Energy-Efficient Designs in Cyber-Physical Systems
with a Control and Optimization Approach
by
Yingsong Huang
A dissertation submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
Auburn, Alabama
May 5, 2013
Keywords: CPS, smart grid, microgrids, wireless networks, video, energy efficiency,
optimization, control, majorization, Lyapunov optimization
Copyright 2013 by Yingsong Huang
Approved by
Shiwen Mao, Chair, Associate Professor of Electrical and Computer Engineering
Prathima Agrawal, Professor of Electrical and Computer Engineering
Jitendra K. Tugnait, Professor of Electrical and Computer Engineering
Tin-Yau Tam, Professor of Mathematics
Abstract
In modern cyber-physical systems (CPS), new dimensions of freedom are enabled to energy
efficient solutions. We first focus on the energy delivery side within the smart grid paradigm for
smart, efficient and reliable energy delivery. We then explore the demand side with ?green? wire-
less networks for multimedia streaming, in response to the drastic increasing demand in multimedia
service in wireless networks.
In this dissertation, we first study energy management systems in smart grid. We design
power scheduling policies for smoothing power profile in power distribution networks. The pro-
posed power scheduling policies allow the operator to deploy generators, transformers and power
transmission lines with smaller capacity in the grid, thus reducing the capital investment. In addi-
tion, the power consumption can be reduced during peak hours, and the average energy generation
cost will also be minimized. We also propose a smart electric energy management system in mi-
crogrids (MGs). With the proposed algorithm, the MG achieves the fundamental requirements in
smart grid with distributed renewable energy integration, energy storage systems management and
residential power quality management, while keeping the compatibility to the legacy grid.
We then propose downlink power control frameworks for streaming multiple variable bit rate
(VBR) videos in wireless cellular networks. We develop both centralized and low-complexity dis-
tributed algorithms, which optimally schedule the transmission power for the BS?s, such that VBR
videos can be delivered to mobile users without causing playout buffer underflow or overflow un-
der wireless channel uncertainty. The proposed solutions achieve the quality of experience (QoE)
requirements of users, as well as keeping the systems ?green?.
In this dissertation, we adopt a control and optimization approach for energy efficient design
in CPS.The synergy of the advanced control and optimization methods in engineering systems
provides new visions for practical solutions to bring a green world in the future.
ii
Acknowledgments
I have received such immeasurable help from many people, and in such diverse ways, to sup-
port me during the development of this dissertation, and more generally, my academic experience.
A few words? mention here cannot adequately capture all my appreciation.
I would like to express my deepest gratitude to my dissertation advisor, Prof. Shiwen Mao, for
his continuing and inspirational guidance, support and encouragement during my Ph.D. program.
I would also like to sincerely thank my dissertation committee, Prof. Prathima Agrawal, Prof.
Jitendra Tugnait, and Prof. Tin-Yau Tam for their valuable comments and time, who contribute
their broad perspective in refining the ideas in this dissertation. I am also indebted to Prof. Alvin
Lim for serving as the university reader, and reviewing my work.
I would like to thank Prof. Stuart Wentworth for his constant support and inspiration in my
instructing the RF Systems Labs. I enjoyed the teaching experience as well as his sense of humor
in the labs.
I want to take this opportunity to recognize all my fellow colleagues in the Electrical and
Computer Engineering at Auburn University: Dr. In Keun Son, Dr. Donglin Hu, Bobby Black,
Jing Ning, Yi Xu, Yu Wang, Zhifeng He and Zhefeng Jiang, for the discussions, cooperation and
assistance during these years. In addition, I am very grateful to Dr. Yihan Li for her hospitality
and help in my study and life at Auburn.
Finally, I would like to extend my heart felt thanks to my parents, without their continuous
support and tremendous care, the achievements of this dissertation could not be possible.
iii
Table of Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Enabling Energy Efficient Cyber-Physical Systems . . . . . . . . . . . . . . . . . 1
1.2 Smart Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Traditional Electricity Grid . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Smart Grid Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Microgrid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.4 Smart Energy Management Systems . . . . . . . . . . . . . . . . . . . . . 9
1.3 Energy Efficient Multimedia Networks . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Multimedia Networks Architecture . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Energy Efficiency in Video Coding . . . . . . . . . . . . . . . . . . . . . 13
1.3.3 Energy Efficiency in Video Transmission . . . . . . . . . . . . . . . . . . 15
1.3.4 Joint Video Coding and Transmission Design . . . . . . . . . . . . . . . . 17
1.3.5 Video over Emerging Wireless Networks . . . . . . . . . . . . . . . . . . 18
1.4 Key Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5 Overview of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2 Smooth Electric Power Scheduling in Power Distribution Networks . . . . . . . . . . 25
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.1 Load Demand Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
iv
2.2.2 Cumulative Demand and Supply Curves . . . . . . . . . . . . . . . . . . . 29
2.2.3 Smooth Power Scheduling Problem . . . . . . . . . . . . . . . . . . . . . 31
2.3 Majorization Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Smooth Electric Power Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.1 SEPS-DL Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.2 Performance of SEPS-DL . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4.3 Extension to the General Case . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.4 Electric Power Allocation Among Individual Users . . . . . . . . . . . . . 39
2.5 Simulation Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Adaptive Electricity Scheduling in Microgrids . . . . . . . . . . . . . . . . . . . . . . 51
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.3 Lyapunov Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Optimal Electricity Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.1 Properties of Optimal Scheduling . . . . . . . . . . . . . . . . . . . . . . 62
3.3.2 MG Optimal Scheduling Algorithm . . . . . . . . . . . . . . . . . . . . . 64
3.3.3 Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.4.1 Algorithm Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4.2 Comparison with a Benchmark . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4 Overview of Green Video Streaming over Celluar Networks and Variable Bit Rate Video 76
v
4.1 Green Video Streaming over Celluar Networks . . . . . . . . . . . . . . . . . . . 76
4.2 VBR Video System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5 Downlink Power Allocation for Stored Variable-Bit-Rate Video in Cellular Network . . 81
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2 System Model and Problem Formation . . . . . . . . . . . . . . . . . . . . . . . . 82
5.3 Two-Step Downlink Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . 85
5.4 Distributed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.6 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6 Downlink Power Control for Variable Bit Rate Video over Multicell Wireless Networks 100
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.2.1 Network and Video System Model . . . . . . . . . . . . . . . . . . . . . . 101
6.2.2 Problem Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2.3 Existence of Feasible Solutions . . . . . . . . . . . . . . . . . . . . . . . 105
6.2.4 Comparison with a Lazy Scheme . . . . . . . . . . . . . . . . . . . . . . 107
6.3 Centralized Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.3.1 Reformulation and Linearization . . . . . . . . . . . . . . . . . . . . . . . 108
6.3.2 Branch-and-Bound Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3.3 Enhancement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.4 Distributed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.5.1 Centralized Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.5.2 Distributed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6.5.3 Empirical Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . 122
6.6 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
vi
6.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7 Energy Efficiency on Downlink Multiuser VBR Video Streaming . . . . . . . . . . . . 126
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.2.1 Network and Video Source Model . . . . . . . . . . . . . . . . . . . . . . 127
7.2.2 Power-Aware Transmission Scheduling . . . . . . . . . . . . . . . . . . . 129
7.3 Problem Reformulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.3.1 Majorization Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.3.2 Schur-convexity of Problem (7.9) . . . . . . . . . . . . . . . . . . . . . . 131
7.4 Power Allocation Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.4.1 Power Minimization Algorithm . . . . . . . . . . . . . . . . . . . . . . . 132
7.4.2 Optimality Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.4.3 Multiuser Video Transmissions . . . . . . . . . . . . . . . . . . . . . . . 135
7.4.4 Application to Interactive Video Streaming . . . . . . . . . . . . . . . . . 137
7.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.5.1 Simulation Results for Interactive Video Streaming . . . . . . . . . . . . . 143
7.6 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8 Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
