)

Vsq
0

Vsmax

60°

Vsp

Figure 2.10: Linearization of Vsp and Vsq

−2Vsmax ≤ Vsq +

−2Vsmax ≤ Vsq −

√

3Vsp ≤ 2Vsmax

√

3Vsp ≤ 2Vsmax

(2.47)

(2.48)

Therefore, the linearized equivalent injected power of UPFC are:

Pi = Bij Vsp

(2.49)

Qi = Iq + Bij Vsq

(2.50)

Pj = −Bij Vsp

(2.51)

Qj = −Bij Vsq

(2.52)

where Iq is restricted by 0 ≤ Iq ≤ Iqmax , Vsp and Vsq are constrained by equations (2.46-2.48)

25

2.5
2.5.1

Model of UPFC Benefit Study
Maximize Revenue for Constant Loading Conditions

The objective function of this model is to find an optimal size of UPFC that is capable to minimize
generation cost for the studied load scenario. Numerous work has been done on the linearization
of generator cost. This comprehensive exam will not cover this topic. Assume that the generator
cost has been linearized and expressed as:

F i = ci P i + d i

(2.53)

where ci and di are constants for generator i; Pi is the real power value generated by generator i.
Therefore, the model of problem regarding minimization of generation cost by installing UPFC
is described as:

26

Objective:

NG
j=1 (cj PGj + dj )t + cupf c Supf c

Min:

(2.54)

Subject to:
Pi =
Qi = −

N
m=1 Vm Gim −

m=i Bim δm − (Bii − bii )δi

N
m=1 Vm Bim {i ∈ φb , i = s, r} (2.56)

m=i Gim δm − (Gii − gii )δi −
N
m=1 Vm Gsm −
N
m=1 Vm Grm −

Ps + Bsr Vsp =
Pr − Bsr Vsp =

Qs + Iq + Bsr Vsq = −
Qr − Bsr Vsq = −

m=s Bsm δm − (Bss − bss )δs

(2.57)

m=r Brm δm − (Brr − brr )δr

(2.58)

m=s Gsm δm − (Gss − gss )δs −
m=r Grm δm − (Grr − grr )δr −

Pjmin ≤ Pj ≤ Pjmax
Qmin
≤ Qj ≤ Qmax
j
j
min ≤ V ≤ V max
Vm
m
m
−π ≤ δm ≤ π

{i ∈ φb , i = s, r} (2.55)

N
m=1 Vm Bsm (2.59)
N
m=1 Vm Brm

(2.60)

{j ∈ φG }

(2.61)

{j ∈ φG }

(2.62)

{m ∈ φb }

(2.63)

{m ∈ φb }

max ≤ P ≤ S max
−Sij
{i, j ∈ φb }
ij
ij
max ≤ P − 0.58Q ≤ S max
−1.15Sij
{i, j ∈ φb }
ij
ij
ij
max ≤ P + 0.58Q ≤ S max
−1.15Sij
{i, j ∈ φb }
ij
ij
ij

(2.64)
(2.65)
(2.66)
(2.67)

−Vsmax ≤ Vsq ≤ Vsmax
√
−2Vsmax ≤ Vsq + 3Vsp ≤ 2Vsmax
√
−2Vsmax ≤ Vsq − 3Vsp ≤ 2Vsmax

(2.69)

−Iqmax ≤ Iq ≤ Iqmax

(2.71)

27

(2.68)

(2.70)

where,
δi−δj
Pij = Z
{i, j ∈ φb }
ij
Vi −Vj
Vi Bij
{i, j ∈ φb }
Qij = − 2 + Z
ij
And, NG is the number of generators; N is the number of buses in the system; φb is the set of
buses in the system; φG is the set of generators in the system; cj and dj are cost coefficients for
generator j; PGj is the real power generation from the generator unit j; t is the time duration of
the constant load scenario; cupf c is the cost of UPFC in the time period t; Supf c is the installation
capacity of UPFC capacity; Vm is the voltage magnitude at bus m; δi is the voltage angle at bus
i; s is the number of the bus where UPFC shunt converter is installed; r is the number of the bus
that is connected to the other end of transmission line, where UPFC series converter is installed;
Pi and Qi are the real and reactive power generation from generator at bus i; Bsr is the bus
k is V cos ϕ ; V
admittance of transmission line connecting bus s and r; Vsp is Vs sin ϕs ; Vsq
s
s smax
is maximum voltage magnitude that series converter can provide; Iq is reactive current provided
by shunt converter; Iqmax is the maximum reactive current that shunt converter can provide.
Supf c can be calculate by equation (2.72). The first part describes the capacity of series conmax is the current capacity
verter and the second part describes the capacity of shunt converter. Isr
of transmission line where UPFC is installed.

max + I
Supf c = Vsmax Isr
qmax

(2.72)

The details of number of variables and constraints involved in this problem are shown in Table
2.1 and Table 2.2 respectively.
Table 2.1: Number of Variables
Variable
No.

δm
N

Vm
N

Pj
NG

Qj
NG

Vsq
1

28

Vsp
1

Vsmax
1

Iq
1

Iqmax
1

Table 2.2: Number of Constraints
Constraint type
Equality
constraint

Inequality
constraint

Constraints
Pi
Qi
PGj
QGj
Vm
k
δm
Pij ,Qij
Vsp ,Vsq
Iq

No. of constraints
N
N
2NG
2NG
2N
2N
6NT
6
2

This optimization model requires to (NG × 2 + N × 2 + 5) variables and (6 × N + 4 × NG +
6 × NT + 8) constraints. NT is the number of transmission lines.

2.5.2

Maximize Revenue for Time-Varying Loading Conditions

The objective function of problems of this category is to maximize the gain obtained by installing
UPFC minus the investment of UPFC. It is equivalent to the problem that minimizes the generation
cost plus the investment of UPFC. The problem can be mathematically formulated as follows:

29

Objective:

Nl
k=1

Min:

NG
k
k
j=1 (cj PGj + dj )t + cupf c Supf c

(2.73)

Subject to:
N
k
k
k
{i ∈ φb , i = s, r} (2.74)
m=1 Vm Gim − m=i Bim δm − (Bii − bii )δi
N
k
k
k
Qk
i = − m=i Gim δm − (Gii − gii )δi − m=1 Vm Bim {i ∈ φb , i = s, r} (2.75)
k
k
k
k = N
(2.76)
Psk + Bsr Vsp
m=1 Vm Gsm − m=s Bsm δm − (Bss − bss )δs
k
k
k
k = N
(2.77)
Prk − Bsr Vsp
m=1 Vm Grm − m=r Brm δm − (Brr − brr )δr
Pik =

k
k
Qk
s + Iq + Bsr Vsq = −
k
Qk
r − Bsr Vsq = −

k
k
m=s Gsm δm − (Gss − gss )δs −
k
k
m=r Grm δm − (Grr − grr )δr −

Pjmin ≤ Pjk ≤ Pjmax
max
Qmin
≤ Qk
j
j ≤ Qj
min ≤ V k ≤ V max
Vm
m
m
k ≤π
−π ≤ δm

N
k
m=1 Vm Bsm (2.78)
N
k
(2.79)
m=1 Vm Brm

{j ∈ φG }

(2.80)

{j ∈ φG }

(2.81)

{m ∈ φb }

(2.82)

{m ∈ φb }

max ≤ P k ≤ S max
−Sij
{i, j ∈ φb }
ij
ij
max ≤ P k − 0.58Qk ≤ S max
−1.15Sij
{i, j ∈ φb }
ij
ij
ij
max ≤ P k + 0.58Qk ≤ S max
−1.15Sij
{i, j ∈ φb }
ij
ij
ij
k
k ≤Vk
≤ Vsq
−Vsmax
smax
k
k + √3V k ≤ 2V k
−2Vsmax
≤ Vsq
sp
smax
√
k
k − 3V k ≤ 2V k
−2Vsmax
≤ Vsq
sp
smax

(2.83)
(2.84)
(2.85)
(2.86)
(2.87)
(2.88)
(2.89)

k
0 ≤ Vsmax
≤ Vsmax

(2.90)

−Iqmax ≤ Iqk ≤ Iqmax

(2.91)

30

where,
δik −δj k
{i, j ∈ φb }
Zij
Vik −Vjk
Vik Bij
k
+ Z
{i, j ∈ φb }
Qij = − 2
ij
max + I
Supf c = Vsmax Isr
qmax
k =
Pij

And, Nl is the number of load scenario under study; NG is the number of generators; N is the
number of buses in the system; φb is the set of buses in the system; φG is the set of generators in
k is the real power generation from
the system; cj and dj are cost coefficients for generator j; PGj
the generator unit j in load scenario k; cupf c is the price rate of UPFC in terms of $/p.u.; Supf c
k is the voltage magnitude at
is the UPFC capacity; tk is the time duration of load scenario k; Vm
bus m in load scenario k; δik is the voltage angle at bus i in load scenario k; s is the number of the
bus where UPFC shunt converter is installed; r is the number of the bus that is connected to the
other end of transmission line, where UPFC series converter is installed; Pik and Qk
i are the real
and reactive power generation from generator at bus i in load scenario k; Bsr is the bus admittance
k is V sin ϕ in scenario k; V k is V cos ϕ in
of transmission line connecting bus s and r; Vsp
s
s
s
s
sq
k
scenario k; Iqk is Iq in load scenario k; Vsmax
is Vsmax in load scenario k. Vsmax is the
maximum magnitude of series voltage that installed UPFC can provide; Iqmax is the maximum
max is the current
current magnitude of reactive power that UPFC shunt converter can provide; Isr
capacity of transmission line where UPFC is installed.
The details of number of variables and constraints involved in this problem are shown in Table
2.3 and Table 2.4 respectively.
Table 2.3: Number of Variables
Variable
No.

k
δm
N × Nl

k
Vm
N × Nl

Pjk

Qk
j

NG × Nl

NG × Nl

31

k
Vsq
Nl

k
Vsp
Nl

Iqk
Nl

Vsmax
1

Iqmax
1

Table 2.4: Number of Constraints
Constraint type
Equality
constraint

Inequality
constraint

Constraints
Pik
Qk
i
k
PGj
Qk
Gj
k
Vm
k
δm
k ,Qk
Pij
ij
k
k
Vsp ,Vsq
k
Vsmax
k
Iqmax

No. of constraints
Nl × N
Nl × N
Nl × 2NG
Nl × 2NG
Nl × 2N
Nl × 2N
Nl × 6NT
6Nl
2Nl
2Nl

The problem of this category is more complex to solve than the one described in Section 2.5.1,
because all the system states in different load scenarios are coupled by Vsmax and Iqmax . It
is required to solve this linear optimization problem with Nl × (2NG + 2N + 3) + 2 states and
Nl × (6N + 4NG + 6NT + 10) constraints in one shot. NT is the number of transmission lines in
the system. The limitation is that the size of memory required to store data is growing quadratically
with the number of load scenarios.

2.6
2.6.1

Simulation Result
Maximize Revenue for Constant Loading Conditions

The simulation of maximizing revenue for constant load scenario is conducted on IEEE 24-bus
reliability test system (IEEE-RTS) [27], which is a small-scale power system with 24 buses, 32
generators and 38 transmission lines. It was found that in the original IEEE-RTS system, the transmission system is strong compared to the generation system, and consequently little transmission
congestion is encountered. Hence it was suggested [29] that the system be modified by multiplying the generation by a factor of 2 and the loads by a factor of 1.8 while keeping the transmission
32

capacities the same. This resulting modified RTS, sometimes referred to as MRTS, is used in this
work. RTS system has 32 generators. Depending on the cost curves and locations of the generators,
they are categorized into 14 groups.
The number of variables and constraints involved in this simulation are listed in Table 2.5 and
2.6. There are totally 81 variables and 436 constraints.
Table 2.5: Number of Variables
Variable
No.

δm
24

Vm
24

Pj
14

Qj
14

Vsq
1

Vsp
1

Iq
1

Vsmax
1

Iqmax
1

Table 2.6: Number of Constraints
Constraint type
Equality
constraint

Inequality
constraint

Constraints
Pi
Qi
PGj
QGj
Vm
δm
Pij ,Qij
Vsp ,Vsq
Iq

No. of constraints
24
24
28
28
48
48
228
6
2

If considering the maximum loading condition of IEEE-RTS, whose total amount of load is
2850 × 1.8 = 5130 MW, the simulation results are summarized in Table 2.7 for 2-month, 1-year,
3-year, 5-year and 10-year utilization of UPFC respectively. It is shown for different locations,
the size of optimal UPFC and the potential revenue after installing UPFC are completely different.
Therefore, it can be very beneficial to conduct benefit study before installing UPFC. The results
also show that if the longevity of UPFC can be increased, the revenue is increased considerably.
The column “Max Savings” shows the savings that if no transmission constraints exist, the amount
of generation cost that can be reduced. If the UPFC is installed in branch 16–19 and the utilization
of the UPFC is five years, total generation saving is $105.6 million dollars, which is about 12.76%
33

of the maximum savings that can be achieved in the case of no transmission line constraints.
Table 2.7: Simulation Result for Constant Loading Conditions
No. of year

Max Savings

2 months

$27.58
Million

1 year

$165.51
Million

3 years

$496.5
Million

5 years

$827.5
Million

10 years

$1655.1
Million

Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)
Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)
Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)
Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)
Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)

15–16
0.0644
0
$0.2578
$0.05
0.067
0
$0.2679
$1.61
0.0674
0
$0.2695
$5.3
0.0674
0
$0.2695
$9.1
0.0674
0
$0.2695
$18.5

16–19
0.2996
0
$1.1982
$2.34
0.3317
0
$1.3269
$20.05
0.3329
0
$1.3317
$62.8
0.3329
0
$1.3317
$105.6
0.3329
0
$1.3317
$212.5

In case 2, the reactive power load is multiplied by a factor of 3.2 instead of 1.8 to stress the
reactive power regulation on the system. The 155 MW generator connected to bus 15 and 155 MW
generator connected to bus 16 are also restricted from participating in the voltage regulation. Simulation results are shown in Table 2.8. From the table, it can be seen that for short-time utilization
of UPFC, it is not economical to install shunt converter. The reason is because in this study, the objective function is to reduce generation cost taking into account of the investment of UPFC; while
shunt converter is mainly responsible for voltage regulation, which has very limited influence on
reducing generation cost. However, when the time of utilization grows, the value produced by each
VA of shunt converter increases, then it becomes economical to install shunt converter.
Fig. 2.11 displays the relationship between the capacity of UPFC and the investment. The

34

Table 2.8: Simulation Result For Constant Loading Conditions with Deficient Reactive Power
Generation
No. of year

Max Savings

2 months

$27.89
Million

1 year

$167.43
Million

3 years

$502.1
Million

5 years

$836.8
Million

10 years

$1673.8
Million

Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)
Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)
Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)
Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)
Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)

35

15–16
0.0678
0
$0.2712
$0.09
0.1983
0
0.7930
2.6
0.1983
0
$0.7930
$9.1
0.1983
0
$0.7930
$15.8
0.1983
0
$0.7930
$32.2

16–19
0.2936
0
$1.1743
$2.36
0.2936
0
$1.1743
$20.1
0.2944
0.222
$2.0656
$62.6
0.2944
0.222
$2.0656
$105.8
0.3732
1.2003
$6.2941
$216.5

investment is increased when the installed capacity of the UPFC increases. Fig. 2.12 displays
the relationship between the capacity of UPFC and the benefit obtained by the UPFC when the
utilization of UPFC is five years and it is installed in branch 16–19. It can be seen that the benefit
is achieved maximum at a specific capacity. Installing more or less capacity of UPFC than the
optimal capacity will decrease the benefit; therefore, it is very meaningful to conduct benefit study
to obtain the optimal capacity of UPFC required to install. Fig. 2.13 presents the production costs
saved by UPFC and the payments for the equipment financed for 10 years at 5% interest. The
figure shows that after 10 years, the production cost can be saved about 200 million dollars when
installing UPFC at branch 16–19.