8.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
A Proofs in Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
A.1 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
A.2 Proof of Theorem 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
B Proofs in Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
B.1 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
vii
B.2 Derivation of Equation (3.16) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
B.3 Proof of Lemma 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
B.4 Proof of Lemma 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
B.5 Proof of Lemma 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
B.6 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
B.7 Proof of Theorem 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
B.8 Proof of Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
C Proofs in Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
C.1 Proof of Lemma 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
C.2 Proof of Lemma 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
C.3 Proof of Lemma 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
C.4 Proof of Theorem 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
C.5 Proof of Theorem 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
D Proofs in Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
D.1 Proof of Lemma 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
D.2 Proof of Theorem 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
E Proofs in Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
E.1 Proof of Theorem 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
E.2 Proof of Corollary 7.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
F Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
viii
List of Figures
1.1 Cyber-physical system - Integration of communication, control and computation [1]. . 2
1.2 Traditional electricity grid with unidirectional power and information flow. . . . . . . 4
1.3 Smart grid with plug-and-play interfaces and two-way flow of power and information. 6
1.4 Microgrid: Localized cluster of DRERs, ESS?s, information networks and residents. . 8
1.5 Energy management systems enhanced by smart grid paradigm. . . . . . . . . . . . . 9
1.6 Overview of the wireless video networking system. . . . . . . . . . . . . . . . . . . . 12
1.7 Block diagram of a typical video encoder. . . . . . . . . . . . . . . . . . . . . . . . . 12
1.8 Block diagram of a typical video decoder. . . . . . . . . . . . . . . . . . . . . . . . . 12
1.9 Protocol layers and system controls involved in wireless video transmission. . . . . . . 16
1.10 Overview of the dissertation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 Illustration of the electricity distribution network. . . . . . . . . . . . . . . . . . . . . 27
2.2 Cumulative demand and supply curves. . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Lorenz Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4 Illustrate the operation of the SEPS-DL algorithm. . . . . . . . . . . . . . . . . . . . 36
ix
2.5 Smooth power scheduling with priority load. (Note that although not deferrable, the
cumulative priority load can also be represented with two curves as shown in (a),
where the maximum and minimum curves meet at the corners.) . . . . . . . . . . . . 39
2.6 Comparison of the power schedules achieved by SEPS-DL an GSEPS. . . . . . . . . . 43
2.7 Convergence of the individual power allocation achieved by DUMLC. . . . . . . . . . 44
2.8 Peak electric powers achieved by SEPS-DL, GSEPS, SUDP and UMRP. . . . . . . . . 46
2.9 Load factor achieved by SEPS-DL, GSEPS, SUDP and UMRP. . . . . . . . . . . . . . 46
2.10 Variances of the power schedules achieved by SEPS-DL, GSEPS, SUDP and UMRP. . 47
3.1 Illustrate the microgrid architecture. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2 The system model considered in this chapter. . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Average QoSEs of three residents (V = Vmax). . . . . . . . . . . . . . . . . . . . . . . . 68
3.4 Energy levels of three Li-ion batteries (V = Vmax). . . . . . . . . . . . . . . . . . . . . . 69
3.5 QoSEs for three residents with different service contracts (V = Vmax/2). . . . . . . . . . . . 70
3.6 MG operation traces of the proposed algorithm for the 5-day period. . . . . . . . . . . 71
3.7 MG operation traces of proposed algorithm for the 7-day period. . . . . . . . . . . . . 72
4.1 Perceived quality of VBR and CBR videos: Football video coded with an H.264 codec. 77
4.2 Frame size of VBR and CBR videos: Football video coded with an H.264 codec. . . . 78
4.3 Transmission schedules for VBR video session n. . . . . . . . . . . . . . . . . . . . . 79
5.1 Normalized capacity curves and inflection points for a two-user system, where link 1
has better quality than link 2, i.e. A1 <A2. . . . . . . . . . . . . . . . . . . . . . . . 88
x
5.2 Topology of the cellular network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.3 The sizes of the first 100 frames of the NBC News sequence. . . . . . . . . . . . . . . 94
5.4 Transmission schedule for video NBC News to user 2. . . . . . . . . . . . . . . . . . 95
5.5 Convergence of power allocation and Lagrange multipliers. . . . . . . . . . . . . . . . 96
5.6 Average playout buffer utilization for the entire video sequence (10000 frames). . . . . 97
5.7 Average playout buffer utilization for frames 2000 to 2500. . . . . . . . . . . . . . . . 98
6.1 A multicell wireless network with concurrent VBR video sessions. The inter-cell in-
terference experienced by the central cell user is illustrated. . . . . . . . . . . . . . . 103
6.2 Four-point polyhedral outer approximation for um = log(1 +?m), 1 <?minm ??m ?
?maxm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3 The cumulative overflow, transmission, and consumption curves when transmitting
Star Wars at link 1 with centralized algorithm in the seven-cell network. . . . . . . . . 118
6.4 The cumulative overflow, transmission, and consumption curves when transmitting
Star Wars at link 1 with Accelerated centralized algorithm in the seven-cell network. . 118
6.5 The cumulative overflow, transmission, and consumption curves when transmitting
Star Wars at link 1 with DCPC in the seven-cell network. . . . . . . . . . . . . . . . . 119
6.6 Convergence of the branch-and-bound algorithm in time slot 1 when all the I-frames
are transmitted and all the buffers are empty (i.e., the worst case scenario). . . . . . . . 120
6.7 Rate sums with the algorithms with a seven-link network. . . . . . . . . . . . . . . . . 120
6.8 Convergence of transmit powers of DCPC with a seven-link network. . . . . . . . . . 121
xi
6.9 Convergence of bit rates of DCPC with a seven-link network. . . . . . . . . . . . . . . 121
6.10 Average buffer utilization of the four schemes. . . . . . . . . . . . . . . . . . . . . . 123
6.11 Illustration of underflow events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.12 Average power consumption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.1 Cellular network and video streaming system model. . . . . . . . . . . . . . . . . . . 129
7.2 Two cases for determine the next transmission rate. . . . . . . . . . . . . . . . . . . . 133
7.3 Simulation results: transmission curves of Star wars. . . . . . . . . . . . . . . . . . . 139
7.4 Simulation results: average power consumption. . . . . . . . . . . . . . . . . . . . . . 140
7.5 Cumulative power consumptions achieved by PMA-1 and the conventional scheme. . . 141
7.6 Power consumption for each frame achieved by PMA-1 and the conventional scheme. . 142
7.7 Transmission curves achieved by PMA-1 and the conventional scheme. . . . . . . . . 143
7.8 Transmission curves for real-time interactive video streaming: NBC News and LAP =
16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
7.9 Transmission curves after the user skips the frames in [10s, 20s]: Star Wars. . . . . . . 145
7.10 Transmission curves for the original non-skipped video streaming: Star Wars. . . . . . 146
A.1 Illustration for the optimality proof of SEPS-DL algorithm . . . . . . . . . . . . . . . 155
xii
List of Tables
2.1 Notation Table for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.2 Average operating cost of power generation . . . . . . . . . . . . . . . . . . . . . . . 48
3.1 Notation for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.1 Notation for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.1 Notation Table for Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2 Number of underflow events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
7.1 Notation Table for Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
7.2 VBR Video Trace Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
7.3 Simulation Results: Playout Buffer Underflow Rates . . . . . . . . . . . . . . . . . . 143
7.4 Playout Buffer Underflow Rates for Interactive VBR Videos with Different LAPs . . . 144
xiii
Chapter 1
Introduction
1.1 Enabling Energy Efficient Cyber-Physical Systems
Innovations in energy supply, delivery and consumption in physical systems are the most
effective ways to fight against global warming and environmental degradation. The synergy of
communication, networking, control, computation intelligence and physical components greatly
extends the autonomy, efficiency, flexibility, reliability and adaptability of physical systems. Such
an enhanced system as illustrated in Fig. 1.1 is termed a cyber-physical system (CPS). The emerg-
ing CPS will significantly enhance the capabilities of physical engineered systems and change
how people interoperates with the physical world through real-time embedded systems by sensing,
computation, optimization and control over communication networks, which makes the engineer-
ing systems reliable, secure, efficient and smart. Examples include smart electricity grid, smart
building, smart manufacturing, and smart transportation [2]. The next generation of CPS brings
new dimension of freedom to the energy efficient solution to the practical physical systems.
In this dissertation, we investigate energy efficient design of smart grid and information and
communications technology (ICT) infrastructures. Electric power system is generally identified as
the largest greenhouse gas emission source created by human-beings. Under traditional electricity
grid structure, only 1/3 of fuel energy, mainly from fossils, is converted to electricity and almost
8% is then further lost during the transmission on the power transmission lines. In addition, to
manage the peak demand, about 20% of generation capacity is reserved. Those reserved capacity
is only used in 5% of the time [3]. Thus, an efficient, reliable, flexible and economic electricity de-
livery system is needed for the new era. The next generation of electricity delivery network, called
smart grid [4?6], is an evolution of the 20th centuary traditional grid, which is expected to solve
the major inefficiency of the traditional electricity grid. A smart grid with two-way flow of both
1
10100101010
0101000001000
0101110110010
1010111010101
1001010101011
11010101010 Communication
Control
Computation
Cyber Physical
System
Information
Figure 1.1: Cyber-physical system - Integration of communication, control and computation [1].
electricity and information provides full visibility and automatic control over the components and
services in power networks. The ubiquitous sensing, monitoring and automatic control enable the
ability for responding to a wide range of conditions and events, which allows the integration of the
computational intelligence into the system to efficiently schedule power generation, transmission,
distribution and usage. The smart grid can also faciliate the penertration of renewable energy, such
as photovoltaics, wind, geothermal, and biofuels, which will greatly reduce resource depletion,
increase sustainablity, lower greenhouse gas emissions and reduce air pollution.
Besides the enhancement of electricity delivery networks, it is equally important to implement
energy efficient design at the demand side. Since the electricity supply continuously matches the
demand under the current operation strategy of traditional grid, the emissions of power plants in-
crease as the demand increases. However, it is not environment friendly to proportionally boost the
electricity supply along with demand, especially when we consider that the nation wide electricity
demand is estimated to increase by 41% by 2030 [7]. To alleviate the total energy consumption,
there is a great need for energy efficient infrastructures, devices and consumption patterns. One of
the fastest energy consumption growth comes from today?s ICT infrastructure, which may be re-
sponsible for more than 10% of the total electrical power consumption [8], due to the tremendously
wide spread of the Internet and mobile communication networks. According to a recent study by
2
Cisco, mobile data traffic will be expected to grow to 6.3 Exabytes per month by 2015, a 26?fold
increase over 2010 [9]. In addition, mobile video will generate much of the mobile traffic growth
through 2015. Of the 6.3 Exabytes per month wireless data crossing the mobile network by 2015,
4.2 Exabytes will be related to video.
Furthermore, it is reported that the energy consumed by end-user equipment only contributes
around 7% of the entire consumption, while the remaining 93% is consumed by mobile network
components [10], of which more than 50% is used by the base station (BS) equipment [11]. There-
fore, considerable savings on electrical bills could be achieved for wireless operators when the
power of BS?s is minimized for video streaming. The reduced electricity consumption will also
bring about important improvement in the overall carbon footprint of the wireless industry and
achieve the goal of ?green? communications.
In this dissertation, we examine energy efficient design in CPS from two sides: energy deliv-
ery networks and energy demand using multimedia wireless networks as an example. We inves-
tigate the problems with a control and optimization theoretic approach, which involves Lyapunov
optimization [12], majorization [13], nonlinear and convex optimization [14]. The synergy of these
advanced mathematical tools brings about new visions for energy efficient solutions to practical
engineering systems with performance bounds to bring a green world in the future.
1.2 Smart Grid
1.2.1 Traditional Electricity Grid
The electric power delivery system, i.e. electricity grid, was named as the greatest engineering
achievement of the 20th century [15]. Generally, the electricity grid consists of three parts: power
generation plants, power transmission networks, and power distribution networks, as shown in
Fig. 1.2. The traditional grid is strictly a hierarchical system, in which the power plant is at the
upstream to provide electric power to the user load at the downstream. The electric power is
generated at central power plants normally driven by combustible engines that are mostly fueled
by coal and gas. Due to economic and environmental considerations, the plants are usually located
3
Coal plant Step up
transformer
Step down 
transformer
Transmission networks Distribution networks
Information flow
Electricity flow
Figure 1.2: Traditional electricity grid with unidirectional power and information flow.
far away from users, and they are connected to users by transmission and distribution networks.
The generated electric power is stepped up to a higher voltage (? 110 kV) by step up transformers
to overcome the power transmission loss over a long distance. The transmission networks route
the power to the substations, in which the voltage is stepped down from the transmission level
to medium-voltage distribution level (< 33 kV). After the voltage regulation by the substations,
the power flow is then forwarded into the distribution networks. Finally, when the power flow
arrives at a service location, the voltage is further stepped down to the service level required by
end users, which is generally 120 V, 240 V, or 480 V. It can be noticed that the electricity and
information in this type of grid flows in a unidirectional fashion. The generated power strictly
flows from the plant to the end users, while the information at the downstream is collected by
the upstream components. For example, a distribution control center can acquire load information
from end users, but the users generally have no idea about the status of the power generation and
transmission.
The electricity grid must be operated to achieve real-time balance between generation and
load. Otherwise, the grid frequency will drift up or down from the nominal value (typically 50Hz
or 60 Hz). Today, the overall daily load profile in a given service area can be predicted well,
and the day-ahead generating schedule can be developed based on the prediction. Thus, electricity
4
generation adopts a ?load following? strategy. Due to the limitation of unidirectional flow structure,
loads are not generally controlled directly, except for the case when there is insufficient generation
available on peak time, and then the ?load shifting? operation is executed to encourage users to
shift load from on-peak to off-peak periods.
Increasing environmental concerns urge the high penetration of green and renewable energy
resources, such as solar, wind, geothermal, tidal, and etc. The load-following strategy with the
unidirectional electricity and information flows becomes awkward to meet the penetration of the
new renewable resources into the grid. The electricity generated from renewable resources, such
as wind and solar, is generally random, due to complex fluctuations of weather condition. Thus, it
is hard to accurately predict the generation even in a short period. In addition, renewable energy
sites are largely geographically distributed, due to the distribution of the renewable resources, the
unidirectional electricity and information flow cannot provide the needed services and fast response
to ensure that power generation matches load in real time, which may fail the load following
strategy. To embrace the green electrification age, automated, distributed and advanced energy
delivery networks should be promoted, which are enhanced by the two-way flow of electricity and
information empowered by digital computation, communication and control technologies.