Figure 2.11: UPFC investment versus capacity

2.6.2

Maximize Revenue for Time-Varying Loading Conditions

The simulation of maximizing revenue is also conducted on IEEE 24-bus reliability test system
(IEEE-RTS) [27]. It has load data of 364 days. This simulation produces load data of the 365th
day the same as the load of the first day. Since they are neighbors in the calendar, their load profile
may share similarity. Therefore, there are totally 8760 load points in term of hours for a year. The
number of coefficients for the inequality constraints can be calculated by the number of inequality
constraints multiplying the number of states. If all 8760 load points are considered, the number
36

Figure 2.12: UPFC benefit versus capacity

RETURN ON INVESTMENT (millions US $)

250

X: 10X: 10
Y: 216.5
Y: 216.5

200

PRODUCTION COSTS SAVED BY
UPFC THROUGH ALLEVIATION OF
TRANSMISSION CONGESTION

150
100

PROFIT

50

PAYMENTS FOR EQUIPMENT
FINANCED FOR 10 YEARS AT
5% INTEREST

0
0

2

4

YEARS

6

X: 10
X: 10
Y: 10.25
Y: 10.25

8

Figure 2.13: Production savings over equipment cost

37

10

of data required to store for the coefficients of inequality constraints is about 2.35 × 1012 . If the
type of data is double and the simulation platform is 32-bit, then the size of memory required
is about 8763 GB, which is tremendously huge. In order to reduce the memory required for the
calculation, this simulation makes approximations by categorizing these 8760 load points into 6
levels and computes the weight for each load level. For example, if load demand is between 0.9
p.u. – 1.0 p.u., it belongs to category 0.9 − 1.0. Table 2.9 shows the result of the load categories.
“Average” column lists the average of load demands belonging to each category. “No. of Points”
column lists the number of points in each category. “Weight” column is calculated by the number
of points for each category dividing total number of points. After the categorization, there are six
load scenarios. The objective function in this simulation can be described as:
Table 2.9: Load Categories
Category
0.9 − 1.0
0.8 − 0.9
0.7 − 0.8
0.6 − 0.7
0.5 − 0.6
0 − 0.5

6

F =

Average
0.9275
0.8388
0.7497
0.6484
0.5467
0.4443

No. of Points
115
972
1456
2029
1888
2300

Weight
0.0131
0.1110
0.1662
0.2316
0.2155
0.2626

14

k +d )×ω ×N
(cj PGj
year × 8760 + cupf c Supf c
j
k
k=1 j=1

(2.92)

where ωk is the weight for each scenario listed in Table 2.9; Ny ear is the number of years that
UPFC can work. Ref. [28] summarizes cost functions for traditional FACTS devices, where cupf c
can be obtained. However, in this simulation cupf c = $0.04/VA, since the transformer-less UPFC
built in Michigan State University achieves this target.
The number of variables and constraints involved in this simulation are listed in Tables 2.10
and 2.11 respectively. There are totally 476 variables and 2628 constraints.
In case 1, simulations are conducted for 1-year, 3-year, 5-year, 10-year and 20-year utilization

38

Table 2.10: Number of Variables
Variable

k
δm

k
Vm

Pjk

No.

144

144

84

Qk
j
84

k
Vsq
6

k
Vsp
6

Iqk
6

Vsmax
1

Table 2.11: Number of Constraints
Constraint type
Equality
constraint

Inequality
constraint

Constraints
Pik
Qk
i
k
PGj
Qk
Gj
k
Vm
k
δm
k ,Qk
Pij
ij
k
k
Vsp ,Vsq
k
Vsmax
k
Iqmax

39

No. of constraints
144
144
168
168
288
288
1368
36
12
12

Iqmax
1

of UPFC respectively. Simulation results are shown in Table 2.12. From the results, it can be
seen that as the time of utilization increases, the optimal capacity of UPFC will increase. It is
quite reasonable, because as the time of utilization increases, the value produced by each VA is
increased, which increases optimal value of the capacity of UPFC. However, there will be an
upper bound on the optimal value, because any capacity value larger than this upper bound will
not provide more controllability for the grid from UPFC. Note that no matter how many years the
UPFC can be utilized, the shunt converter is always zero. The reason is because shunt converter is
mainly responsible for reactive power and voltage regulation. In this case, the reactive power from
the generators is sufficient enough to satisfy reactive load demand and regulate bus voltage. It does
not need any shunt converter to participate in the reactive power regulation.
Table 2.12: Simulation Result of Case 1
No. of year

Gen. cost

1 year

$496.05
Million

3 years

$1488.1
Million

5 years

$2480.2
Million

10 years

$4960.5
Million

20 years

$9921.0
Million

Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)
Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)
Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)
Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)
Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)

17–22
0
0
$0
$0
0.43
0
$1.7194
$0.1
0.798
0
$3.1929
$2.4
0.8075
0
$3.2293
$8.0
0.9095
0
$3.6371
$19.4

16–19
0.2935
0
$1.1732
$0.2000
0.357
0
$1.4279
$3.3
0.3595
0
$1.4389
$6.4
0.3595
0
$1.4389
$14.4
0.364
0
$1.4554
$30.2

In case 2, the reactive power load is multiplied by a factor of 3.5 instead of 1.8 to stress the
reactive power regulation on the system. The 155 MW generator connected to bus 15 and 155
40

MW generator connected to bus 16 are also restricted from participating in the voltage regulation.
Simulation results are shown in Table 2.13.
Table 2.13: Simulation Result of Case 2
No. of year

2.7

Gen. cost

1 year

$497.9
Million

3 years

$1493.7
Million

5 years

$2489.5
Million

10 years

$4979.0
Million

20 years

$9958.0
Million

Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)
Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)
Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)
Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)
Series Capacity (p.u.)
Shunt Capacity (p.u.)
Investment (Million)
Revenue (Million)

15–16
0.273
0
1.093
1.78
0.277
0
$1.1080
$7.5
0.2835
0.4212
$2.8183
$13.8
0.318
0.7092
$4.1081
$31.5
0.3095
0.8699
$4.7168
$67.1

16–19
0.2835
0
$1.1343
$1.37
0.358
0
$1.4310
$6.9
0.358
0
$1.4310
$12.4
0.3595
0.5009
$3.4425
$27.2
0.3635
0.9636
$5.3092
$58.9

Summary

This chapter utilizes linearization techniques and develops a fast and reliable model for UPFC
benefit study. The contributions are listed as follows:
1. Developed a linearized model of UPFC taking both series and shunt converters into account.
2. Developed a complete linearized model and algorithm for UPFC benefit study, including
model of maximizing revenue for constant loading condition and model of maximizing revenue for time-varying loading conditions. This model is benefit for fast, accurate and reli41

able characteristics. Because of this linear character, this algorithm is guaranteed to find the
global optimal solution, which none of AC models and algorithms can guarantee.
3. The linearized algorithm developed in this chapter is a general one, which is not limited for
UPFC benefit study, but can also be extended to other FACTS devices.
Future work for this chapter includes:
1. Develop an algorithm that examines all the load scenarios within acceptable size of memory.
This will cover two possible solutions, 1) combining nonlinear techniques and the algorithm
developed in this chapter to reduce the size of memory. 2) decoupling coefficients matrix of
equality and inequality constraints to divide the memory size into small pieces and examines
these pieces one by one.
2. Simulate nonlinear model and DC model for comparisons in terms of calculation time and
accuracy.
3. Extend the developed algorithm to other FACTS devices.

42

Chapter 3
Optimal UPFC Placement and Parameter
Selection to Minimize Generation Costs
3.1

Introduction

Chapter 2 presented a linearized methodology for UPFC benefit study in a pre-specific location. It
showed that by deploying UPFC in certain locations, the congestion of transmission lines can be
greatly alleviated and the economic profit is quite considerable because of the reduced generation
cost. However, in practice, sometimes the location of the UPFC may not be given. Since the
investment of UPFC requires a large capital cost and not every branch would provide benefit by
installing UPFC, it is quite necessary to conduct research to determine the optimal location of
placing UPFC.
The influences of installing UPFC have area characteristics. In some areas where the power
transmissions are highly congested, the generation cost can be greatly improved by installing
UPFC. However, in some other areas where transmission systems are not congested, installing
UPFC may not provide any benefit. Previously, many researchers have realized the importance of
finding an optimal place to install UPFC so that the benefit can be maximized [30–33]. A simple
way to determine the optimal place of installing UPFC is to install it in every branch of the studied

43

power system model and calculate the benefit after installing UPFC. The branch which offers the
largest benefit after installing UPFC is the optimal place. This method could be very computational
expensive if the studied system is large with thousands or tens of thousands of branches.
If locations are coded as integer numbers, optimal placement and parameters selection problem
of UPFC becomes a mixed-integer optimization problem. Because of the discrete characteristics of
integers, traditional gradient based optimization approach usually fails in solving integer problems.
Besides, differs from other mixed-integer optimization problems, the integers in this problem are
names. They do not have any mathematical meanings. They can not be compared depending on
their values. Therefore, the most popular integer optimization method, branch and bound, fails.
Refs. [30–32] use various intelligent methods, such as genetic algorithm or particle swarms, to
search for the optimal locations. These methods can find the optimal results; however, the efficiencies of these methods are not satisfying when dealing with large systems. This chapter studies
the sensitivity of original power system without installing UPFC and proposes a methodology to
rename/renumber the branches so that the efficiencies of intelligent methods can be greatly improved.

3.2
3.2.1

Methodology
Problem Analysis

The way of numbering branches is quite critical for the efficiency of intelligent methods, such as
genetic algorithm. The reasons can be explained as follows: the solutions of genetic algorithm
are expressed as chromosomes and these chromosomes are composed of genes. The principle of
genetic algorithm is that after the selection, chromosomes with good genes are kept while those
with bad genes are abandoned. The evolution of chromosomes will lead to the best chromosome in
the end, which is the optimal solution. By looking deeply into this process, there are two important
and interesting points which should be paid attention on: 1. The evolutionary of chromosomes is
usually slow and step by step. It always goes from worse chromosomes to better ones and then
44

more better ones. After many generations of evolution, it can finally reach the best one. The
probability of evolution from worse chromosomes directly to the best one is quite low. 2. Better
chromosomes and the best chromosomes always share some common genes. Therefore, when
these genes are found and kept, the chromosomes are evolving to the best one.
In the placement of UPFC problem, the number of branches are names. They do not have
any mathematical meaning. If they are not carefully numbered, it is very high likely that the best
solution is surrounded by bad solutions. In this case, the evolution to the best solution is very low
efficiency. Fig. 3.1 below gives three examples of coding scenarios. Black area is the total solution
space; Magenta area is the collection of good solutions with smaller fitness function values; Red
dot is the optimal solution with the smallest fitness function value. In Fig. 3.1(a), locations are
coded randomly. Good solutions and the optimal solution are spread out randomly in the solution
space. In this case, the good solutions and optimal solution do not share any common genes. The
population will be very difficult to converge to the optimal solution. Hence, it is very inefficiency
for the GA to find the optimal solution. In Fig. 3.1(b), the optimal solution is coded outside the
region of good solutions. It is very likely that the final population will converge to a sub-optimal
solution in the magenta area while missing the optimal solution. Fig. 3.1(c) is a good coding
strategy. Initially, the population is evenly distributed over the solution space; but after a certain
generations, the population will find the good solutions and converge to the magenta good solution
area and finally find the optimal one inside the magenta area. Therefore, it can be deduced that
if good solutions are mapped together and share a few common genes, they are more easily to be
found.

3.2.2

Sensitivity Analysis and Branch Numbering

In a stressed power transmission system, some of the transmission lines are carrying power up to
their limits. These transmission line constraints prevent power transfered from cheap generation
area to expensive generation area through these constrained lines. Therefore, total system generation cost is increased. UPFC has the ability to insert a voltage vector in a transmission line
45

(a)

(b)

(c)
Figure 3.1: (a) Bad solution space; (b) bad solution space; (c) good solution space

46

to regulate the real power transfers, and can also independently control reactive power injections.
Hence, it can redirect the power flows going through other unconstrained transmission lines to the
expensive generation area and decrease the generation cost.
Installing UPFC in the congested areas has more effect to improve generation cost than installing UPFC in non-congested areas. Sensitivity analysis of power system before installing UPFC
can provide a way to find congested areas, which are good candidates to install UPFC and potential
good solutions described in section 3.2.1. The model of power system before installing UPFC can
be described as:

Min:
Subject to:

F (x, u)

(3.1)

gi=1,2,3...m (x, u) ≤ 0

(3.2)

hj=1,2,3...n (x, u) = 0

(3.3)

where F(x,u) is the objective function; gi (x, u) is the inequality constraint; hj (x, u) is the equality
constraint. The Lagrangian function can be expressed:
m
L = F (x, u) +

n
µi gi (x, u) +

i=1

λj hj (x, u)

(3.4)

j=1

where µi and λj are Lagrange multipliers for inequality and equality constraints respectively. µi
is called marginal cost of the inequality constraint. It reflects the change of objective function
by changing the constant of inequality constraint slightly. A large value of µi implies the corresponding inequality constraint is very binding while a zero µi means the corresponding inequality
constraint is unbinding. By examining the marginal cost of transmission constraints, power flow
congestions can be located.
The principle to optimally number branches is as follows: first, identify the congested branches
by running base-case optimal power flow model without UPFC and examine high marginal cost
branches; second, locate the branches adjacent to these congested branches. These selected branches
47

including high marginal cost branches and their adjacent branches are potentially optimal place to
locate UPFC. The procedures of numbering the branches can be described as follows:
(1) Run the base-case optimal power flow without installing UPFC and rank the marginal cost
of branches;
(2) Number the branch which has the largest marginal cost but not numbered yet;
(3) Consecutively number the branches which are not numbered yet and adjacent to the numbered branch in step 2;
(4) If the largest marginal cost of remaining unnumbered branches is less than a threshold value,
number the all the remaining unnumbered branches consecutively based on their locations
and the process is ended; otherwise go to step 2.
The flowchart of the procedures is shown in Fig. 3.2.

3.2.3

Configuration of the Algorithm

Genetic algorithm is being utilized and combined with linear programming to find the optimal
placement and parameters for UPFC. The combined algorithm is designed and shown in Fig. 3.3.
The genetic algorithm performs as the engine of the algorithm. It picks up locations for the linearized model, receives the fitness function values of picked locations, compares them and picks
up new locations until the fitness function values converge.

3.2.3.1

Fitness Function

Normally, fitness function is formulated from objective function of the optimization problem. For
a minimization problem, a popular way to construct fitness function, also known as penalty based
fitness function, is to sum the objective function and the penalty of the violation of constraints.
The performance of penalty based fitness function is highly dependent on the selection of penalty
coefficients. An unsuitable penalty coefficients may result in the delay of convergence or even
48

Mark all the branches as
“un-numbered”
Run the OPF
without UPFC
Rank the marginal
cost of branches
Number the “un-numbered”
branch with largest marginal cost
Number the branches adjacent to the
numbered branch in the last step

Is the largest marginal
cost of “un-numbered”
branch < ɛ

Yes

No
Number the remaining “un-numbered”
branches based on locations
End

Linearized UPFC
Optimization Model

Selector;
Mutation;
Crossover
Genetic Engine

Fitness function values

Locations

Figure 3.2: Flow chart of numbering branches

Figure 3.3: Configuration of the algorithm

49

making the the problem diverge. A penalty-parameter-less fitness function can be used to avoid
the selection of penalty coefficients. It is described as:

f=













fmin for 1 < l < lmax
F + c ∗ (l − lmax )

if

l > lmax

F

if

l=0

(3.5)

where lmax is the number of locations to install UPFC; F is a large value, larger than the biggest
value of the fitness function values when l is between 1 and lmax , so that once variable l is not
selected in a valid position to install UPFC, the fitness function will get a large value and selector
will not choose this point to survive; c is a coefficient penalizing the selection of l outside feasible
positions.