1.2.2 Smart Grid Evolution
Smart grid is a 21st century evolution of electricity delivery systems. Smart grid enhances the
traditional power grid through communication, computation, and control technologies throughout
the processes of electricity generation, transmission, and distribution. A key feature of smart grid is
the two-way flow of electricity and realtime information through communication networks, which
offers many benefits and flexibilities to both electricity consumers and providers. The US 2009
Recovery Act indicates that a smart grid will replace the traditional power grid system to improve
energy efficiency and advance the liberalization of energy in North America [16].
The smart grid is illustrated in Fig. 1.3. In contrast to the traditional power grid, the power
generation and power flow patterns in the smart grid exhibit more flexibility. The initial concept
5
Wind 
farm
Nuclear 
plant
Photovoltaic cells
Energy
Storage
Operations
Residential 
user
Residential 
user
Commercial
user
Industrial
user
Electricity 
market
PHEV
Elect
ricity 
flow
Elec
tri i
ty fl
ow
Info
rma
tion
 flowInformat
ion flo
w
Smart 
Grid
Solid state 
equipment
Figure 1.3: Smart grid with plug-and-play interfaces and two-way flow of power and information.
of smart grid begins with the introduction of automated meter infrastructure (AMI) systems in
distribution networks. AMI enables the utilities to monitor the demand status of end users and
impose certain control on the consumption and costs [3, 4, 17]. Future evolution with the inte-
gration of various new power electronics and information techniques provides real time sensing,
monitoring and control for every corner of the power delivery system. For example, in power de-
livery networks, phasor measurement unit (PMU) are being deployed to synchronized measure the
real-time phasor data at multiple points in the grid [18]. Solid state transformer (SST) can respond
to signals from a facility or a household to change the voltage and other electric characteristics in
the system [19]. On the user side, local renewable resources generation and storage systems, smart
meters and smart facilities empower the pervasive sensing, monitoring and control of the power
flow and power usage in response to the utility supply and market price fluctuations.
One of the most important benefit from smart grid is that the two-way flow of electricity
and information facilitates the deployment and management of the distributed renewable energy
resource (DRER), such as wind farms and solar photovoltaic cells. Unlike the load following
strategy, in which the supply continuously matches the demand, the real time information exchange
6
among DRERs, central power plants and end users provides a new way to match the demand to the
available supply by regulating the power generation, as well as controlling the load service level
of the users.
The deployment of the DRERs fundamentally alters the operation of power generation, which
is conventionally controlled centrally. Furthermore, the increasing popularity of plug-in hybrid
electric vehicles (PHEVs) serves as distributed energy storage system for residential users by
vehicle-to gird (V2G) technology [20]. To cope with distributed generation, a concept of vir-
tual power plant (VPP) is introduced [21], which clusters numerous DRERs with a total capacity
comparable to a traditional power plant. The group of DRERs is managed by a central controller
and appears like a virtual central power plant to the grid. VPP provides a promising paradigm to
replace a conventional power plant by a cluster of local DRERs with more flexibility and efficiency.
It can be seen from Fig. 1.3 that the smart grid is organized like the Internet, which may be
called the ?Energy Internet? [22], in contrast to the strictly hierarchical structure of the traditional
grid in Fig 1.2. All the components in the power delivery systems, including generation, transmis-
sion,distribution and consumption can be deployed and managed through plug-and-play interfaces.
By the full duplex of electricity and information flows, configuration of the devices in the system
may be customized to respond the grid status in real time. For example, energy storage systems
may cooperate with DRERs to balance the supply and demand according to the power generation
conditions. On the other hand, users may customize their demand for low cost energy consumption
by responding to the realtime market price. The concept of Energy Internet envisions the highly
flexible smart grid framework to facilitate a green and sustainable energy-based society, mitigating
the growing energy crisis, and reducing the impact of greenhouse gas emissions.
1.2.3 Microgrid
The existing traditional electricity grid has often been cited as the most complex engineering
system ever built. Thus to fundamentally overhaul the existing infrastructure is either unimple-
mentable or economically inefficient. The transition to the smart grid would favor the strategy
7
MicrogridMGCC Energy
Storage
Wind 
turbinesPhotovoltaic cellsPower flow
Information 
flow
Figure 1.4: Microgrid: Localized cluster of DRERs, ESS?s, information networks and residents.
based on the reuse and upgrade from the existing grid by adding capabilities and functionalities in
a sustainable growth fashion. In a long period, the smart gird would coexist with the traditional
grid, and also provide certain backward compatibility with the legacy systems.
With the concept of plug-and-play interface in the smart grid, a new grid paradigm called
microgrid (MG) is regarded as a promising component for future smart grid deployment. Micro-
grids are interconnected networks that provide a localized cluster of renewable energy generation,
storage, distribution for local demand, to achieve reliable and effective energy supply with small
scale implementation of smart grid functionalities [4, 23]. A typical MG is shown in Fig. 1.4,
which includes DRERs, energy storage systems (ESS?s), wired/wireless networks for information
delivery, an MG central controller (MGCC), and local users. An MG is centrally controlled and
managed by the MGCC [23], which may exchange information with the local users via two-way
information networks, such as a wireless network or a power line communication (PLC) system.
There is a single common coupling point with the macrogrid. When disconnected, the MG works
in an islanded mode, in which DRERs and ESS?s continuously provide electricity to satisfy the
local demands. When connected to the macrogrid, the MG may request extra electricity from the
macrogrid or sell the excess energy back to the market [3].
The MGs are designed with the fundamental elements of smart grid, such as the integration
of DRERs, intelligence core, two-way electricity and information flow, self-healing, and demand
side management. This simplified design of the integration of DRERs and the ability to isolate
8
Electricity 
market
Power Networks
AMI
ControlInformation 
Networks
Energy market information
Network status
Generation Information
Demand Information
Energy 
Management 
Figure 1.5: Energy management systems enhanced by smart grid paradigm.
the MG from the macrogrid in disturbance will yield highly reliable electricity supply. The island
operation mode has the potential to provide a higher local reliability and efficiency than that pro-
vided by the general macrogrid. In addition, the MG works with the plug-and-play interface to the
macrogrid, which minimizes the regulation effort of DRERs in the cooperative power scheduling
and management with the macrogrid and enables sustainable evolution to the emerging smart grid.
1.2.4 Smart Energy Management Systems
The two-way flow of electricity and information infrastructure enables various innovative
functions and management principles in practicing dynamic energy management systems, which
significantly alter the nature of future power system operation and consumer behavior. It involves
the incorporation of smart energy management based on advanced control and communications
capabilities, computational intelligence, and smart devices that make the electricity gird ?smart?,
as shown in Fig. 1.5. The real time information flow from the grid, such as active/reactive power,
voltage, phase, user demand, as well as energy price on the market, brings great flexibility in
designing the new smart energy management system. With smart energy management systems,
the grid achieves energy savings, operation cost reduction, demand and supply balance, emission
control, peak load reduction and elimination of many inherent inefficiencies that may be caused by
the conventional ?load-following? strategy.
9
Among various designs of smart energy management, demand response (DR) serves an im-
portant function in smart grid. Demand response is the mechanisms to manage the demand from
user side in response to the supply condition. Unlike the ?load-following? strategy, which con-
tinuously matches the supply to the demand, demand response enables the ?generation-following?
strategy to match the demand to the available supply by controlling the service level, thus achieving
better overall capacity utilization [24]. Currently, the research in demand response mainly focuses
on two branches: direct load control and real time pricing.
Direct load control takes advantage of the scheduling flexibility of certain loads, which may
be scheduled on and off remotely without degrading the satisfaction of end users. It is estimated
that up to 33% residential loads, such as dishwasher, washer/dryer, and PHEV charging, could
be rescheduled at some level without major impact on users [24]. Unlike the traditional energy
management, the smart demand response mechanism has the capability to aggregate and precisely
control the service level of individual load according to the grid status. The application of direct
load control not only sheds load during peak demand hours, but also intends to actively promote
new types of grid services that could reshape a demand profile to a nicely smoothed demand profile.
Real time pricing is another important method that encourages the users to reshape their elec-
tricity consumption pattern by various price strategies. The utilities change energy price based on
the fluctuation in the cost of generation, the aggregated load demand, and other realtime states of
the grid, thus providing immediate financial incentives to the users to regulate the demand side
applications and perform load shifting. The pricing strategy jointly optimizes the cost of genera-
tion, user electricity payment and user electricity utilization, which increases economic and energy
efficiency and delivers the fair prices to both the utilities and users.
In smart grid, both approaches rely on the information exchange between energy providers
and consumers. The control center monitors the real time states of the power networks, as well as
transmits control commands to the users by various communication options, including PLC, wide
area networks (WAN) and local area networks (LAN). Smart meters provide the interface between
LAN and home area networks (HAN), and serve as the gateway for security authentication and
10
command interpretation to ensure the command issued from the trusted controller. Smart meters
may plan the service level of each smart load/facility, and send out the control commands with the
service level of individual load through HAN. After the smart load obtains the command, it adjusts
its service level according to the new requirement. The process resorts to the support of information
transmission protocols, security authentication, energy plan intelligent core in smart meter, and
remote control for smart facilities, which are still under development and standardization.
1.3 Energy Efficient Multimedia Networks
1.3.1 Multimedia Networks Architecture
A typical wireless video system is illustrated in Fig. 1.6, which generally consists of the
video encoder/ decoder, the wireless/wireline networks, and mobile receiving/playout devices. To
achieve QoE guaranteed and/or energy efficient wireless multimedia systems, various schemes
have been developed, each of which focuses on one (or several) component(s) of the system .
Video contents provided by commercial video providers or individuals are encoded into com-
pressed frames with different codecs. The picture coding basics are detailed reviewed in [25, 26].
The algorithms in the codecs play large role in the quality of the video, which is directly related to
the QoE. Typical encoder and decoder block diagrams are illustrated in Fig. 1.7 and Fig. 1.8, re-
spectively. These frameworks have been adopted in many video coding standards, such as H.264,
and MPEG-4, which consist of motion estimation and compensation, discrete cosine transform
(DCT), quantization, entropy coding, inverse quantization, and inverse DCT (IDCT).
After source coding, the encoded frames are then packetized by the network transmission
protocols. The video packets are streamed toward the destination through wired and/or wireless
networks. When frames arrive at the destination, the codec decodes the received frames. Since
the video packets could be corrupted or lost during transmission, error control and concealment
techniques at the codec may be applied to mitigate the impact of transmission errors. Then the
reconstructed video frames are played out on screens at the receiving device. Multimedia-aware
network protocols design has gained consideration for multimedia application support [27?33].
11
The Internet
Mobile UserVideo Content Transmitter Wireless Network
High speed wired link 
Figure 1.6: Overview of the wireless video networking system.
Motion 
Compensation
DCT 
Transform Quantilization
Entropy 
Coder
Inverse 
Quantilization
IDCT 
Transmform
Bit 
stream 
out
Motion 
Esitmation
Figure 1.7: Block diagram of a typical video encoder.
IDCT 
Transmform
Bit stream in
Inverse 
Quantilization
Entropy 
Decoder
Reconstruction & 
Interpolation
Figure 1.8: Block diagram of a typical video decoder.
During the video streaming process, energy is largely consumed by video codec encoding,
network transmissions, receiver decoding, error mitigation, and playout. The energy consumption
incurred at the encoder and decoder are mainly due to the processing of video data at the end
12
nodes, while encoding is usually more computation and energy intensive than decoding. The
codec computation and network transmission use up the largest part of the overall energy, and are
also critical components for the achievable QoE of video service [34]. It is important to address
the ?green? communication problem in video streaming by exploring the energy savings in both
video codec and video transmission.
1.3.2 Energy Efficiency in Video Coding
In the past decades, the advances of wireless communications and networking technologies
are much significant than that of the battery technology. Consequently, how to prolong the battery
life of mobile devices becomes one of the major environmental and economical concerns. We focus
on power efficient codec design in this section and will explore the energy efficient transmission
in the next section.
As the increasing demand of high quality video, high compression efficiency codecs are de-
signed to enable higher resolution, which significantly increases the complexity of encoding algo-
rithms. Moreover, the stringent delay requirements of video service usually keep the video device
processor constantly busy for managing the high computation tasks. The processor may consume
as much as 2/3 of the total power of a mobile device [35]. Thus it is important to balance video
quality and the computational complexity to achieve power-aware video coding.
Typical encoder and decoder block diagrams are illustrated in Fig. 1.7 and Fig. 1.8, respec-
tively. Such approaches have been adopted in many video coding standards, such as H.264,
and MPEG-4. The framework consists of motion estimation and compensation, DCT, quanti-
zation, entropy coding, inverse quantization, and IDCT. According to recent research [36], mo-
tion estimation/compensation constitutes more than 40% of the CPU workload, DCT/IDCT and
quantization/inverse-quantization makes up over 16% of the CPU workload, and the entropy en-
coder, whose computational complexity largely depends on the coding bit rate, composes less than
10% of the CPU workload. Thus, it is important to explore the energy efficiency of these compo-
nents that consume the most part of the processing power at a video codec.
13
There are many power-aware codec designs to balance power consumption and video quality.
The main idea behind these techniques is that the diverse complexity of the video content may
require different levels of compression. To achieve a certain video quality, slow motion and simple
scenes require much less computation than high motion sports and movie streams. Thus it is theo-
retically feasible to obtain the optimal power efficiency by dynamically adjusting the computation
complexity of the codec components for different videos, frames, macro blocks (MBs) and blocks,
while keeping the video quality relatively constant at a certain level.
In [37], the authors present a configurable coding scheme, which adjusts the codec control
parameters to achieve an optimal operation point on complexity-distortion curves based on ex-
haustive search and the Lagrangian multiplier method. In [38], a power-aware motion estimation
algorithm is presented, which is adaptive to the battery status by a content-based subsample algo-
rithm. When the battery is in the full capacity, all the processing elements in the motion estimation
function are turned on to provide the best quality. On the other hand, when the battery capacity
is decreased, some processing elements are disabled to extend the battery life with little quality
degradation. In [39], the authors extend the functions of DCT/IDCT in a framework to decrease
the power consumption by skipping the low energy MBs in DCT and all zero coefficients input
data in IDCT. The combined method reduces, on average, 94% of power dissipation.
Another class of power-aware video codec design aims to dynamically adjust the voltage and
frequency of the CPU for energy conservation. Various dynamic voltage scaling (DVS) algorithms
are provided to determine the minimum energy consumption for processing video tasks under
stringent delay requirements. With a DVS enabled processor, the voltage level and associated
clock frequency are adapted to the time-varying video processing workload to save energy. The
trade-off between reducing voltage level/clock frequency and increasing processing time is the
core in the DVS-based design.
In [40], the authors derive the optimal voltage scheduling with linear programming. The
algorithm calculates the optimal scheduling offline with knowledge of the precise complexity and
arrival time of each decoding job, which may not be easy to acquire in real time. A heuristic
14
algorithm is then introduced by predicting the stochastic complexity of the workloads. In [41],
a DVS algorithm is presented that adjusts both the clock frequency and the voltage level of the
CPU to achieve energy efficiency for video content processing, while maintaining the QoS of
the video. A comprehensive statistical analysis of the CPU workload is presented in [42] for
multimedia applications. The statistical results show that there is large room for DVS to reduce
energy consumption for multimedia streaming and the processor workload can also be accurately
predicted with a moderate effort. The DVS system is based on the control theoretic framework. A
PID-based DVS controller is developed to achieve a penalty controllable energy reduction, which
can be incorporated into an online algorithm.