3.2.3.2

Number of Genes

The number of genes to construct the chromosome is determined by:

logl2max −1 < B ≤ logl2max

(3.6)

where lmax is the number of locations to install UPFC.
3.2.3.3

Crossover

Crossover is to produce two new children by reorganizing the information from two parents. A
single-point crossover is shown in Fig. 3.4. A random crossing site is chosen and two parents
exchange the genes on the right side of the selected site to create two new children. If a suitable
crossover place is selected, good genes can be combined from parents to form better children.
On the other hand, if the selection place is not appropriate, the children can be worse than the
parents. Children that inherit good genes from their parents will survive and reproduce in the next
generation while those not inheriting good genes will die out after the selection operation.

50

1100 11011
1000 01001

1100 01001
1000 11011

Parents

Children

Figure 3.4: Crossover strategy
3.2.3.4

Mutation

Mutation operator, which changes 0 to 1 and vice versa with probability pm , is applied to introduce diversity to the existing population. It can realize a local search by producing a point in the
neighborhood of current point.

1100 01001
1100 00001
Figure 3.5: Mutation strategy

3.2.3.5

Selection

The selection operator selects good chromosomes in a population and forms a mating pool to
reproduce children. This dissertation utilizes the strategy that sorts parents and children together
based on the fitness function values and selects the half population with smaller values.

3.3

Case Study

To validate the effectiveness of the proposed genetic-linear combination algorithm for the optimal
placement and parameter selection of UPFC, IEEE 300 test system [69] is studied. It has 411 transmission lines and 69 generators. In the original IEEE 300 bus system, the limits of transmission
lines are too loose that none congestion exists. To stress the transmission system, the constraints
of transmission lines are modified to 0.26/x p.u., where x are the reactance of transmission lines.

51

There are 411 transmission lines; therefore, the number of possible locations to install UPFC
is 822. It can be calculated from Eq. (3.6) that the number of genes needed is 10. The crossover
probability and mutation probability are taken as 0.9 and 0.2 respectively. The coefficients in the
fitness function take values as follows: M = 1012 , c=1011 .
The simulation results of base case are shown in Table 3.1. It can be seen from Table 3.1 that
the benefits brought by installing UPFC are very competitive. For example, the benefit investment
ratio of half a year utilization of UPFC is about 5.71. This table also clearly presents that the series
part of UPFC is more significant in reducing operating cost than the shunt part. If the longevity of
UPFC is increased, the economic benefit of UPFC is also increased considerably.
Table 3.1: Simulation Optimal Placement and Parameter Selection of UPFC
No. of year
0.5 year
1 year
5 year
10 year

Location
121–119
191–225
191–225
191–225

Series(pu)
0.1749
1.6693
1.6749
1.6749

Shunt(pu)
0
0
0
0

Invest(M)
0.70
6.68
6.70
6.70

Benefit(M)
3.99
9.76
70.56
157.91

Table 3.2 shows the benefits of subsequent investments. It displays that the benefits of subsequent investment will decrease, since the congestion problem is relieved by the previous installation of UPFC.
Table 3.2: Simulation Results of Subsequent Investment in the Case of 5-Year Utilization of UPFC
Investment
1
2
3

Location
191–225
192–191
121–119

Series(pu)
1.6749
1.5070
0.1749

Shunt(pu)
0
0
0

Invest(M)
6.70
6.03
0.70

Benefit(M)
70.56
67.35
45.49

Fig. 3.6 compares two strategies of coding branch numbers. The utilization of UPFC is assumed to be five years. The program runs 200 generations each time and 100 times each strategy.
The average of each generation of the 100 runs of each strategy are calculated and compared. The
red curve represents the average fitness value of each generation when the branch numbers are
52

7

8

x 10

7
6
5
4
3
2
not optimal numbering
optimal numbering

1
0
0

50

100

150

200

Figure 3.6: Average value of the population for the genetic algorithm with optimal branch numbering and not optimal branch numbering.
not optimized. The blue curve shows the average fitness function values of each generation when
using optimal numbering strategy. It can be seen that good coding of branch number helps the
program to converge to the optimal solution quicker. When using the optimal numbering strategy
proposed in this section, the genetic algorithm converges in about 100 generations; however, for
the one without optimal numbering strategy, the program converges after 200 generations.

3.4

Conclusion

This chapter develops an efficient approach combining genetic algorithm and the proposed linearized framework to find the optimal location and parameters of UPFC. It presents the benefits of
installing UPFC versus investments by applying the proposed approach to the IEEE 300 bus system. The test results present the considerable benefits brought by installing UPFC. It also shows the
diminishing returns when subsequent investments are made one after another. The contributions
of this chapter is listed as follows:
1. Developed a combined algorithm using genetic algorithm and linearized optimal power flow
framework for the optimal location and parameter selection of UPFC.
53

2. Developed an optimal branch numbering strategy to accelerate the convergence speed of the
proposed UPFC location and parameter selection strategy.
3. The proposed location and parameter selection approach is a generalized approach and it can
also be extended to other FACTS devices.

54

Part III
Distributed Methods for the Control in
Microgrid

55

Chapter 4
Distributed Power Balance Control
4.1

Introduction

Normally, a microgrid can be operated in two modes: grid-connected mode and islanded mode.
In grid-connected mode, all micro-sources are expected to supply power at their rated capacity to
minimize power imported from the grid, and load can consume at its demand, since the power
available in the grid is sufficient. While in islanded mode, micro-sources and load cannot generate
or consume simply following their willing, since the grid is unavailable to consume or provide
extra power for microgrids. It is required that the power provided by DGs must satisfy the total
demand of load plus line loss. Otherwise, some load have to be shed in order to match power
generation and power consumption. DGs and load need to be coordinated to keep the balance
that at any instantaneous time, the power generation and consumption should be equal within the
microgrid.
In large power system, system operators are employed to be responsible to dispatch generation
or curtail load to keep power balance and maintain safe operation of the system. However, because
of small size and large number of DGs, it is not feasible and economic to hire system operators
to coordinate DGs and load to achieve power balance; While on the other hand, DGs with power
electronic interfaces lack inertia and respond fast to the system disturbance which would easily

56

affect system transient and voltage stability; therefore, a fast reacting system operator is expected
in microgrids.
For power balance control, a well-known method proposed previously to maintain power balance is P/f and Q/V droop control [34–38]. This method imitates intrinsic characters of synchronous generators. It obtains control signals locally and does not require communications with
other components in the microgrid. It benefits from “plug and play”. However, it relies on the
output-voltage settings of DGs; therefore, its performance may be compromised if a tight voltage
regulation is expected [42]. Besides, droop control strategy affords no control of load, is therefore
effective only when total generation capacity exceeds total load, which is not always the case in
microgrids. MAS, one of the most popular distributed control entities, could behave as a system
operator to manage power balance operation in microgrid [39–41]. Benefits from MAS include
accomplishing “plug and play” attributes, avoiding single point failures, and realizing distributed
control.
Refs. [43, 44] proposed an effective algorithm based on the Average Consensus Theorem to
calculate system net power and maintain power balance of the microgrid. The algorithm in [43, 44]
is implemented in a completely decentralized manner, but may take long to converge for large
systems. In this chapter, we propose a novel MAS-based methods that accomplish exact power
balance in three sweeps, regardless of system size. In the proposed, the information flows in
parallel and the results are also obtained in a non-iterative way; therefore, this algorithm achieves
superior performance in terms of speed without any convergence issues.
In order to maintain power balance, dispatch generation or demand within microgrids, the entire process contains three parts: determine information flow route, obtain net real and reactive
power, and re-assign generation and load values. Since there is no central agent in the decentralized architecture of multi-agent system, no single agent unilaterally governs others to achieve a
global goal. All agents are at the same hierarchical level to obey specific rules to communicate so
that all those autonomous agents can coordinate to manage microgrids. Consider a system connected as Figure 4.1. It is a simple but typical structure and embodies a ring. It is easy to find

57

that a communication algorithm applied in this structure can be extended to a more complicated
structure which may have multiple rings or radial lines. In this figure, each node represents an
agent, and the position of the node shows topological position of the corresponding agent. An arc
connecting two nodes implies that the agents represented by these two nodes are neighbors and
they can communicate with each other to exchange information and data. Two agents without a
line connecting them are not considered neighbors and can not communicate directly.

1
2
4
3
5
7

6

Figure 4.1: Multi-agent system structure.
To explain the following ideas, some definitions are provided first to clarify the process. The
agent that is processing system information is called current agent. If agent A1 transmits information to agent A2 , then A1 is called A2 ’s parent agent, and A2 is called A1 ’s child agent. View,
either local or global, is agent’s knowledge of system information. This information consists of
maximum real and reactive power generation capacity PG , QG , dispatchable real and reactive
power generation capacity PDG , QDG , vital real and reactive load demand Pv , Qv , and non-vital
real and reactive load demand Pnv , Qnv . Define view mathematically as vector u.
u = [u1 , u2 , u3 , u4 , u5 , u6 , u7 , u8 ]T
= [PG , QG , PDG , QDG , Pv , Qv , Pnv , Qnv ]T

58

(4.1)

4.2
4.2.1

Parallel Method: Minimal Spanning Tree
Stage 1: Discovery of Minimal Spanning Tree

This process is intended to organize decentralized multi-agent system and facilitate to steer information flow. However, three major problems are encountered to discover power information of
the microgrid. Firstly, a sophisticated communication protocol should be defined to conduct information flow so that all the nodes in the system can be spanned. Secondly, this protocol is able
to route the information flow so that every node receives and processes information only once.
Thirdly, real time control of multi-agent system requires this protocol to be able to discover system
information in parallel, so that it can quickly react to the disturbance in the system. To overcome
these problems, a minimal spanning tree can be constructed in MAS to direct the information flow.
The process can be summarized as follows:
(a) A token is generated by a starting agent.
(b) Every agent who receives the token, memorizes its parent agent ID, then it transmits the
token to all its other neighbors, and stores its child agent IDs.
(c) Any agent, who receives multiple tokens simultaneously, keeps one and discards others. At
the same time, removes child-parent relationships with those whose tokens are discarded.
To demonstrate this algorithm, let us also consider the example shown in Fig. 4.1. Since all
agents are identical, starting agent can be selected randomly. In this case, let us simply choose
agent 1 as starting agent. Initially, agent 1 will generate a token and transmit it to agent 2. Then
agent 2 receives the token and sends it to its neighbors, agent 3 and agent 4. After that, agent 3
will send the token to its neighbors, agent 5 and agent 7. In parallel, agent 4 will send the token to
agent 5. Now for agent 5, it receives two tokens, so it discards one. Let us assume it discards the
token coming from agent 3. Lastly, the token will flow from agent 5 to agent 6. Until now, all the
agents in the system have been discovered. During this process, all agents will store their parent

59

1

1

1

2

2

2

4

4

4

3

3

3
5

Removes
7 redundancy

5
6

7

5
7

6

6
(c)

(b)

(a)

Figure 4.2: (a) Token transmission route. (b) Minimal spanning tree constructed. (c) Information
flow path for Stage 2.
agent and child agent IDs. Their relationships are shown in Table 4.2.1. Fig. 4.2 (a) and (b) depict
the transmission path of the token and the minimal spanning tree established by Stage 1.
Note that although removal of any one of the arcs 2-3, 2-4 or 4-5 instead of arc 3-5 has the
same effect of establishing a minimal spanning tree, the sequence of discovery described above
ensures the shortest communication time.
Table 4.1: Parent Child Relationship Diagram
Agent
Parent ID
Child ID

4.2.2

1
–
2

2
1
3,4

3
2
7

4
2
5

5
4
6

6
5
–

7
3
–

Stage 2: Information Feedback Process

Once the information flow path is established in Stage 1, the next step is intended to collect system
generation and load data. This algorithm can be stated as follows:
(a) When no new neighbors are discovered, each agent now responds its parent agent with local
view of the system.

60

(b) Every agent who receives system view information from all its child agents, processes this
data as shown in (4.2) below and transmits updated view information to its parent agent.

u = ulocal +

ui

(4.2)

i∈φ
where u is current agent’s updated view; ulocal is current agent’s local view; φ is the set of child
IDs; ui is child agent i’s updated view.
Stage 2 ensures that each local node view information flows back from child agent to its parent
agent. Finally, the starting agent can obtain system global view and calculates net power based on
(4.3). Fig. 4.2 (c) shows the information flow path of Stage 2.








 Pnet   1 0 1 0 −1 0 −1 0 
u

=
0 1 0 1 0 −1 0 −1
Qnet

(4.3)

where Pnet and Qnet are system net real and reactive power; u is updated view of starting agent.

4.2.3

Stage 3: Generation and Load Dispatch

In Stage 2, starting agent obtains global view of system generation and load demand. Based on
the information collected, decentralized agents can dispatch generation or curtail load from parent
agents to child agents. This process is summarized as:
(a) When receiving system net power values, each agent first adjusts local available generation
or load to minimize the unbalanced power. Mathematically, it is described by (4.4)–(4.6).
Minimize:
c | = |I p − I
|Inet
DG + Inv |, I ∈ {P, Q}
net

61

(4.4)

Subject to:

0 ≤ IDG ≤ IDG

(4.5)

0 ≤ Inv ≤ Inv

(4.6)

c is system net P or Q after dispatch of current agent; I p is system net P or Q
where Inet
net
obtained from parent agent; IDG and Inv are dispatched power values for current agent.
Note that only dispatchable generators and non-vital load can be dispatched.
(b) If local availability is not enough, each agent, based on the updated system view obtained in
Stage 2, will split the net power values according to its child agents’ dispatchable capability,
and transmit new values to its child agents. Mathematically, it is stated as:









i =
Inet








i
IDG
c
c
m Inet if Inet ≥ 0
IDG
m∈φ
, I ∈ {P, Q}
i
Inv
c
c
m Inet if Inet < 0
Inv
m∈φ

(4.7)

i is net P or Q for child agent i; I i is u or u , updated dispatchable generation
where Inet
3
4
DG
c is the net P or
information that agent i obtained in Stage 2. φ is the set of child agents. Inet
i is u or u , updated non-vital load information that child
Q obtained by current agent; Inv
7
8
agent i obtained in Stage 2.
The information flow path in this step is in the opposite direction to that of Stage 2.