In summary, the power-aware codec design focuses on the tension between video quality and
power consumption based on content diversity. The existing power-aware schemes extend the tra-
ditional video codec functions by jointly considering the video content and power constraint. The
algorithms aim to adjust the codec parameters to minimize the power consumption while preserv-
ing good video quality. In addition, the hardware support for DVS technologies enables adaptive
adjustment of the clock frequency and operating voltage level of the CPU, to accommodate varying
codec workload. It should be noted that the power-aware codec design needs to jointly adjust large
number of configurable parameters, which provides the context for applying effective globally
optimal techniques and algorithms.
1.3.3 Energy Efficiency in Video Transmission
A typical video transmission path is shown in Fig. 1.9. The video frames generated by the
codec are packetized and delivered through the network protocol stack (UDP/RTP/IP). The link
layer schedules the packets with a MAC protocol (e.g., TDMA, FDMA, CDMA, or CSMA/CA)
and passes the frames down to the physical layer, where channel coding, modulation and power
allocation may be applied to overcome the time-varying and unreliable wireless channels. At the
15
Application
Protocol stack
TCP/UDP/RTP
IP
Link
Physical
Codec design
Power-aware 
transmission
Power-aware 
routing
Scheduling
Modulation, 
Channel coding 
and Power control
System controller
Application
Protocol stack
TCP/UDP/RTP
IP
Link
Physical
Codec design
Power-aware 
transmission
Power-aware 
routing
Scheduling
Modulation, 
Channel coding 
and Power control
System controller
Figure 1.9: Protocol layers and system controls involved in wireless video transmission.
receiver, received video packets are decoded. The video bit stream is restored and then decom-
pressed by the decoder. Error concealment techniques may be applied to mitigate the impact of
delayed and corrupted video frames.
The key challenge of video over wireless networks is the time-varying wireless channel,
which has a gain that varies over time due to channel fading, shadowing, and inter channel in-
terference [43]. This causes random packet losses and delays. Thus, fixed resource allocation or
scheduling schemes may not be sufficient to achieve the best video quality or energy efficiency. An
adaptive, video content-aware resource allocation scheme is necessary for supporting energy effi-
cient video streaming over wireless networks. While it is important to increase the bandwidth and
throughput in wireless networks [44, 45], energy efficiency is also a critical factor for the success
of multimedia applications over wireless networks.
It is reported that BS equipment consumes more than 50% of the total power in a typical cel-
lular network. The energy efficiency and power control in cellular networks thus demand careful
reexamination to achieve the goal of green communications. Power control in cellular networks
has been widely studied for more than 15 years for voice or data applications. Many effective
power control algorithms are proposed in the literature (see [46?50]), and the closely related ad-
mission control problems are also investigated [51?53]. The problem of video communications in
wireless cellular networks brings about many new challenges to the BS power control problem,
16
since the power allocations need to achieve not only the target Signal to Interference-plus-Noise
Ratio (SINR), but also the QoS or QoE of the streamed videos [54?58].
1.3.4 Joint Video Coding and Transmission Design
As shown in Fig. 1.9, the wireless video system is a complex system with many closely
coupled control knobs and parameters. Clearly, a cross-layer design that jointly optimizes multiple
parameters in different layers has the potential of achieving better energy efficiency and video
performance, comparing to the traditional layered approach.
Consider power allocation as an example. Normally, a lower transmission rate requires a
smaller transmit power, as well as high compression ratio at the video codec. However, as dis-
cussed in the previous section, a higher compression ratio incurs more intensive computations and
consumes more power at the codec. Apparently, a cross-layer design framework would be useful
in this case, where video coding in the application layer and the transmission schemes in the lower
network layers are jointly considered and optimized [56,59]. Specifically, the lower layer network
protocols now have the opportunity to exploit the information from the video content and source
coding parameters to optimize the transmission strategy. On the other hand, video coding in the
application layer may also take advantage of the channel and network information, and thus can
select the coding parameters to provide the best coding quality and be adaptive to the status of wire-
less networks and channels. This approach also provides future support for the ?content-centric?
multimedia network design [60?63].
A large number of designs that jointly consider video coding and transmission have been
proposed in the literature. In [64], the authors adopt joint source coding and channel coding to
minimize the total power consumption, while keeping the end-to-end video quality at a fixed level.
A framework of joint source-channel coding and power adaptation is presented in [65], where error
resilient source coding , channel coding and transmission power adaptation are jointly designed to
optimize video quality, given constraints on the total transmission energy and delay. An algorithm
17
is introduced to find the minimal energy source coding and power allocation by adaptively allo-
cating resources to different video segments based on their relative importance. In [66], the RF
front-end circuit energy is controlled for wireless video transmission by adjusting parameters in
physical layer and MAC layer. In [67], the authors investigate the transmission over bandwidth-
limited multi-acess wireless uplink channel. Energy-efficient video communication is obtained by
jointly adapting video summarization, coding schemes, modulation schemes, and packet transmis-
sion. In [23],the authors present a framework for joint network optimization, source adaption and
deadline-driven scheduling for multi-user video streaming over wireless networks. Both the physi-
cal layer and application layer are jointly considered to maximize the total users? reception quality
under the power consumption constraint. In [68], the authors investigate the joint optimization
among source coding at application layers, ARQ scheme at data link layers and adaptive modula-
tion and channel coding at physical layer. Within the delay-distortion framework, the parameters
of above layers are jointly optimized to achieve the best quality of the received videos.
1.3.5 Video over Emerging Wireless Networks
The video streaming problem has also been considered for several emerging wireless net-
working paradigms. Visual sensor networks (VSN) also represent an important application of en-
ergy efficient multimedia networks. VSN consists of a large number of low-power camera nodes,
which integrate the image sensor, embedded processor, and wireless transceiver. The development
of VSN has brought about many potential applications, such as surveillance, environmental moni-
toring, smart homes/cities, and visual reality [69], to name a few. Due to the battery limitation, the
life time of VSN camera nodes is limited by their energy consumption in wireless channel sens-
ing, transmission, and video and image data processing. Energy efficiency is a critical issue in the
design of VSN nodes, since they may not be recharged as often as smart phones, and are expected
to operate over extended periods of time (e.g., on two AA batteries for one year [70]). Therefore
power efficient designs are highly preferable at all the protocol layers in VSNs.
18
Comprehensive surveys of VSNs can be found in [69,70]. It has been shown that power-aware
routing is highly effective in prolonging the lifetime of wireless sensor networks [71]. In [72,
73], the authors investigate the directional-control data fusion scheme to reduce the amount of
sensory data transmission in sensor networks. When processing video data is allowed within the
network, data fusion can be employed to reduce the redundancy among multiple video streams
along the routing path, thus reducing the volume of transmitted video data and saving energy at
the intermediate nodes. Power-aware transport layer designs are mainly based on de facto standard
of TCP. In [74, 75], the authors incorporate a new error-recovery mechanism into TCP to avoid
unnecessary retransmissions caused by AIMD, especially when the network is disconnected or
there are losses due to high bit error. This scheme is shown to prolong the lifetime of wireless
sensor networks.
In [76?79], the authors investigate the problem of video transmission over cognitive radio
networks, where secondary users sense the licensed channels and aim to exploit the transmission
opportunities in the spectrum holes. The uncertain channel availability condition brings about
many unique challenges. These works investigate the challenging problem of video over cognitive
radio networks with a cross-layer optimization approach, which leads to effective centralized or
distributed algorithm design with performance guarantees.