4.3

Multi-Agent System Implementation and Simulation

This subsection will describe the implementation of the MAS framework described above and
compare its performance with other available methods proposed previously in the literature. In
practice, the framework would comprise individual agents programmed to perform as described in
62

Sections 4.2.1, 4.2.2 and 4.2.3, and located at their respective positions in the actual network. Their
interactions, as described in Sections 4.2.1, 4.2.2 and 4.2.3, would be conducted through parallel
threads. However, in the laboratory, the entire system was simulated in a MATLAB environment,
on a single computer with a single thread. This was implemented through “virtual agents” in the
MATLAB code, using code representing agent functions at each node. In order to simulate multiple
threads of multi-agent system in a single-thread program, this platform requires an extra variable
for the time pointer. At a specific time t, the program holds t, and uses one thread to compute
multiple threads sequentially. After computing all the threads, the program goes to t + ∆t, where
∆t is time step. Fig. 4.3 displays the time pointer in this process. The pointer at time t will not
move to the next time t + ∆t until the processor computes all the necessary behaviors conducted
by every agent at time t.

t

t+∆t

Agent 3

Agent 2

…

Agent 1

Agent 3

Agent 2

Agent 1

Time pointer

…

t+2∆t

Figure 4.3: Simulation of multiple threads on a single-thread platform.
Power balance control is critical for stable operation of microgrids; therefore, it has the highest
priority in the proposed multi-level control architecture.
To demonstrate the speed and effectiveness of the proposed power balance control strategy for
MAS, IEEE 14-bus, 30-bus, 57-bus and 118-bus systems are tested. The system configurations
and data can be obtained from [69].
If two nodes are adjacent to each other through direct electrical connection, their agents are
neighbors and they can communicate directly with each other. The simulations are conducted by
selecting agent 1 (corresponding to bus 1) as starting agent. Simulation results are shown in Table
4.3. A hop represents the path between two neighbor nodes. Note that this lists the maximum
number of hops required for the power balance control; because for the last sweep, once the net
63

power reaches zero, power balance is achieved and net power information does not need to be
transmitted further. However, in this simulation, power balance is obtained when the dispatch
information has reached all the agents.
Table 4.2: Simulation Results for Power Balance Control
Period
Hops

IEEE 14 bus
12

IEEE 30 bus
18

IEEE 57 bus
30

IEEE 118 bus
42

The time to achieve power balance control can be estimated from the number of hops shown in
Table 4.3. The actual execution time of the algorithm depends on the specific implementation, in
terms of hardware and software. Ref. [43] uses equation (4.8) to estimate the communication time.
n × nd × nb
T = i
R

(4.8)

where ni is the number of hops for the algorithm; nd is the number of data points required to be
transmitted; nb is the number of bits used to represent each data point; R is the communication
speed in bits/second. For instance, for IEEE 118 bus system, the total number of hops is 42, with
14 hops per sweep. If the token in the first sweep is represented by 8 bits, and each data point in
the second and third sweeps is represented by 16 bits, then for a network speed of 10Mbit/s, the
execution time can be estimated as (14×1×8+14×8×16+14×2×16)/10 000 000 = 0.000235
s.
Ref. [43] used the IEEE 162-bus system [69]. We compare here the performance of the proposed algorithm with that in [43] for the same system. The agent at bus 1 is chosen as starting
agent. It takes 9 hops for each sweep, requiring 27 hops in all to reach power balance. Assuming
the same number of bits and communication rate as those in [43], the estimated time can be computed as (9 × 1 × 8 + 9 × 8 × 16 + 9 × 2 × 16)/10 000 000 = 0.0001512 s. However, [43] reports
that it took their algorithm 0.1283 s to converge. Note that the times calculated here and in [43]
do not consider time delays or other factors. They vary for different implementations of hardware
and software. However, this approximation clearly shows the speed advantage for the proposed
64

algorithm, which results from the parallel nature of information flow in this algorithm.
Further discussion: the speed of the proposed algorithm depends highly on the coupling of the
agents. For example, for the two 7-node MAS structures shown in Fig. 4.4, if the starting agent
is chosen as agent 1, the first requires only 1 hop for each sweep, while the second requires 6
hops for each sweep. Therefore, for the same scale MAS, the tighter coupling (higher number of
branches) the MAS has, the faster the solution can be reached. Most MAS systems will comprise a
combination of the two, and the execution time will depend on how tightly coupled the agents are.

3

2

4
1

2

3

4

5

6

1
7

7

5
6

(a)

(b)

Figure 4.4: Two MAS structures: (a) completely parallel resulting in tightest coupling; (b) completely sequential resulting in loosest coupling.

4.4

Summary

This chapter proposed a distributed power balance control algorithm based on a decentralized
multi-agent system. The contributions of this chapter are listed as follows:
1. The information flows in parallel by this algorithm; therefore, it achieves superior speed
performance and is applicable for on-line power balance control;
2. The proposed algorithm implements power balance control in a non-iterative way; therefore,
it does not have any convergence problem.

65

Chapter 5
Distributed Power Balance Control
Considering Voltage Regulation and Losses
5.1

Introduction

Chapter 4 proposes a fast and reliable distributed power balance control strategy. It can achieve superior performance in terms of speed without any convergence problems, due to the characteristics
that its information flows in parallel and the results are obtained in a non-iterative way. However, it
does not consider losses in the transmission lines; therefore, it may introduce considerable power
balance error in the resistance-dominant network, such as distribution system or microgrids. Furthermore, this algorithm does not consider the effect of voltage regulation by generators, which
is commonly used in most power systems. Ref. [43, 44] shares the same shortcomings as chapter
4. To overcome the mentioned drawbacks, this chapter proposes a distributed multi-agent based
power flow algorithm, in which each agent has its local power flow equation and updates state information simultaneously with limited data from their immediate neighbors. Ref. [75] proposed a
distributed power flow algorithm for multi-agent platform; however, its algorithm updates state information sequentially from one agent to another, which limits its speed, especially in multi-agent
framework, where communication speed rather than computation speed is the bottleneck that re-

66

stricts the speed of algorithm. The distributed power flow algorithm proposed in this chapter fully
makes use of communication time, and updates state information synchronously among agents,
which offers considerable speed advantage. Based on the proposed power flow algorithm, this
chapter also shows that real and reactive unbalanced power can be calculated at slack node. This
net power acquisition is implemented fully distributed; therefore, it provides a possible solution
for distributed load shedding or restoration.

5.2

Net Power Acquisition by Gauss Method

Assume in each bus of the microgrid, there is a Node Agent. It has information about local generation, such as real power capacity, reactive power capacity, voltage magnitude that is going to
be maintained; meanwhile, it also has local load information, such as real and reactive power demands by the load. If two nodes are connected electrically, then the corresponding agents are
marked as neighbors. Agents with neighborhood can communicate with each other; those without
neighborhood relationship cannot communicate.
Assume N nodes in the system, with net power input Pn , Qn . If it is a load node, then Pn and
Qn are negative; if it is a generator node, then Pn and Qn are positive.

5.2.1

Distributed Construction of Ybus

Conventionally, to perform power flow, we need to first build the bus admittance matrix, Ybus ,
which contains system structure and admittance information. In multi-agent based power flow, we
still need system information for power flow calculation. However, this approach differs from traditional centrally computational method, in this distributed based framework, Ybus is constructed
locally. Each agent only obtain part of Ybus information. The construction principle is shown
below:
For agent i, it constructs ith row of Ybus matrix:

67

Yi = [Yi1 Yi2 ...YiN ]

(5.1)

N
Yii =

yin

(5.2)

n=1

Yin = −yin

when (n = i)

(5.3)

where, Yi is the system parameters constructed by agent i for power flow calculation; Yii is the
ith element of Yi ; Yin is the nth element of Yi , when n = i. yii represents equivalent admittance
between bus i and ground. yin when n = i, is equivalent admittance between bus i and bus n.
Recall that in Ybus matrix, if bus i and bus n are not connected, Yin = −yin = 0. Since
neighborhood is determined by electrical connection, it also means if agent i and agent n are not
neighbors, then Yin = 0, expressed in (5.4).

Yin = 0 if (n ∈
/ N (i))

(5.4)

where, N(i) is a set of agents that are neighbors of agent i.
Construction principles based on equation (5.1)-(5.4) shows that all the information required
to build system parameters by agent i are yin , where n = 1, 2, ..., N. Agent i can obtain these
parameters locally without global information.

5.2.2

Agent Based Power Flow

Let us select node 1 as slack node. When calculating the power flow, its voltage magnitude and
angle are maintained constant. Other nodes use their original real and reactive power inputs (PQ
node) or real power input and voltage magnitude requirement (PV node) to calculate voltage magnitude (PQ node), voltage angle (PQ or PV node) or reactive power demand (PV node).

68

Local power flow equation for agent m is (5.5)


N

∗ =V 
Sm = Vm Im
m
n=1

∗
Ymn Vn 

(5.5)

where Sm is the complex power input to node m; Vm is the voltage vector at node m; Im is
the current input into node m. Ymn is built following principles of section 5.2.1. Rewrite this
equation:




N
 S∗

 m

Ymn Vn 
Vm =
 ∗ −
Ymm  Vm

n=1
n=m
1

(5.6)

where, m = 2, 3, ..., N
From (5.4), Ymn = 0, if agent n is not a neighbor of agent m. Then we obtain:




Vm =

∗
 Sm
 ∗ −
Ymm Vm
1


Ymn Vn 

(5.7)

n∈N (m)

where, N(m) is a set of agents that are neighbors of agent m.
To solve power flow, Gauss method is adopted. Differs from traditional Gauss-based power
flow, in this multiagent framework, each agent solves its own nodal power flow locally by exchanging the required data with its neighbors. Writing the local power flow equation using the
iterative form of Gauss method, we get:



∗(k)
1  Sm
(k+1)
(k) 
Vm
=
−
Ymn Vn 

∗(k)
Ymm
Vm
n∈N (m)

(5.8)

where, k is the iteration number.
The local power flow equation (5.8) shows that each agent only needs to exchange with its
neighbors the voltage values from the last iteration in order to update voltage data in its local
calculation. It does not need system global information or information beyond its neighbors to
69

calculate its local power flow.
Now, if node m is a PV node, then it first has to estimate Qm :




(k) ∗
(k)
(k) ∗ 
Ymn Vn
+ Vm
Ymm Vm


(k+1)
 (k)
Qm
= Im Vm

(5.9)

n∈N (m)
(k+1)
where Qm
is the reactive power needed to maintain pre-specific voltage magnitude in the
(k + 1)th iteration. Then it updates the voltage:



(k+1)
1  Pm − jQm
(k) 
(k+1)
−
Ymn Vn 
Vm
=

∗(k)
Ymm
Vm
n∈N (m)




(k+1)
(k+1) 

Vm
= Vm



Vm

(k+1) 
Vm

(5.10)

(5.11)

(k+1)
(k+1)
(k+1)
(k+1)
If Qm
≥ Qmax
= Qmax
≤ Qmin
= Qmin
m , then Qm
m . If Qm
m , then Qm
m .
This means the generator does not have enough reactive power capacity to further maintain voltage
magnitude. So the node changes to a PQ node, and voltage magnitude will vary. Voltage update is
now updated by (5.12).

(k+1)
1  Pm − jQm
(k) 
(k+1)
Vm
=
Ymn Vn 
−

∗(k)
Ymm
Vm
n∈N (m)


5.2.3

(5.12)

Net Power Acquisition

After agent-based Gauss power flow calculation method, each agent could obtain local voltage
magnitude and angle, and they satisfy the following equation:


N

P m + jQm = Pm + jQm = Vm 
n=1

70

∗
Ymn Vn 

(5.13)

where P m and Qm are calculated real and reactive power; Pm and Qm are original planned real
and reactive power input.
Also we could calculate real and reactive power input at node 1:


N

P 1 + jQ1 = V1 
n=1

∗
Y1n Vn 

(5.14)

where P 1 and Q1 are calculated real and reactive power at node 1.
Losses of lines are categorized by relationship with nodes. If a line connects both node m and
node n, then we define half of that line loss is contributed to node m, the other half of that line loss
is contributed to node n. If a line connects node m and ground, then we say that all of the line loss
is contributed to node m. Then local losses related to node m can be calculated by equation (5.15).

loss + jQloss = y
∗ 1
Pm
mm Vm Vm +
m
2

N
n=1
n=m

∗ +V V∗ −V V∗ −V V∗]
ymn [Vm Vm
n n
m n
n m (5.15)

loss and Qloss stand for real and reactive power losses related to node m respectively.
where Pm
m
So the total losses in the system can be expressed as:

71

P loss + jQloss =

N

loss + jQloss
Pm
m

m=1
N
=
m=1
N
=
m=1
N
=
m=1
N
=
m=1
N
=
m=1

∗ +1
ymm Vm Vm
2

N

∗ +V V∗ −V V∗ −V V∗]
ymn [Vm Vm
n n
m n
n m

n=1
n=m
N
∗
∗ − V V ∗]
ymm Vm Vm +
ymn [Vm Vm
m n
n=1
n=m
N
N
∗
Vm Vm
ymn +
ymn [−Vm Vn∗ ]
n=1
n=1
n=m
N
∗
Ymm Vm Vm +
Ymn Vm Vn∗
n=1
n=m
N
Ymn Vm Vn∗
n=1

(5.16)

where P loss and Qloss are total system real and reactive power losses.
Therefore, total system generation deficiency or surplus can be calculated:
N

(Pm + jQm ) − (P loss + jQloss )
m=1
N
= P1 − P 1 + jQ1 − jQ1 +
(P m + Qm ) − (P loss + jQloss )
m=1

∗
N
N
= P1 − P 1 + jQ1 − jQ1 +
Vm 
Ymn Vn 
m=1
n=1
N
N
Ymn Vm Vn∗
−
m=1 n=1

∆P + j∆Q =

= P1 − P 1 + jQ1 − jQ1
72

(5.17)

where ∆P and ∆Q are generation deficiency or surplus for the system, compared with load demands taking line losses into consideration. Note that if ∆Q ≥ 0, the reactive power is automatically balanced, because voltage regulator at slack bus will control the reactive power output of the
generator to balance reactive power. However, if ∆Q ≤ 0, load dispatch is required to maintain
reactive power balance.
From equation (5.17), it shows that the power deficiency considering line losses in the system
can be calculated at slack node. Agent at node 1 knows generator capacity or load demand P1 and
Q1 , also it can calculate P 1 and Q1 from voltage and angle data in power flow calculation. Then
it can obtain power shortage in the system.

5.3

Multi-Agent Implementation Algorithm

Normally, multi-agent system is formed by two or more agents. Each agent in multi-agent system
can perceive environments, communicate with their immediate neighbors and perform locally to
realize their functions, and therefore system overall function can be realized. Based on the mathematical foundations stated in section 5.2.2 and 5.2.3, microgrid generation deficiency or surplus
can be computed in multi-agent environment by the following algorithms:
Generation Surplus or Deficiency Algorithm
//Initialization:
forall Agent Ai do
//set up neighborhood:

set up N (Ai )
//Input system data:

gather system admittance data yin
//obtain local Y row:

Compute Yi by (5.1), (5.2) and (5.3)
//Input node information:

Ai ← Vi , for Ai is responsible for a slack bus
73

Ai ← Pi , Qi , for Ai is responsible for a PQ bus
Ai ← Pi , Vi , Qmax
, for Ai is responsible for a PV bus
i
repeat
//Update voltage for PQ agents:

forall Ai ∈ PQ Agents do
ask for V from neighbors
compute Vi by (5.8)
//Update voltage for PV agents:

forall Ai ∈ PV Agents do
ask for V from neighbors
if Qi ≤ Qmax
, then compute Vi by (5.9), (5.10) and (5.12)
i
else compute Vi by by (5.11)
until ∆V < V
//Compute real and reactive power difference at slack agent

for slack agent A1 do
compute calculated real and reactive power by (5.14)
compute real and reactive power deficiency or surplus by (5.17).

5.4

Load and Generation Dispatch

In the dispatch process, initially, all the agents are marked as “unprocessed”. Once the slack bus
agent obtains net power ∆P and ∆Q of the system, MAS system can dispatch generation or load
following a sequential three-step algorithm:
1. Process. Modify local control variables to minimize ∆P and ∆Q and mark current agent as
“processed”;
2. Next. Transmit net power values to the neighbor who is marked “unprocessed”.

74

3. Return. If the current agent has no neighbors or all the neighbors are marked as “processed”,
the return net power values to its parent agent.
The proposed three-step algorithm is capable to route the information flow to span all the agents
in MAS in a sequential way and coordinate them for power balance control. Figure 5.1 displays
one of the information flow paths of the load and generation dispatch for the system structure
shown in Figure 4.1. It can be computed that the maximum number of hops required for load and
generation dispatch process is 2 × (N − 1), where N is the number of agents in the system.

1
1
12

2

2
3
6

4

11
10
7
5

4

7

3
9

5
8

6

Figure 5.1: Information flow path for multi-agent system
Discussions: Since load and generation dispatch process does not consider losses, it is recommended that after a certain number of agents finishing their dispatch process, MAS can select a
current agent with generators as new slack agent and implement distributed power flow again and
obtains updated net power; then it continues load and generation dispatch process among those
agents which are still marked “unprocessed”.

5.5

Simulation Result

To demonstrate the proposed multi-agent based load shedding algorithms, a five bus system from
[74] is tested.

75

5.5.1

Five-Bus System

Original system configuration is shown in Fig. 5.2. Bus input data and Line input data are modified
and displayed in Table 5.5.1 and 5.5.1. All the data have be transformed into per unit.
1
G1

5

3

4
SG2

SG1

G2
A5

A1

A3

A4
SL3

SL5
2
A2
SL2

Figure 5.2: Tested five-bus system

Table 5.1: Bus Input Data for Five-Bus System
Bus
1
2
3
4
5

Type
slack
PQ
PV
PQ
PQ

V
1.0
1.05
-

δ
0
0
-

(PG , QG )
(2, 1)
(0, 0)
(5.4, -)
(0, 0)
(0, 0)

(PL , QL )
(0, 0)
(5, 2.8)
(3, 0.4)
(3, 1)
(1.8, 1.0)

QG
+
4.0
-

QG
−
−2.8
-

Table 5.2: Line Input Data for Five-Bus System
Bus to Bus
1−5
2−4
2−5
3−4
4−5

5.5.1.1

R’
0.0015
0.009
0.0045
0.00075
0.00225

X’
0.02
0.1
0.05
0.01
0.025

G’
0
0
0
0
0

B’
0
1.72
0.88
0
0

Multi-agent Based Construction of Ybus

Based on the construction principles expressed as (5.1)-(5.4), 1st row of Ybus built by Agent A1
is:

76

Y12 = Y13 = Y14 = 0
1
Y15 = −
= −3.73 + j49.72
0.0015 + j0.02
Y11 = 3.73 − j49.72
All the data needed to compute local Y1 can be obtained locally by Agent A1 . Other agents
can establish their own local Yi vectors in the same manner. Results of Ybus matrix is shown in
Table 5.5.1.1.
Table 5.3: Local Yi Vectors
A1
A2
A3
A4
A5

5.5.1.2

Yi
[3.73 − j49.72 0 0 0 −3.73 + j49.72]
[0 2.68 − j28.46 0 −0.89 + j9.92 −1.79 + j19.84]
[0 0 7.46 − j99.44 −7.46 + j99.74 0]
[0 −0.89 + j9.92 −7.46 + j99.44 11.92 − j147.96 −3.57 + j39.68]
[−3.73 + j49.72 −1.79 + j19.84 0 −3.57 + j39.68 9.09 − j108.58]

Multi-Agent Based Power Flow Calculation

(0)
(0)
(0)
(0)
Assume flat start, V2 = 1.0, V3 = 1.05, V4 = 1.0, V5 = 1.0. The first iteration of Gauss
method for multiagent system is as follows:

Agent 2:



∗(0)
1  S2
(1)
(0) 
V2 =
−
Y2n Vn  = 0.947∠ − 10.30◦

∗(0)
Y22
V2
n∈N (2)

77

Agent 3:




(0) ∗
(0)
(0) ∗ 
Y3n Vn
+ V3
Y33 V3
 = 4.91

(1)
 (0)
Q3 = Im V3
n∈N (3)

(1)
− Q3L = 4.0 − 0.4 = 3.6
Q3 > Qmax
3
Therefore:
(1)
Q3 = 3.6



(1)
1  P3 − jQ3
(1)
(0) 
V3 =
−
Y3n Vn  = 1.039∠1.11◦

∗(0)
Y33
V3
n∈N (3)
Agent 4:

∗(0)
1  S4
(0) 
(1)
Y4n Vn  = 1.033∠ − 1.13◦
−
V4 =

∗(0)
Y44
V4
n∈N (4)


Agent 5:



∗(0)
1  S5
(1)
(0) 
V5 =
Y5n Vn  = 0.996∠ − 0.93◦
−

∗(0)
Y55
V5
n∈N (5)
If we define convergence criterion as a voltage tolerance of 0.001 pu shown in (5.18), it needs
39 iterations to converge. The first five iteration results are given at Table 5.5.1.2.
(k)
(k+1)
Vi − Vi
≤ 0.001 ∀i

5.5.1.3

Netpower Acquisition

After system node voltages are obtained, total netpower can be computed at slack agent:

78

(5.18)

Table 5.4: Results for the First Five Iteration
No.
1
2
3
4
5

Agent 2: V2
0.947∠ − 10.30◦
0.920∠ − 10.81◦
0.916∠ − 12.79◦
0.905∠ − 13.18◦
0.904∠ − 13.97◦

Agent 3: V3
1.039∠1.11◦
1.05∠0.13◦
1.05∠ − 0.06◦
1.05∠ − 1.28◦
1.05∠ − 1.56◦

Agent 4: V4
1.033∠ − 1.13◦
1.019∠ − 1.22◦
1.026∠ − 2.48◦
1.022∠ − 2.74◦
1.022∠ − 3.81◦

Agent 5: V5
0.996∠ − 0.93◦
0.996∠ − 3.14◦
0.986∠ − 3.21◦
0.987∠ − 4.02◦
0.983∠ − 4.16◦

Agent 1:


N

P 1 + jQ1 = V1 
n=1

∗
Y1n Vn  = 7.34 + j1.36

Ps + jQs = P1 − P 1 + Q1 − Q1 = −5.34 − j0.36
Therefore, considering generators and transmission network characteristics (PV, PQ, power
flow), line losses, the deficiency of generation in the system is -5.34-j0.36. If simply neglecting
these factors, netpower is calculated as:

Ps + jQs =

(PG − PL ) + j(

QG − QL ) = −5.4 − j0.2

Because of the voltage magnitude requirements at node 3, generators at node 3 will not simply
provide maximum reactive power, even though reactive power in the system is deficient. Therefore,
neglecting system operation principles and generation characters, the net reactive power estimation
are not accurate. In this case, its error is above 44%. Real power difference between this two
methods mainly resulted from systen losses. In microgirds, with low voltage level and mainly
resistant network, this loss sometimes could not be omitted.

5.5.2

14-Bus System, 30-Bus System and 57-Bus System

To test feasibility of the proposed algorithm, 3 different systems with 14 buses, 30 buses and 57
buses obtained from university of washington [69] are studied individually. Time expense for
79

multi-agent system’s convergence to the result are calculated.

5.5.2.1

14-Bus System

It costs multi-agent system 175 iterations to converge to the result if the voltage acceptable error
is set to be 10−7 . If the communication rate is still 10 Mbit/s and each data is represented by 16
bits and each iteration is required to transmit two data, voltage magnitude and angle, then the time
used for convergence is 175 × 16 × 2/10 000 000 = 0.56 × 10−3 s. Total generation capacity is
272.4 + j138 MVar, total complex load is 259 + j73.5 MVar, total generation is 272.4 + j103.74.
Net power obtained at by slack node agent is 0.008 − j0.114 MVar. The number of hops required
for load and generation dispatch is 26; therefore, the dispatch takes 26 × 2 × 16/10 000 000 =
0.832 × 10−4 s.

5.5.2.2

30-Bus System

In this case, multi-agent system needs 411 iterations to converge to the result if the voltage acceptable error is set to be 10−7 . If the communication rate is still 10 Mbit/s and each data is represented by 16 bits, the time used for convergence is 411 × 16 × 2/10 000 000 = 1.3152 × 10−3
s. Total generation capacity is 300.2 + j178 MVar, total complex load is 238.4 + j126.2 MVar,
total generation 300.2 + j135. Net power obtained at by slack node agent is 0.76 − j0.61 MVar.
The number of hops required for load and generation dispatch is 58; therefore, the dispatch takes
58 × 2 × 16/10 000 000 = 1.856 × 10−4 s.

5.5.2.3

57-Bus System

In this case, multi-agent system needs 499 iterations to converge to the result if the voltage acceptable error is set to be 10−7 . If the communication rate is still 10 Mbit/s and each data is represented
by 16 bits, the time used for convergence is 499 × 16 × 2/10 000 000 = 1.5968 × 10−3 s. Total
generation capacity is 928.9 + j499 MVar, total complex load is 1250 + j336.4 MVar, total generation is 1278.97 + j198.44. Net power obtained at by slack node agent is −350.07 − j147.18 MVar.
80

The number of hops required for load and generation dispatch is 112; therefore, the dispatch takes
112 × 2 × 16/10 000 000 = 3.584 × 10−4 s.
The comparison with the algorithm proposed in chapter 4 is shown in Table 5.5.2.3.
Table 5.5: Comparisons Between Power Balance Control Algorithms
Algorithms
No loss

With loss

5.6

Name
Net power
Dispatch
Total Time
Net power
Dispatch
Total Time

Unit
Hops
Hops
Second
Iteration
Hops
Second

14 bus
8
4
0.672 × 10−4
175
26
6.432 × 10−4

30 bus
12
6
1.008 × 10−4
411
58
1.5008 × 10−3

57 bus
20
10
1.68 × 10−4
499
112
1.9552 × 10−3

Summary

This chapter proposes a multi-agent based distributed power flow algorithm, in which each agent
is only required to get access to the local data without any knowledge of global information. By
communicating local control data with its immediate neighbors, each agent can obtain its state
information. The contributions of this chapter includes:
1. It offers an innovative way to obtain state information of power system in a complete distributed way, in which each node in the system is only required to get access to the local
information without any awareness of global information.
2. In this algorithm, real and reactive net power can be calculated at slack bus, which can be
used as a theoretical foundation for distributed load shedding or restoration process.
3. This algorithm is also applicable for real-time power balance control of microgrids.

81

Chapter 6
Distributed Economic Dispatch
6.1

Introduction

The algorithms in Chapter 4 ensure stable steady state operation, and are adequate for most microgrids. However, if a microgrid is amenable to optimal dispatch, i.e., if its generating capacity
exceeds its load and has some DGs that are dispatchable, then the framework proposed in this
chapter can also perform economic dispatch. This can be achieved by applying the following algorithms, which modify the dispatched power values obtained in Chapter 4 to realize minimum
generation cost.
In traditional bulk power system, economic dispatch is usually conducted in a central controller,
which has access to global system information. However, characterized as flexible and “plug and
play”, microgrids are preferable to be controlled in a distributed manner. Ref. [64, 65] presented,
respectively, a two-level incremental cost consensus algorithm, and an evolutionary programming
method for economic dispatch based on MAS environment. These are both dedicated algorithms
for solving economic dispatch problems, but are complex and slow to converge. This chapter
presents a robust and fast algorithm for on-line economic dispatch of distributed generators in microgrids based on a decentralized architecture of multi-agent system. In this architecture, all DGs
are equipped with identical agents, which could only perceive local information or obtain neces-

82

sary information from their neighbors. By competition and corporation with each other, agents
intend to maximize their own profit, therefore an optimal global solution could be obtained. In this
algorithm, total generation cost always changes in a non-increasing direction. This decentralized
algorithm also allows the system to continue operation in the presence of single point failure. If
any agent in the system fails, only the cost in that node is not optimized; while the cost for the
remaining system can still be minimized.
This chapter will first deduce necessary conditions for minimal cost operation of microgrids.
Then it proposes an innovative multi-agent communication algorithm to implement the minimal
principle in multi-agent environment. A proof of convergence is also presented. In the end, a
five-agent, IEEE-118 systems are investigated to validate the proposed multi-agent system.

6.2

Economic Dispatch Model

Assume there are n generators in a microgrid. Each generator has a convex cost function Fi (Pi ) in
terms of real power output Pi . If system losses are not considered, the economic dispatch problem
can be stated as follows: given total system load, schedule real power output of each generator so
that the total generation cost is minimized while satisfying upper and lower output constraints of
generators, as well as real power balance requirement. Mathematically, this optimization problem
can be stated as follows:
Minimize:

n
F =

Fi (Pi )

(6.1)

i=1
Subject to:
n
PD =

Pi

i=1
min
Pi
≤ Pi ≤ Pimax
where
83

(6.2)
(6.3)

Fi

= cost function of generation unit i

Pi

= real power output of generation unit i

PD

= total load in microgrid

Ploss = total loss in microgrid
Pimin = lower limit of generation unit i
Pimax = upper limit of generation unit i
The Lagrange function can be constructed as (6.4):
n

n
Fi + λ(PD −

L=

n
Pi ) +

λi (Pimin

i=i

i=1
i=1
n
λi (Pimax − t2i − Pi )
+s2i − Pi ) +
i=1

(6.4)

where
λ, λi , λi = Lagrange multipliers
si , ti

= real variables that convert inequality constraints to
equality (i.e., slack and surplus variables)

Based on the necessary condition, at the minimum cost point, the partial derivatives of the
Lagrange function should be zero. Therefore:

∂Fi
∂L
=
− λ − λi − λi = 0
∂Pi
∂Pi
∂L
= 2λi si = 0
∂si
∂L
= 2λi ti = 0
∂ti

84

(6.5)
(6.6)
(6.7)

When Pimin < Pi < Pimax , then si = 0, ti = 0. Based on (6.5)–(6.7), it is easy to obtain
∂F
λi = 0, λi = 0, i = λ
∂Pi

(6.8)

Alternatively, when Pi = Pimin or Pi = Pimax , according to (6.5)–(6.7):
∂Fi
− λ − λi = 0 when Pi = Pimin
∂Pi

(6.9)

∂F
ti = 0, λi = 0, i − λ − λi = 0 when Pi = Pimax
∂Pi

(6.10)

si = 0, λi = 0,

Equations (6.8)–(6.10) describe necessary conditions to obtain minimum generation cost. This
just implies that at the optimal operating point, generators in the system are dispatched at equal
incremental cost, except those units that have already reached upper or lower limits.
Now the power balance requirement in the microgrid can be satisfied by multi-agent control as
described in Chapter 4. Each agent obtains an assignment that satisfies power balance described
by equation (6.11), but it is not the optimal value that minimizes generation cost.
n

(0)
Pi = PD

(6.11)

i=1
(0)
where Pi = initial generation assignment from power balance control for generation agent i.
Realize that this initial value is within generation limits of generation agent i.
Two neighbor agents m and n can meet optimal conditions (6.8)–(6.10) by implementing the
behavior described by (6.12) (6.13):
(k+1)
(k+1)
Fm (Pm
) = Fn (Pn
)

(6.12)

(k+1)
(k+1)
(k)
(k)
Pm
+ Pn
= Pm + P n

(6.13)

where Fm = dFm /dPm . Agent behavior as in (6.12) and (6.13) provides a means of achieving
85

minimum generation cost.
However, if agent m reaches its limit before it satisfies (6.12), then (6.12) does not need to be
satisfied. Agent m chooses its limit as new assignment, and agent n calculates its assignment by
(6.14). A similar logic applies to agent n.
(k+1)
(k)
(k)
(k+1)
Pn
= Pm + P n − P m

(6.14)

(k+1)
limit , the limit can be the upper bound or lower bound of agent m, depending
where Pm
= Pm
on which one prevents agent m from having the same incremental cost as agent n. This behavior
prevents generation output from violating generation limits, thus equation (6.17) is always valid.