1.4 Key Contributions
In this dissertation, we address the problem of energy efficient design in CPS, including smart
grid and multimedia communication networks, with a control and optimization approach. The key
contributions are summarized as follows.
First, We explore the problem of smooth electric power scheduling in power distribution net-
works [80]. The smooth power profile greatly simplifies the strategy for balancing the supply and
demand in the grid. Moreover, since electricity generation and transmission systems are generally
designed to accommodate peak electric power [4], the smooth demand profile has the advantage of
19
optimizing the assets and operation cost of the grid. We introduce a deterministic model to char-
acterize the complex relationship between demand and supply. The deterministic model adopts
cumulative electricity demand/supply curves, which characterize the time varying demand/supply
relationship. A constrained nonlinear optimization problem is then formulated aiming to min-
imize the electric power variation, as well as satisfy users? power usage quality. We develop
majorization-based algorithms for deriving smooth power schedules for the networks. We also
design a distributed algorithm for supplying the power among the users. Although many existing
work reveals the intrinsic connection between pricing policies and demand response, few of the
existing work explicitly address the problem of smooth electric power scheduling. It is shown that
the simple off-peak pricing scheme may not be effective in mitigating the demand peak problem,
because simply shifting the off-peak period may generate a new rebound peak [81]. To the best
of our knowledge, this is the first work that directly addresses the smooth optimal energy schedul-
ing with majorization theory in power distribution networks. The solutions show the deterministic
performance for smooth power scheduling, peak power and operating cost reduction through the
enhancement of bidirectional communication flow, smart meters and smart facilities.
We next propose a comprehensive design of an energy management system in MG by taking
advantage of the plug-and-play interfaces of smart grid [82?84]. We jointly consider renewable
energy penetration, ESS management, residential demand management, and utility market partici-
pation in the MG and introduce the model of Quality of Service in Electricity (QoSE). The QoSE
concept takes into account minimization of the MG operation cost, while maintaining the power
usage quality of residents. We transform the QoSE control problem and ESS management problem
into queue stability problems by introducing the QoSE virtual queues and battery virtual queues.
The Lyapunov optimization method is applied to solve the problem and generate the online opti-
mal ESS charge/discharge algorithm, adaptive residential load service and cost effective operation
strategy on utility market, with hard performance bounds, which do not require any statistics and
future knowledge of the electricity supply, demand and price processes. The proposed policy ef-
fectively reduces the MG operation cost and maintains the QoSE for the residents. With this new
20
energy management framework, the MG achieves the fundamental requirements of smart grid in
DRERs integration, ESS?s management, residential power quality management, and maintains the
compatibility to the legacy grid.
Furthermore, we investigate the energy efficient design on the demand side as applications
in wireless communication and networks. Among various green communication technologies, we
focus on the energy efficiency of base stations for downlink video streaming. This is due to the
expected surge in wireless video data, as well as the drastic increase in the deployment of BS?s.
Therefore, any small improvement in the energy efficiency of wireless video streaming system will
be amplified by the huge volume of wireless video data and number of BS?s deployed, and will
result in considerable environmental impact.
Specifically, we design the energy efficient streaming for variable bit rate (VBR) video over
wireless networks. VBR video offers stable and superior quality over constant bit rate (CBR)
videos, however, the complexity statistics of the VBR video frames introduces great challenge
in wireless network design. We first present analytical frameworks for streaming multiple VBR
videos in a wireless cellular networks, where downlink capacities are limited by inter-cell/intracell
interference [55, 56, 58]. By jointly considering the deterministic model for VBR video traffic,
stringent playout delay constraint, BS peak power constraint, wireless channel uncertainty and fi-
nite playout buffer at the mobile users, we formulate the video streaming systems as nonlinear
optimization problems with the objective to maximize the throughput under the QoE and power
constraint. For the intracell interference situation, we analyze the convexity conditions of the
problem and propose a two-step approach to maximize the streaming transmission throughput un-
der power constraint, while maintaining the QoE. We also develop a distributed algorithm based
on the dual decomposition technique. The more challenging problem involved in intercell inter-
ference is solved by a centralized branch-and-bound algorithm incorporating the Reformulation-
Linearization Technique, which can produce optimal bounded solutions. We also propose a low-
complexity distributed algorithm with fast convergence. The proposed solutions effectively make
21
use of the power in BS?s to stream the VBR video in cellular networks, while preserving the QoE
requirement.
We further study the energy efficient downlink multi-user VBR streaming in the wireless cel-
lular networks with orthogonal channels by directly minimizing the power consumption to achieve
green multimedia communications [57,85?87]. We present a cross-layer optimization and schedul-
ing framework with the objective to minimize the BS?s power consumption during steaming period
while maintaining the QoE of video users. We develop a majorization-based solution approach to
solve the formulated problem. We prove the proposed algorithm is unique and global optimum,
and demonstrate that the proposed algorithm is also smoothness optimal. These research projects
may bring about a new paradigm for the design of future green wireless multimedia networks.
1.5 Overview of the Dissertation
In this dissertation work, we focus on the energy efficient design in CPS from two sides: elec-
tricity delivery networks and wireless multimedia networks, which are integrated by the method-
ology of control and optimization theoretic design as indicated in Fig. 1.10. The rest of the disser-
tation is organized as follows.
We present the smooth electric power scheduling in power distribution networks in Chapter 2.
We introduce an electricity supply/demand model that takes into account of time-varying demands
and their deadlines. We formulate a constraint nonlinear optimization problem and incorporate
the theory of majorization to develop algorithms that can compute smoothness optimal schedules.
After the smooth power schedule is obtained, a distributed user benefit maximization load control
scheme is used to allocate the scheduled power to individual users, while maximizing their level
of satisfaction. We demonstrate the efficacy of the proposed algorithms by extensive simulations.
In Chapter 3, we propose a smart energy management framework in MG based on the concept
of QoSE. The MGCC aims to minimize the MG operation cost and maintain the outage probability
of quality usage, i.e., QoSE, below a target value, by scheduling electricity among renewable
energy sources, energy storage systems, and macrogrid. We formulate the problem to a constrained
22
Control Optimization
Cyber-Physical 
Systems
Energy 
Efficiency
Smart Grid ICT
Chapter 2. Smooth Power 
Scheduling in Power 
Distribution Networks
Chapter 3. Adaptive 
Electricity Scheduling in 
Microgrids
Chapter 5. Downlink 
Power Allocation for VBR 
Video in Cellular Network
Chapter 6. Downlink 
Power Control for VBR 
Video over Multicell 
Networks 
Chapter 7. Energy 
Efficiency on Downlink 
Multiuser VBR Video 
Streaming
MajorizationLyapunov Optimization Linear Optimization
RLT Branch and BoundNonlinear OptimizationConvexity
Decomposition
Figure 1.10: Overview of the dissertation.
stochastic programming problem and apply Lyapunov optimization technique to derive an adaptive
electricity scheduling algorithm by introducing the QoSE virtual queues and energy storage virtual
queues. We derive several hard performance bounds for the proposed algorithm and evaluate its
performance with trace-driven simulations.
In Chapter 5, we shift the energy efficient design from electricity delivery networks to the
demand side. We investigate the power control for the downlink VBR video streaming in the
cellular networks with intracell interference. We consider a deterministic model for VBR video
traffic and finite playout buffer at the mobile users. The objective is to derive the optimal downlink
power allocation for the VBR video sessions, such that the video data can be delivered in a timely
fashion without causing playout buffer overflow and underflow. The formulated problem is a non-
linear nonconvex optimization problem. We analyze the convexity conditions for the formulated
problem and propose a two-step approach to solve the problem. We also develop a distributed al-
gorithm based on the dual decomposition technique. The performance of the proposed algorithms
are validated with simulations using VBR video traces under realistic scenarios.