6.3

Proof of Convergence

To establish the stability of this algorithm, a convergence analysis is performed, based on a Lyapunov function. Note that a convex and differentiable cost function Fi : χ → R satisfies the firstorder convexity condition:

Fi (y) ≥ Fi (x) + Fi (x)(y − x), ∀x, y ∈ χ,

(6.15)

Define a Lyapunov function V: χN ⊂ RN → R of the form shown in (6.16):
(k)
Fi (Pi )

V (P(k) ) =

(6.16)

i∈ν
(k)
where P(k) is a vector consisting of nodal generation values in iteration k; Pi is the ith element
of P(k) representing generation value for agent i; ν is the set of agents in the system.
Note that Fi is the cost function for generator unit i, so it is non-negative if Pi is within
generation limits. Therefore, we have:

V (P(k) ) ≥ 0
86

(6.17)

Assume in iteration k + 1, agent m and agent n accomplish behaviors described by (6.12),
(6.13). Then the change of the lyapunov function would be:
V (P(k+1) ) − V (P(k) )
(k+1)
(k+1)
(k)
(k)
= Fm (Pm
) + Fn (Pn
) − Fm (Pm ) − Fn (Pn )
(k+1)
(k+1)
≤ Fm (Pm
) + Fn (Pn
)
(k+1)
(k+1)
(k)
(k+1)
− Fm (Pm
) + Fm (Pm
) P m − Pm

(6.18)

(k+1)
(k+1)
(k)
(k+1)
− Fn (Pn
) + Fn (Pn
) Pn − P n
(k)
(k+1)
(k+1)
(k)
(k+1)
(k+1)
= −Fm (Pm
) P m − Pm
− Fn (Pn
) Pn − Pn
=0
If in iteration k + 1, agent m reaches its limit before it satisfies (6.12), then agent m will choose
its limit and agent n will implement behavior (6.14). To proof the convergence of this behavior,
let us study the case that the marginal cost of agent m is smaller than that of agent n at iteration k,
described by equation (6.19). A similar proof can apply in the case that the marginal cost of agent
m is larger than that of agent n at iteration k.
(k)
(k)
Fm (Pm ) < Fn (Pn )

(6.19)

If Fm and Fn are convex cost functions, generator m will increase its generation to increase
its marginal cost; while generator n will decrease its generation to decrease its marginal cost. The
k and P k+1 , P k and P k+1 are shown in equations (6.20) and (6.21)
relationships between Pm
m
n
n
respectively. Figure 6.1 shows the relationships between them.
(k)
(k+1)
Pm < Pm

87

(6.20)

(k)
(k+1)
Pn > Pn

Fm

(6.21)

Fn
Pnmin

Pmmax

Pm(k)

Pm(k+1)

Pn(k+1)

Pm

Pn(k)

Pn

Figure 6.1: Cost curve for agent m and agent n
Since agent m reaches its maximum limit before reaching identical marginal cost as agent n,
its marginal cost at iteration k + 1 will be still smaller than the marginal cost of agent n, expressed
in equation (6.22).
(k+1)
(k+1)
Fm (Pm
) < Fn (Pn
)
Hence, the change of lyapunov function is

88

(6.22)

V (P(k+1) ) − V (P(k) )
(k)
(k+1)
(k+1)
(k)
= Fm (Pm
) + Fn (Pn
) − Fm (Pm ) − Fn (Pn )
(k+1)
(k+1)
≤ Fm (Pm
) + Fn (Pn
)
(k+1)
(k+1)
(k)
(k+1)
− Fm (Pm
) + Fm (Pm
) Pm − P m
(k+1)
(k+1)
(k)
(k+1)
− Fn (Pn
) + Fn (Pn
) Pn − Pn

(6.23)

(k+1)
(k)
(k+1)
(k+1)
(k)
(k+1)
= −Fm (Pm
) Pm − Pm
− Fn (Pn
) P n − Pn
(k+1)
(k)
(k+1)
(k+1)
(k)
(k+1)
= Fm (Pm
) Pn − Pn
− Fn (Pn
) Pn − Pn
=

(k+1)
(k+1)
Fm (Pm
) − Fn (Pn
)

(k)
(k+1)
Pn − Pn

<0
Therefore, the convergence of the designed algorithm is guaranteed.
It is easy to verify that the final result P(P1 ,P2 ,...PN ) satisfies Karush-Kuhn-Tucker Conditions
shown in equations (6.24-6.29).
2N
∇F (P ) −

uj ∇gj (P ) − v∇h(P ) = 0

(6.24)

j=1
where,

∇gj (P ) = Pjmax − Pj ≥ 0,

min ≥ 0,
∇gj (P ) = Pj−N − Pj−N

j = 1, 2, ..., N

j = N + 1, N + 2, ..., 2N

(6.25)

(6.26)

N
Pi − PD

h(P ) =
i=1
89

(6.27)

uj gj (P ) = 0,

uj ≤ 0,

j = 1, 2, ..., 2N

j = 1, 2, ..., 2N

(6.28)

(6.29)

Therefore, the final result is a KKT point. According to KKT Sufficiency Theorem [67], since
F (P ) is convex, every gj (P ) is concave, and h(P ) is linear, the KKT point is the global optimal
point.
The procedure for decentralized agents to achieve optimal economic dispatch is as follows:
(a) Agent m selects its neighbor agent n.
(b) Both agents implement behaviors (6.12) and (6.13), if one agent encounters a limit, then
it uses this limit as its new assignment, meanwhile, the other agent implements behavior
(6.14).
(c) If the assignments of all agents converge within the accepted tolerance, optimization process
is terminated. Otherwise, steps (a) through (c) are repeated.
Figure 6.2 displays the flow chart for economic dispatch in multi-agent platform.
Generation agents can implement behaviors described above simultaneously; hence, the convergence time for this algorithm is more competitive than single-thread optimization. However,
to implement this algorithm, generation agents may be required to discover extra neighbors, apart
from topological neighbors. Because this algorithm is based on communications between generation agents, topologically, it is possible that one generation agent may not have any generation
agent as a neighbor; then its cost can not be optimized. Also note that not all microgrids permit
economic dispatch; the algorithm presented in this section enhances the general framework for the
control of microgrids, and remains available should a microgrid evolve to where it does permit
economic dispatch.

90

Agent m
Select a neighbor Agent n
Implement behaviors:

dFm ( Pm( k ) ) dFn ( Pn( k ) )
=
dPm
dPn

Pm( k ) + Pn( k ) = Pm( k −1) + Pn( k −1)
No Agent m violates
limit?
Yes

Pm( k ) = Pmlim it
Pn( k ) = Pm( k −1) + Pn( k −1) − Pmlim it
No

Agent n violates
limit?
Yes

Pn( k ) = Pnlim it
Pm( k ) = Pm( k −1) + Pn( k −1) − Pnlim it

Convergence?

No

Yes
End
Figure 6.2: Flow chart for economic dispatch
91

6.4

Simulation Results

Economic dispatch is an additional service provided by MAS where the capability exists. It can
help to optimize total generation cost, but it cannot change total generation amounts determined
by power balance control.
To validate the proposed agent-based economic dispatch algorithm, a small prototype microgrid with five DGs are studied. Assume the cost function satisfy polynomial equation expressed
in (8.7). Other forms of cost functions also can be optimized by the proposed algorithm, as long
as they are convex. The microgrid cost data are shown in Table 6.4. Generation 4 is a renewable
source, so it is operated in maximum power point. If not equipped with storage device, its output
power is preferable not to be regulated. The neighbor relationships for this small microgrid is
shown in Fig. 6.3. A line connecting two agents indicate neighbor relationship between them.

F = a0 + a1 P + a2 P 2

(6.30)

Table 6.1: Generation Data for the Microgrid
Gen. No.
1
2
3
4
5

Pmax (kW)
400
1000
800
600
900

Pmin (kW)
200
300
100
600
150

a0
8
12
9
10

a1
0.096
0.072
0.064
0.084

a2
0.00012
0.00008
0.0001
0.00014

Initially, let us assume total load and line losses are 2600 kW. Initial values for Generation
1 to Generation 5 are 300 kW, 500 kW, 400 kW, 600 kW and 800 kW respectively. Fig. 6.4
shows the dispatched results for each generation. Only 4 iterations are needed to achieve optimal
solution. It is seen in Fig. 6.5 that marginal costs are converged to an identical value in the end to
achieve minimal operation cost except the one of the undispatchable source, Generation 4. Fig. 6.6
displays total cost in microgrid for the same amount of generation is reduced from $339 to $304.
The proposed economic dispatch algorithm is also applied to IEEE 118-bus test system. This
92

2

1

3
4
5
Figure 6.3: Neighborhood relationship diagram

Dispatched Generation amount
800

700

Generation (kW)

600

500

400

300

200

0

2

4

6

8

10
Iteration

12

14

16

18

Figure 6.4: Generation dispatch result for five-agent system

93

20

Marginal cost ($/kW)
0.35

0.3

Marginal cost ($/kW)

0.25

0.2

0.15

0.1

0.05

0

0

2

4

6

8

10
Iteration

12

14

16

18

20

18

20

Figure 6.5: Marginal cost for five-agent system
Total generation cost ($)
340

335

330

Total cost ($)

325

320

315

310

305

300

0

2

4

6

8

10
Iteration

12

14

16

Figure 6.6: Generation cost for five-agent system
94

system has 54 generators; therefore, 54 agents are involved in this control. Total load demand is
4.377 × 103 MW. Assume that each agent has six direct neighbors. (The number of neighbors
in this case is not initially known, as explained in the last paragraph of Section 6.3.) Generation
cost and limit data are obtained from [70]. Initially, assume that the 54 generators share the load
in proportion to their capacities. If the convergence tolerance is set to 1 × 10−4 pu, it takes
46 iterations to converge. Simulation results are shown in Fig. 6.7. If each agent stores cost
coefficients and generation upper and lower limits of its neighboring generators, then the only
data required to be communicated is the generation value. Therefore, an estimated time can be
computed as 46 × 1 × 16/10 000 000 = 0.0000736 s. This time does not include the computational
time; however, considering the computation required in this algorithm is not complex and the
computation time is usually much smaller than communication time, the total computation time
will not be significant.

200
150
100
50
0

10

20

30

40

50

Total generation cost ($/h)

Marginal cost ($/MWh)

5

250

Iteration
(a)

1.5

x10

1.45
1.4
1.35
1.3
0

10

20

30

40

50

Iteration
(b)

Figure 6.7: Simulation results for economic dispatch
Further discussion: Fig. 6.7(b) shows that the cost decreases rapidly in the first few iterations,
while in subsequent iterations the improvements diminish. In this example, the saving is about
$1.5 × 104 /h in the first 12 iterations; however, in the last 34 iterations, the saving is only about
$5×102 /h. Therefore, if the tolerance is set appropriately larger, the time required for the algorithm
can be reduced tremendously, while the saving is not compromised too much. In this case, if the
tolerance is set to 1 × 10−3 , it takes 12 iterations to converge. So the time is reduced by 73.91%

95

while the saving in cost is only reduced by about 3.34%.

6.5

Summary

This chapter proposes a potential solution for distributed economic dispatch realization in a decentralized multi-agent platform. A decent communication algorithm based on the consensus theorem
is proposed. In this algorithm, all agents are completely identical and autonomous. They do not
have access to system global information; however, by making use of their local information and
limited communications with their immediate neighbors, these agents are able to compete and
corporate with each other to achieve a global minimum operation cost. The contributions of this
chapter are listed below:
1. It proposes a potential distributed solution for economic dispatch by a decentralized multiagent system.
2. Proof of convergence and proof of global optimization are presented in this chapter.
3. Because of the synchronous characters, this algorithm is capable to satisfy speed requirement
of on-line economic dispatch.
4. This algorithm also allows the system to continue operation in the presence of single point
failure. If any agent in the system fails, only the cost in that node is not optimized; while the
cost for the remaining system can still be minimized.

96

Chapter 7
Distributed Economic Dispatch Considering
Network Constraints
7.1

Introduction

In chapter 6, it proposed a distributed synchronous algorithm to optimize generation cost without
considering network constraints. It is realized in a decentralized multi-agent platform, in which
each agent can only get access to local information. By collaborating and competing with its
neighbors, each agent is trying to minimize its own generation cost while satisfying total demand
requirement; therefore, total generation cost is minimized. It is shown that this algorithm is guaranteed to converge to the global optimal point. The speed of the algorithm is demonstrated to
be capable for on-line economic dispatch. However, this algorithm does not consider network
constraints, such as voltage magnitude requirements of buses or capacity limits of transmission
lines. In a stressful power system, these constraints can have considerable influences on generation dispatch, which can not be overlooked. In this chapter, it proposes an innovative distributed
economic dispatch algorithm considering network constraints for a decentralized multi-agent system, in which each agent is responsible for local control variables and can observe partial or full
system states. The proposed algorithm can also optimize total generation cost in a non-increasing

97

direction. Simulation results show that if each agent can observe full system states, the multi-agent
system can achieve global optimum; if each agent can only observe local system states, multi-agent
system can still optimize generation cost, but it may be trapped by a local optimum.

7.2

Problem Formulation

Conventional economic dispatch is implemented in a central controller, which has global information of the system. The problem can be formulated as:

Min:
Subject to:

F (x, u)

(7.1)

g(x, u) ≤ 0

(7.2)

h(x, u) = 0

(7.3)

However, in the decentralized multi-agent platform, there is no central unit that has global
information of the system. Each agent is only limited to access its local information and necessary
information from its immediate neighbors. Consider a system of N agents, where each agent i,
∈ A = [1, 2, ..., N ] is responsible for an area Ai of the system. In each area, the agent has a set
of control variables, ui ∈ Rni , which can be real and reactive power output of generators. It also
has information of local states, xi = [xi1 , xi2 , ..., xim ], which can be bus voltage magnitudes,
bus angles. Based on its available state information and state information from its neighbors, each
agent can compute real and reactive power flow through transmission lines in its area. Two agents
are allowed to have the observability over the same bus or the same transmission line, but each
control variable is only belonged to one local agent. To clarify the decentralized algorithm, some
notations are defined first. N(i) is the set of agents that are neighbors of agent i; xB
i is a vector of
the states for the boundary buses that agent i can observe; fB
i is the real and reactive power flowing
through the boundary of agent i;

98

In order to optimize generation cost, each agent i and its immediate neighbors are required to
implement the following optimization for their control variables:

Fm ({um })

Min:

(7.4)

m∈{i,N(i)}
Subject to:

B
gAm ({xAm }, {uAm }|{xB
Am }, fAm ) ≤ 0
B
hAm ({xAm }, {uAm }|{xB
Am }, fAm ) = 0

Am = Ai + AN(i) (7.5)
Am = Ai + AN(i) (7.6)

where, m is the set of agent i and its immediate neighbors; Fm is the cost function for the control
variables in the area of agent m; Am is the area that combines the areas that agent i and its
immediate neighbors are responsible for; xAm and uAm are the states and control variables inside
area Am ; gAm and hAm are lists of inequality and equality constraints on xAm and uAm , given
B
that the states xB
Am and real and reactive power flows fAm of the boundary of Am are held
constant.
Equations (7.4)–(7.6) can be understood intuitively in this way, each time agent i and its immediate neighbors form an area, and optimize the cost inside this area while maintaining the states and
power flows through transmission lines on the boundary constant; therefore, it does not interfere
the states and control variables in other areas. This algorithm is also applicable to be implemented
in parallel, as long as the parallel formed areas do not have overlap with each other.

7.3

Proof of Convergence

To analyze the stability of the proposed algorithm, this Section performed a convergence analysis
based on a Lyapunov function.
Define a Lyapunov function V: χN ⊂ RN → R of the form shown in (7.7):

99

(k)
Fi (ui )

V (uk ) =

(7.7)

i∈ν
(k)
where u(k) is a vector consisting of system control values in iteration k; ui is control variable
values in iteration k for agent i; ν is the set of agents in the system.
Since Fi is the cost function of generator units, Fi ≥ 0. Then, we have
V (uk ) ≥ 0

(7.8)

If in iteration k, agent i and its immediate neighbors N(i) implement the behaviors in described
by equations (7.4)–(7.6). Then we have:
(k+1)
Fi (ui
)−

V (u(k+1) ) − V (u(k) ) =
i∈ν

(k)
Fi (ui )

(7.9)

i∈ν

Since equations (7.4)–(7.6) maintain the states and power flows on the boundary of Am ; therefore, it does not interfere the states and control variables in the area other than Am , hence we
have:
(k+1)
Fi (ui
)=
i∈ν,i∈A
/ m

(k)
Fi (ui )

(7.10)

i∈ν,i∈A
/ m

Therefore,
(k+1)
Fi (ui
)−

V (u(k+1) ) − V (u(k) ) =
i∈Am

(k)
Fi (ui )

(7.11)

i∈Am

Since the behaviors (7.4)–(7.6) optimize the generation cost inside area Am , we have
(k+1)
Fi (ui
)≤
i∈Am

(k)
Fi (ui )
i∈Am

100

(7.12)

Therefore,
V (u(k+1) ) − V (u(k) ) ≤ 0

(7.13)

Therefore, the convergence of the proposed algorithm is guaranteed.

7.4
7.4.1

Simulation Results
Linearized Economic Dispatch Considering Network Constraints

To testify the proposed decentralized economic dispatch algorithm, IEEE 14-bus test system is
studied [69]. The simulation conducted in this subsection utilizes linearized optimal power flow
model described in Chapter 2. In this system, it has 5 generators located at bus 1, bus 2, bus 3, bus
6 and bus 8 respectively.
The simulation will test two cases: in the first one, each agent can observe all the system
states; while in the second one, only local states can be observed by the agent. IEEE 14 bus
system has very large capacity limit on transmission lines, therefore, the constraints of transmission
line do not limit the power flow. Only bus limit is active. To examine the effect of constraints
of transmission lines, each simulation case will test two scenarios. The first one is the original
system with transmission lines of large capacities. In the second scenario, the capacity limit of
transmission line connecting bus 1 and bus 2, and the capacity limit of transmission line connecting
bus 1 and bus 5 are changed to 19.99 p.u. and 29.99 p.u. respectively.

4
1

5

2
3

Figure 7.1: Neighborhood relationship for IEEE 14 bus system
101

Case 1: each agent is responsible to control one generator, but can observe all the states of
the system. The neighborhood relationship is shown in Figure 7.1. A straight line connecting two
nodes indicate these two nodes are neighbors. In scenario 1, the generation cost curve is shown in
Figure 7.2. The final optimal generation result is 8.5559 × 103 $/hr. In scenario 2, the generation
cost curve is shown in Figure 7.3. The final optimal generation result is 9.3737×103 $/hr, which is
larger than the result obtained in scenario 1. Because scenario 2 is more constrained than scenario
1, its optimal value is larger.

Generation cost ($)

9200
9100
9000
8900
8800
8700
8600
8500

0

2

4

6

8

10

12

14

16

18

20

Iteration
Figure 7.2: Generation cost curve for the first scenario of case 1

Table 7.1: Observed Area for Each Bus
Agent No.
1
2
3
4
5

Bus No. in the observed area
1, 5, 6
1, 2, 3, 4
3, 4, 5, 6, 7, 8
5, 6, 10, 11, 12, 13, 14
7, 8, 9

Case 2: in this case, each agent is also responsible to control one generator, but it is limited
to observe local states. The neighborhood relationship is also shown in Figure 7.1. The local area
that each agent can observe is shown in Table 7.1. In scenario 1, the optimal generation cost is
102

Generation cost ($)

9520
9500
9480
9460
9440
9420
9400
9380
9360

0

2

4

6

8

10

12

14

16

18

20

Iteration
Figure 7.3: Generation cost curve for the second scenario of case 1

Generation cost ($)

9020
9000
8980
8960
8940
8920
8900

0

50

100

150

200

Iteration
Figure 7.4: Generation cost curve for the first scenario of case 2

103

250

Generation cost ($)

9600
9580
9560
9540
9520
9500
9480
9460

0

5

10

15

20

25

30

35

40

45

50

Iteration
Figure 7.5: Generation cost curve for the second scenario of case 2
8.9187 × 103 $/hr. The generation cost curve is shown in Figure 7.4. In scenario 2, the optimal
generation cost is 9.4614 × 103 $/hr. The generation cost curve is shown in Figure 7.5. It can been
seen that multi-agent system can reduce generation cost in a non-increasing way. However, because
each agent lacks observability over some of states, the optimal solution is easily got trapped by the
local optimum. The optimal value obtained by multi-agent system in this case highly depends on
the initial point and how large area each agent can observe.
If we implement the algorithm with no network constraints described in Chapter 6, the optimal
generation cost is 7.6426 × 103 $/hr; The centralized linearized economic dispatch obtains the
optimal generation cost of 8.5559 × 103 $/hr for scenario 1 and 9.3717 × 103 $/hr for scenario 2.
The reason that the algorithm in Chapter 6 achieves less generation cost than the centralized linearized economic dispatch is that it does not consider network constraints. The fewer constraints it
considers, the less value it obtains. But the value is also more inaccurate. The results are summarized and analyzed in Table 7.2. It can be seen that multi-agent system with full observability of
system states can achieve global optimum. Multi-agent system with partial observability can optimize generation cost, however, because each agent does not get access to full states, the optimal
generation cost is sacrificed and highly depends on the initial point and the observability of each

104

agent.
Table 7.2: Result Summary

Architecture
Centralized
Distributed

7.4.2

Scenarios
Type
Network Constraint
No Network Constraint
Full Observability
Partial Observability

Scenario 1
Gen cost($/hr)
Error
3
8.5559 × 10
0
3
7.6426 × 10
10.67%
3
8.5559 × 10
0
3
8.9187 × 10
4.24%

Scenario 2
Gen cost ($/hr)
Error
3
9.3717 × 10
0
3
7.6426 × 10
18.45%
3
9.3737 × 10
0.02%
3
9.4614 × 10
0.96%

Nonlinear Economic Dispatch Considering Network Constraints

The proposed decentralized algorithm of economic dispatch can also be implemented in the nonlinear optimization models of power system. IEEE 9-bus system is tested to verify the proposed
algorithm. It has three generators located at bus 1, bus 2 and bus 3 respectively. In this simulation,
we assigned three agents, each of which is responsible for one generator. Since there are only three
agents in the system, each time only two agents are communicating and optimize their local area
together.
The simulation will also test two cases: in the first one, each agent can observe full system
states; in the second one, only partial system states can be observed by the agent. Each case will
also have two scenarios: the first one will use the original system; the second will change the
capacity of transmission line connecting bus 1 and bus 4 to 1.5 p.u.
Case 1: each agent is responsible to control one generator, but can observe full states of the
system. In scenario 1, the generation cost curve is shown in Figure 7.6. The optimal generation cost
obtained by these distributed fully observable agents is 5.2967 × 103 . In scenario 2, the generation
cost curve is shown in Figure 7.7. The optimal generation cost obtained is 5.4995 × 103 . Because
the network constraints are more stressful in the second scenario than they are in the first scenario,
the optimal value obtained in the scenario 2 is larger than it is obtained in scenario 1.
Case 2: each agent is responsible to control one generator, but can only observe partial of the
105

Generation cost ($)

5440
5420
5400
5380
5360
5340
5320
5300
5280

0

5

10

15

Iteration

20

25

30

Figure 7.6: Generation cost curve for the first scenario of case 1 for AC models

Generation cost ($)

7500

7000

6500

6000

5500

5000

0

2

4

6

8

10

12

14

16

18

Iteration
Figure 7.7: Generation cost curve for the second scenario of case 1 for AC models

106

states in the system. The local area that each agent can observe is shown in Table 7.3. In scenario
1, the optimal generation cost is 5.3423 × 103 . The generation cost curve is shown in Figure 7.8.
In scenario 2, the optimal generation cost is 5.6187 × 103 . The generation cost curve is shown in
Figure 7.9. Multi-agent system with partial observability can minimize generation cost; however,
it is not guaranteed to achieve global optimal. The optimal value obtained in the end depends on
the initial solution and the size of the area each agent can observe.

Generation cost ($)

5440

5420

5400

5380

5360

5340

0

5

10

15

20

25

30

35

40

45

50

Iteration
Figure 7.8: Generation cost curve for the first scenario of case 2 for AC models

Table 7.3: Observed Area for Each Bus for IEEE 9-Bus System
Agent No.
1
2
3

Bus No. in the observed area
1, 4, 5, 7, 8
2, 4, 6, 7, 9
3, 5, 6, 8, 9

If we implement the algorithm with no network constraints described in Chapter 6, the optimal
generation cost is 5.216 × 103 $/hr; The centralized linearized economic dispatch obtains the
optimal generation cost of 5.2967 × 103 $/hr for scenario 1 and 5.4995 × 103 $/hr for scenario 2.
The results are summarized and analyzed in Table 7.4. It can be seen that multi-agent system with
full observability of system states can achieve global optimum. Multi-agent system with partial
107

Generation cost ($)

5850

5800

5750

5700

5650

5600

0

5

10

15

20

Iteration

25

30

35

40

Figure 7.9: Generation cost curve for the second scenario of case 2 for AC models
observability can optimize generation cost, however, because each agent does not get access to
full states, the optimal generation cost is sacrificed and highly depends on the initial point and the
observability of each agent.
Table 7.4: Result Summary

Architecture
Centralized
Distributed

7.5

Scenarios
Type
Network Constraint
No Network Constraint
Full Observability
Partial Observability

Scenario 1
Gen cost($/hr) Error
5.2967 × 103
0
3
5.216 × 10
1.53%
3
5.2967 × 10
0
3
5.3423 × 10
0.86%

Scenario 2
Gen cost ($/hr) Error
5.4995 × 103
0
3
5.216 × 10
5.16%
3
5.4995 × 10
0%
3
5.6187 × 10
2.17%

Summary

This chapter proposes a distributed algorithm for economic dispatch considering network constraints based on a decentralized multi-agent platform. Convergence of the proposed algorithm is
presented to ensure its stability. By implementing the proposed algorithm, total generation cost
is always changing in a non-increasing direction. The simulation shows that if each agent in the
108

multi-agent system can observe system states globally, the proposed algorithm can help multiagent system achieve global optimum as a centralized controller; if each agent can only observe
local system states, this algorithm can help multi-agent system minimize generation cost, but it
may get trapped by local optimum. In the second case, it is highly suggested that multi-agent system can make use of available information from the system and start with an initial point as better
as possible. The contributions of this chapter are listed below:
1. It proposes a general distributed solution for economic dispatch by a decentralized multiagent system taking network constraints into account. The proposed algorithm can be used
for DC model, linearized AC model or full AC model.
2. Proof of convergence is presented in this chapter.
3. The proposed algorithm is applicable to be implemented in a parallel manner, which potentially provides speed advantage.
4. This algorithm also allows the system to continue operation in the presence of single point
failure. If any agent in the system fails, only the cost for the control variables of that agent
is not optimized; while the cost for the remaining system can still be minimized.

109

Chapter 8
Multi-Agent Implementation and
Multi-Level Control Architecture
8.1

Multi-Level Control Structure

This dissertation proposes a hierarchical control structure displayed in Fig. 8.1 for the control
of microgrids. The upper decentralized multi-agent layer, acting as a quick-responding system
operator, is responsible for power balance and economic dispatch control. The lower local control
layer, in response to the power commands required by MAS, regulates real and reactive power
output of local DGs.
For the MAS power balance control, the fast three-sweep algorithm proposed in chapter 4 can
be used to react to power unbalance in real time, so as to maintain stable operation of microgrids.
The functions of the three sweeps are to determine information flow route, obtain net real and
reactive power, and dispatch generation and load. All the information in these three sweeps is
designed to flow in parallel, so that the required communication can be achieved rapidly in the
proposed MAS architecture. MAS power balance control is critical for stable operation; therefore,
it has the highest priority in MAS computation and control.
Economic dispatch is conducted on-line among agents that are responsible for generation nodes

110

(also called generation agents). Since real power balance is maintained in MAS power balance
control layer, economic dispatch will optimize total generation cost while maintaining total real
power generation.
As shown in Fig. 8.1, each local controller will receive real and reactive power settings from
the MAS layer and implement local control for DGs. However, because only load information is
transmitted and losses are not calculated, the MAS power balance control achieves an approximate
allocation of generation. The losses, which equal the difference in actual generation and load, can

Upper level

be compensated by local controller with reserved generation capacity.
Power balance
control

Generator
Agent

P1, Q1

P2, Q2

Generator
Agent

Load Agent

P4, Q4
P3, Q3

Agent layer

Economic dispatch
P1

Lower level

Generator
Agent

P2

P3

Local control
Local controller
Local controller
Local controller
layer
DGs
DG1

DG2

Electrical layer
DG3

Communication channel
Control link

Figure 8.1: Proposed hierarchical control architecture.

8.2

Multi-Agent Implementation

Multi-agent system is implemented here in Java Agent DEvelopment (JADE) [71, 72] framework,
which complies with the Foundation of Intelligent Physical Agents (FIPA) [73]. FIPA Semantic
Language (FIPA-SL) is adopted in this work for agent communication.
The action of an agent is realized by implementing behaviors. Each behavior is an object of
a class that extends Behaviour class in the jade.core.behaviours package. There are two main
behaviors for the proposed agents: Power Balance and Economic Dispatch.
1. Power Balance: This behavior implements the algorithm described in Chapter 4. It contains
111

three sub-behaviors:
(a) Token Transfer: This sub-behavior implements the algorithm in Section 4.2.1 and
builds parent-child relationships among agents, based on a minimal spanning tree. In
summary, this behavior waits for the first token, rejects all other tokens which come
later, and removes neighborhood relationships with those agents whose tokens are rejected. Then this behavior sends the token to all its other child agents.
(b) Information Feedback: This sub-behavior implements the algorithm in Section 4.2.2
and helps collect view information from child agents to parent agents. When each agent
receives view from all its child agents, this sub-behavior processes the view and sends
the updated view to the parent agent.
(c) Generation or Load Dispatch: This sub-behavior implements the algorithm in Section 4.2.3. It receives net power values from parent agent, adjusts local dispatchable
generation or non-vital load to minimize net power, and splits net power values for its
child agents based on (4.7). When net real and reactive power values are both zeros after local adjustment, this sub-behavior informs neighbors of power balance completion.
If an agent receives information of power balance completion from all its child agents,
this sub-behavior informs parent agent of power balance completion. If an agent receives information of power balance completion from parent agent, this sub-behavior
informs child agents of power balance completion.
2. Economic Dispatch: This behavior, as the name suggests, realizes economic dispatch function by implementing the algorithm proposed in Chapter 6. There are two states of this
behavior:
(a) Inquiry state: The agent sends inquiring and local generation value to one of its neighbors to ask for economic dispatch implementation. If it receives accept and local generation values from its neighbor, it updates its generation value by computing (6.12)

112

and (6.13) and goes to the reply state. If it receives refuse from the neighbor, it goes to
the reply state. During inquiry state, whenever the agent receives other agents’ inquiry
for economic dispatch implementation, it refuses.
(b) Reply state: This state waits for inquiry from other agents. If it receives inquiry, it
replies with accept and local generation values; then it updates local generation by
computing (6.12) and (6.13). There is also a time interval set for reply state. When the
time expires, the behavior goes to inquiry state.

8.3

Simulation Platform of Lower Level

Lower-level simulation platform consists of local control layer and electrical layer. In this simulation, electrical layer is constructed by master-slave organized DGs with PE interfaces. The
configuration of a generic PE interface is shown in Fig. 8.2. L and C are, respectively, the inductance and capacitance of the output stage inductors and capacitors. In master-slave organization,
PE-based DGs in an islanded microgrid can be classified into two types: a master inverter operates
in voltage source mode, providing voltage support for the microgrid; all other (slave) inverters
operate as current sources, supplying specific real and reactive power into the microgrid.

8.3.1

Master Power Electronic Interfaces

The objective of the control strategy for voltage-source PE interface is to output required voltage
at the sending point, which will be used as the reference voltage level for the microgrid. Normally,
sources which are employed to offer voltage support in microgrids have large reserve capacity, so
that it is able to provide or absorb extra power when any disturbance occurs in the microgrids,
thereby maintaining the voltage.
A multi-loop control strategy with an inner current control loop and an outer voltage control
loop is adopted for the voltage-source PE interface shown in Fig. 8.3. The transfer function of the
inner current control loop is described by
113

L

Va

L

Vb

L

Vc
C

C

C

Figure 8.2: Configuration of PE interface.

Ls
vref +

−

PRC

+
+
+ iref
−

Kp

+

−

Inner current loop
iL

+
1/Ls

−

Plant
1/Cs

vo

Kds

Outer voltage loop

Figure 8.3: Control block for voltage-source power electronic interface.

Vo =

Kp

(1 − Kp)Ls
Iref −
I
LCs2 + Kp Kd s + 1
LCs2 + KpKds + 1 L

(8.1)

To improve the response to load variation, a feedforward loop is adopted for the load current.
If Kp = 1, from the transfer function, the coefficient of IL is zero; therefore, the effect of load
current on the output voltage is compensated. The damping ratio of the inner loop is derived as
K Kp
ξ = √d
2 LC

(8.2)

The differential feedback from the capacitor voltage introduces damping into the system, avoiding high resonant peak from a pure LC filter. The damping ratio can be adjusted by adjusting Kd .
Theoretically, the optimal performance can be obtained when ξ = 0.707, from which Kd can be
computed.
In order to smooth the transition from grid-connected mode to islanded mode when the grid
fails, a reference voltage feed-forward loop is utilized to accelerate the response of the control sys114

tem. Traditional stationary PI controllers suffer from steady state error when tracking sinusoidal
waveforms. High gain can decrease the error, but it will also decrease magnitude and phase margin, potentially introducing instability. A popular method of minimizing the steady state error is
to use a PI controller in DQ space. However, it requires computations for the frame transformation. Moreover, sources in microgrids can be single-phase or three-phase. The implementations
of single-phase and three-phase DQ controls are different. This platform utilizes a low-pass to
band-pass technique expressed in (8.3) that can transform a dc compensation network in rotating
frame into an equivalent ac compensation network in stationary frame to achieve identical response
feature in the frequency area of interest [68] (it should be noted that the terms in the parentheses
on the right comprise the argument of the gain function):

GAC (s) = GDC

s2 + ω02
2s

(8.3)

Hence the equivalent PI controller in rotational frame can be expressed in the stationary frame
as:
2Ki s
GP RC (s) = Kp1 +
s2 + ω02

(8.4)

This controller, which is also called Proportional Resonant (PR) controller, can vastly boost
the gain at resonant frequency. It does not require frame transformation and can be uniformly
implemented for both single-phase and three-phase sources. Theoretically, at resonant frequency,
infinite gain can be obtained to remove steady state error. Since PR controller can affect the phase
within the band of resonant frequency, a careful selection of Kp1 and Ki is required to decrease the
bandwidth of PR controller while maintaining sufficient phase margin around resonant frequency.

8.3.2

Slave Power Electronic Interfaces

Slave PE interfaces work in current control mode and supply specific real and reactive power assigned by multi-agent system into the microgrid. In this simulation platform, voltage level is set
115

up by a master PE interface source, and slave PE interfaces do not directly participate in voltage
control. Instead, they regulate their output current to provide specified real and reactive power
output. Since voltage is maintained by the voltage source, in order to produce required real and reactive power output using multi-agent system, current-source PE interfaces can adjust their current
to regulate power.
The controller shown in Fig. 8.4 is utilized for a slave inverter. A proportional resonant controller is also employed to decrease steady state error. However, the classic PR controller as expressed in equation (8.4) can not be applied here, because current control inverter is a first order
system. The plant originally has a −90◦ phase shift, while the classic PR controller will introduce
another −90◦ phase shift at resonant frequency, which will cause the system to be unstable. To
overcome this instability problem, a modified form of PR controller as expressed in (8.5) is utilized, where ωc is the breakpoint frequency of dc transfer function [68]. The new controller can
boost gain considerably at resonant frequency while its phase shift can be adjusted by changing
Ki .
vo

Control loop
Kp1

vo
P*
Q*

Current
calculator

iref +

−

2 K iω c s
s 2 + 2ωc s + ω02

Plant
Cs

+ vref +
+
+

Kp

+

−
1/Ls

+

−
io

Figure 8.4: Control block for current-source power electronic interface.

2Ki ωc s
GM P RC (s) = Kp1 +
s2 + 2ωc s + ω02

(8.5)

A feedforward loop of output voltage achieves acceleration of the controller’s dynamic response to the commands from the multi-agent system, and diminishes the influence of the load’s
effect on the output current. If Kp = 1, the open loop transfer function for the reference current is described by (8.6). Kp1 , Ki and ωc can be designed to obtain sufficient phase margin,
while the gain at fundamental frequency is still boosted sufficiently so that steady state error can

116

be minimized.

Go (s) =

8.4

Kp1 s2 + 2(Ki + Kp1 )ωc s + Kp1 ω02
Ls3 + 2Lωc s2 + Lω02 s

(8.6)

Multi-Level Control Architecture

To demonstrate the effectiveness of the proposed multi-level control architecture, a prototype microgrid shown in Fig. 8.5 is simulated in MATLAB/SIMULINK. MAS is implemented using JADE
described in section 8.2. This study system is connected to a utility through a breaker at point of
common coupling (PCC). The breaker in this simulation is open; therefore the microgrid is operated in islanded mode. (Note that in grid-connected mode, the grid at the PCC would provide
the voltage reference, and all inverters in the microgrid operate in current control mode.) Agents
A1–A7 are responsible for bus 1–bus 7 respectively. The structure of the multi-agent system is
the same as that of Fig. 4.1. Since buses 1, 3, 5 and 7 are equipped with DGs, their agents are
called generation agents. To enable economic dispatch between these four agents, assume that they
establish neighborhood relationships among themselves, and are able to communicate with each
other. All the other neighborhood relationships are established based on electrical connections.
The generator data for this microgrid are shown in Table 8.1. The system consists of four DGs,
among which DG3 is a solar panel. It is controlled at maximum power point, consequently, its
real power output is not dispatchable. DG1 is partially dispatchable, while DG2, DG4 are both
fully dispatchable. Their power outputs can be adjusted through commands from the multi-agent
system. The generator cost function of a micro-source is expressed as:

F = a0 + b 0 P + c 0 P 2

(8.7)

Detailed data for the PE interfaces and local controllers are displayed in Table 8.2. The studied
load profile is shown in Table 8.3. The message traces of token transfer to build minimal spanning
tree structure among MAS are shown in Fig. 8.6. The sequence of agents that receive the token
117

DG1

Bus 1
Bus 2

Load 1
Utility

PCC

Load 2
Bus 4

Bus 3

MV/LV
transformer DG2

Bus 5
Bus 7

Bus 6
DG3

DG4

Load 3

Figure 8.5: Microgrid configuration under study.

Node
DG1
DG2
DG3
DG4

Table 8.1: Generation Data
Max Gen
Min Gen
Coefficient
a0
b0
c0
kW kvar kW kvar ($) ($/kWh) ($/kW2 h)
20
10
10
0
8
0.012
0.008
10
6
0
0
12
0.016
0.01
15
0
15
0
4
0
0
10
4
0
0
6
0.014
0.013

verifies the description in Section 4.2.1. It should be pointed that after receiving the token from A3,
agent A5 refuses the token which comes later from A4 and breaks the neighborhood relationship
with A4. This rule helps remove the communication redundancy and construct minimal spanning
tree structure. After building the minimal spanning tree, the view information is transmitted from
child agent to parent agent shown in Fig. 8.7. Agents A7, A6 and A4 stay at leaves of the minimal
spanning tree; therefore, they initializes the Information Feedback Process described in Section
4.2.2.
The message traces of Generation or Load Dispatch Process during 0 s – 0.05 s are shown
in Fig. 8.8. Total power demand is 40 + j16 kvar, less than total generation capacity, which is
55 + j20 kvar. Therefore, the starting agent A1 will cut off its dispatchable generation 10 + j4

118

Table 8.2: Data for PE Interfaces and Local Controllers
DG1
DG2
DG3
DG4
L
1 mH
0.56 mH 0.4 mH 0.56 mH
C
6.34 µF
11.6 µF
13 µF
11.6 µF
Kp
1
1
1
1
Kd 0.000316 0.000114 0.000102 0.000114
Kp1
3
4
4
4
Ki
20
80
80
80

Period
0s – 0.05s
0.05s – 0.1s
0.1s – 0.15s
0.15s – 0.2s
A1
1
2
3
4
5
6
7
8
9
10
11
12
13
14

Table 8.3: Load Profile
Load 1 (kVA) Load 2 (kVA)
10 + j3
20 + j10
5 + j3
18 + j5
25 + j8
20 + j6
20 + j5
15 + j6
A2

A3

A4

A5

Load 3 (kVA)
10 + j3
10 + j5
13 + j4
13 + j4
A6

A7

Token
Accept
Token
Accept
Token
Accept
Token
Accept
Token
Token
Accept
Refuse
Token
Accept

Figure 8.6: Traces of messages for token transfer.
and send net power value 5 + j0 kW to A2. Since bus 2 does not have generation connected to it,
A2 can not reduce net power. It splits the net power values according to its child agents’ updated
view expressed by (4.7). A4 is the leaf agent and it does not have generation connected to it. From
(4.7), the split net power values it receives are Pnet4 = 0, Qnet4 = 0. The net power for A3 is
5 + j0 kvar. A3’s local dispatchable generation can reduce 5 kW to make net power zero, which
initializes A3 to inform its neighbors of power balance completion. At the same time, A4 detects
119

A1
1
2
3
4
5
6

A2

A3

A4

A5

A6

A7

Inform (view 7)
Inform (view 6)
Inform (view 4)
Inform (view 5)
Inform (view 3)
Inform (view 2)

Figure 8.7: Traces of messages for Information Feedback Process.
that the net power it receives is also zero. Then A4 will inform A2 of power balance completion.
When A2 receives power balance completion from both A3 and A4, it will inform its parent agent
A1 of power balance completion. After power balance, MAS can also conduct economic dispatch.
The optimal generation values are obtained as follows: PDG1 = 10.465 kW, PDG2 = 8.172
kW, PDG3 = 15 kW, PDG4 = 6.3631 kW. Convergence of the proposed economic dispatch
is shown in Fig. 8.9(a). For this small system, only three iterations are needed for convergence.
The global marginal cost is around $0.18/kWh. DG3 is a solar panel. Its output power cannot be
dispatched and so its marginal cost does not change. Fig. 8.9(b) shows the iterations to obtain
the optimal generation values. None of dispatchable generators reaches its real power limit in this
case. Total reactive power demand during this period is 16 kvar; hence, after MAS power balance
control, reactive power generation values for the DGs are: QDG1 = 10 kvar, QDG2 = 2 kvar,
QDG3 = 0 kvar, QDG4 = 4 kvar.
At time t = 0.5 s, load changes. The total real power demand is 33 kW, which is still less
than the total generation capacity. The economic dispatch is displayed in Fig. 8.11. The global
marginal cost converges to $0.106/kWh. DG1 is constrained by its lower limit of real power output;
therefore, its marginal cost converges to $0.172/kWh, larger than global marginal cost. From Fig.
8.10(b), new optimal generation values for this load profile are: PDG1 = 10 kW, PDG2 = 4.4783
kW, PDG3 = 15 kW, PDG4 = 3.5217 kW. The total reactive power demand is 13 kvar. Reactive
power generation dispatch values are: QDG1 = 10 kvar, QDG2 = 0 kvar, QDG3 = 0 kvar,
QDG4 = 3 kvar.

120

A1
1
2
3
4
5
6
7
8
9

A2

A3

A4

A5

A6

A7

Inform (Pnet2 Qnet2)
Inform (Pnet3 Qnet3)
Inform (Pnet4 Qnet4)
Inform (completion)
Inform (completion)
Inform (completion)
Inform (completion)
Inform (completion)
Inform (completion)

Figure 8.8: Traces of messages for Generation or Load Dispatch.

DG4

DG3

DG1

DG1

DG2
DG2

DG4

DG3

(a)

(b)

Figure 8.9: (a) Convergence of marginal cost. (b) Dispatched values for the DGs.
At time t = 1.0 s, total real power demand increases to 58 kW, but total real power generation
capacity is 55 kW. Therefore, MAS power balance control will shed 3 kW load. In this simulation,
load 3 will be curtailed by 3 kW. Since load demand is larger than generation capacity. MAS will
not conduct economic dispatch and all DGs are required to output maximum real power to reduce
load shedding. Total reactive power demand is 18 kvar, still less than total reactive power capacity.
The resulting reactive power dispatch values in this stage are: QDG1 = 8 kvar, QDG2 = 6 kvar,
QDG3 = 0 kvar, QDG4 = 4 kvar.
At time t = 1.5 s, total real demand decreases to 48 kW shown in Table 8.3. Fig. 8.11 displays

121

DG3
DG1
DG1
DG4
DG4
DG2
DG3

DG2

(a)

(b)

Figure 8.10: (a) Convergence of marginal cost. (b) Dispatched values for the DGs.
DG4
DG1
DG3

DG1
DG2

DG2
DG4

DG3

(b)

(a)

Figure 8.11: (a) Convergence of marginal cost. (b) Dispatched values for the DGs.
economic dispatch process. From the figure, it can be seen that real power output of DG2 reaches
its upper limit, therefore its marginal cost is lower than the global value; while for DG1 and DG4,
their output does not reach the limit, so their marginal cost converges. The optimal dispatch values
are: PDG1 = 14.29 kW, PDG2 = 10 kW, PDG3 = 15 kW, PDG4 = 8.71 kW. Total reactive
power demand decreases by 3 kvar; therefore QDG1 drops to 5 kvar.
Fig. 8.12 displays simulation results for the DGs and loads. Fig. 8.13 shows output current
transitions of DGs at the times of load changes.

122

:

:
'*

'*
/RDG
'*
/RDG
/RDG

'*

D

YDU

F

YDU

/RDG

'*
'*

/RDG

'*
/RDG
'*
E

G

Figure 8.12: (a) Real power output of DGs. (b) Reactive power output of DGs. (c) Real power
consumed by Load. (d) Reactive power consumed by Load.
(A)

(A)
DG3

DG1
DG1

DG3
DG4
DG2
DG2
DG4
(s)
(a)

(s)
(b)

Figure 8.13: (a) Output current transitions of DGs at 0.5s. (b) Output current transitions of DGs at
1.0s.

8.5

Summary

In this chapter, multi-agent system is implemented using JADE platform. A comprehensive multilevel control architecture was presented for master-slave organized microgrids with PE interfaced
DGs. The MAS power balance control strategy that can accomplish exact power balance in three
sweeps, regardless of system size, was used as the upper control layer. The lower MAS based
economic dispatch layer, utilizing limited communications between neighboring agents, is able to
conduct economic dispatch on-line. Local control layer complies with the instructions from multiagent system and implement controls locally. These three layers collaborate and interact with each

123

other to act as a united control system, which can operate microgrids in a safe, economic and stable
way. In the presence of markets, the quadratic generation cost function can be replaced by bids,
and the associated objective function, such as social welfare function, can also be optimized by
the proposed MAS. The performance of the proposed strategy was demonstrated on a test system.
Its benefits, both in terms of speed as well as robustness in tracking time-varying loads, were
demonstrated.

124

Part IV
Conclusions

125

Chapter 9
Contributions and Future Work
9.1

Contributions

This dissertation presents linearization and distributed methods for optimization and control of
smart grid and microgrid. The contributions include:
1. This dissertation presents linearization method to linearize optimal power flow problem with
FACTS devices. The linearized model can achieve fast and reliable advantage as traditional
DC model; furthermore, it does not sacrifice accuracy and can obtain full state information.
2. This dissertation presents a combined algorithm using genetic algorithm and linearized optimal power flow framework for the optimal location and parameter selection of UPFC. An
optimal branch numbering strategy is proposed to accelerate the convergence speed of the
algorithm.
3. This dissertation presents a distributed method for the power balance control of microgrids.
The information flows in parallel and results are obtained in non-iterative way; therefore,
the algorithm can achieve superior performance in terms of speed without any convergence
issues.
4. In the context that requires to consider losses and voltage regulation, this dissertation presents
126

a distributed power balance algorithm based on Guess method. The distributed power flow
algorithm fully makes use of communication time, and updates state information synchronously
among agents, which offers potential speed advantage.
5. In the microgrid that is amenable to economic dispatch, this dissertation proposes a distributed economic dispatch algorithm based on the consensus theorem. Convergence of the
proposed algorithm and proof of achieving global optimum are also presented. This algorithm is also capable for on-line economic dispatch.
6. In a stressful power system network, this dissertation proposes a distributed economic dispatch algorithm considering network constraints. Convergence of the proposed algorithm
is demonstrated. The distributed power flow algorithm fully makes use of communication
time, and updates state information synchronously among agents, which offers potential
speed advantage.
7. This dissertation presents the implementation of the proposed multi-agent system in JADE
environment. A comprehensive multi-level control system is proposed in this dissertation
for safe and economic operation of microgrids.

9.2

Future Work

1. Develop an algorithm that examines all the load scenarios within acceptable size of memory.
This will cover two possible solutions, 1) combining nonlinear techniques and the algorithm
developed in this chapter to reduce the size of memory. 2) decoupling coefficients matrix of
equality and inequality constraints to divide the memory size into small pieces and examines
these pieces one by one.
2. Simulate nonlinear model and DC model for comparisons in terms of calculation time and
accuracy.
3. Extend the developed algorithm to other FACTS devices.
127

4. Simulation the distributed economic dispatch algorithm considering network constraints on
a larger system.
5. Complete the distributed control framework by developing distributed state estimator.

128

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129

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