23
In Chapter 6, we study the more challenging problem of power control for streaming multiple
VBR videos in multicell wireless networks under the intercell interference condition and derived
both centralized and distributed algorithm for the solutions. The problem is formulated to find the
optimal transmit powers for the base stations, such that VBR video data can be delivered to mobile
users without causing playout buffer underflow or overflow. We formulate a nonlinear noncon-
vex optimization problem and prove the condition for the existence of feasible solutions. We then
develop a centralized branch-and-bound algorithm incorporating the Reformulation-Linearization
Technique, which can produce (1 ??)-optimal solutions. We also propose a low-complexity dis-
tributed algorithm with fast convergence. Through simulations with VBR video traces under fad-
ing channels, we find the distributed algorithm can achieve a performance very close to that of the
centralized algorithm.
In Chapter 7, we relax the channel constraint in the cellular wireless networks to the orthog-
onal channels and directly address power minimization strategy for multiuser VBR video stream-
ing. We also adopt a deterministic model for VBR video traffic that incorporates video frame and
playout buffer characteristics, and formulate a constrained stochastic optimization problem. We
then develop a majorization-based solution approach. For the case of a single VBR video session
with relaxed peak power constraint, we develop a power optimal algorithm with low complexity.
We prove the power optimality of the proposed algorithm and the uniqueness of the global opti-
mum, and demonstrate that the proposed algorithm is also smoothness optimal. For the case of
multiuser VBR video streaming, we develop a heuristic algorithm that selectively suspends some
video sessions when the peak power constraint is violated. In addition to the traditional VBR video
streaming application, we also consider the case of interactive video streaming, and show that the
proposed schemes can be easily adapted and applied.
We conclude the dissertation and present the future work in Chapter 8.
24
Chapter 2
Smooth Electric Power Scheduling in Power Distribution Networks
2.1 Introduction
The emergence of Smart Grid (SG) brings about many fundamental changes in electric power
systems [4]. Various new power electronics and information techniques are greatly advancing the
control and management of energy and resources in the power system. For example, solid state
transformers (SST) can respond to signals from a facility or a household to change the voltage
and other electric characteristics. On the user side, smart meters and smart facilities empower
the pervasive monitoring and controlling at all levels of power usage in response to power supply
and market price fluctuations [4]. The two-way flows of electricity and information in SG are
instrumental to the control and optimization of energy and resource allocation in the grid to achieve
efficient, green and robust energy systems.
Unlike the traditional grid, in which the electricity supply continuously matches user de-
mands, the next generation power distribution system is based on a network structure [88] and is
capable of allowing users to control their loads in response to the dynamics in the grid. Demand
response (DR) is a technique to balance power generation and demand in the grid [89]. One of
the important targets of DR is to reduce the peak demand by scheduling user requests. With the
two-way information flow among provider, users and the market, various DR schemes based on
real-time pricing and day-ahead load response concepts have been investigated recently [90?93].
Most of the existing DR schemes aim to maximize the social welfare or minimize the electricity
payment under given demand requirements. Although revealing the intrinsic connection between
pricing policies and demand response, the problem of smooth electric power scheduling is not ex-
plicitly addressed, although being the key issue in DR. It is shown that the simple off-peak pricing
25
scheme may not be effective in mitigating the demand peak problem, because simply shifting the
off-peak period may generate a new rebound peak [81].
In this chapter, we address the challenging problem of smooth electric power scheduling in
power distribution networks. The network model is shown in Fig. 2.1. We assume the end-users are
equipped with smart meters and are capable of communicating with the distribution substation and
the distribution control center (DCC) through a communication network, and receiving commands
from the DCC to adjust the user?s electric energy consumption level [3]. The DCC schedules
electricity supply on daily basis, which is further divided into multiple time slots. The electricity
usage requests at each user are classified into two categories: the priority load that must be satisfied
in every time slot, and the deferrable load that should be satisfied before specific deadlines. Users
may set the load for each type according to their preference (e.g., lighting, entertainment, laundry,
or charging a plug-in hybrid electric vehicle (PHEV)) [89]. The DCC aggregates the demand
profiles from the users through the aggregator [94] and smooths the aggregated electric power
supply under the priority load and deferrable load deadline constraints.
In the smooth electric power scheduling problem, the objective is to minimize the power vari-
ation during a daily period, based on the concept of day-ahead load response. A deterministic
electricity supply/demand model is introduced with cumulative electricity demand/supply curves,
which characterize the demand/supply relationship during the day. We find the formulated prob-
lem suits well with the majorization theory, which concerns with the comparison and ordering of
vectors with respect to the distribution of their elements [13]. Majorization has been used in solv-
ing optimization problems in the communications and networking area [57,95,96]. In this chapter,
we present a majorization-based framework to develop two smooth electric power scheduling al-
gorithms with low computational complexity. After the smooth electric power profile for the entire
network is obtained, a user benefit maximization load control algorithm will be executed to allocate
the total amount of supply to the individual users, while maximizing their satisfaction of electricity
usage. The proposed algorithms can achieve the minimum peak power, thus requiring smaller ca-
pacity for the generators, transmission lines and transformers to support the same demand. Since
26
Power 
Generator
Distribution 
Control Center
Distribution 
Substation
Power line Network switch Information networkCircuit breaker
Figure 2.1: Illustration of the electricity distribution network.
electrical generation and transmission systems are generally designed to accommodate peak elec-
tric power [4], the smooth electric power schedule has the potential of optimizing the deployment
and operation cost of the grid.
The remainder of this chapter is organized as follows. We first present the system model
and problem statement in Section 2.2. The smooth electric power scheduling algorithms are de-
scribed in Section 2.4 and their performance evaluated in Section 2.5. Related work is discussed
in Section 2.6 and Section 2.7 concludes this chapter.
The notations used in this chapter are summarized in Table 2.1.
2.2 Problem Statement
2.2.1 Load Demand Profile
We consider a power distribution network with two-way flows of electricity and information.
We assumeN users in the power distribution network, which may generate residential, commercial
27
Table 2.1: Notation Table for Chapter 2
Symbol Description
R set of the electricity consumption users in the power
distribution networks
N total number of local electricity consumption user
L total number of slots
? slot period
pn(t) power consumption of user n in slot t
en,p(t) priority load electric energy of user n in slot t
en,d(t) deferrable load electric energy need to be fulfilled
since last deadline of user n in slot t
pmaxn (t) maximum power consumption of user n in slot t
pminn (t) minimum power consumption of user n in slot t
en(t) electricity usage of user n in slot t
emaxn (t) maximum electricity usage of user n in slot t
eminn (t) minimum electricity usage of user n in slot t
En total electricity usage during L slots for user n
Emax(t) maximum electricity usage for the residential area in slot t
Emin(t) minimum electricity usage for the residential area in slot t
E(t) scheduled electricity usage for the residential area in slot t
vectorWmax cumulative upper bound of electricity usage during L slots
vectorWmin cumulative lower bound of electricity usage during L slots
vectorW cumulative scheduled electricity usage during L slots
? total electricity usage for all users during L slots
vectorP feasible power supply scheduling during L slots
vectorP? smooth optimal power supply scheduling during L slots
Un(?) utility function for user n
h(t) electricity price at time slot t
? Lagrange multiplier
L Lagrange function
?(l) stepsize of step l in equation (2.7)
?(l) stepsize of step l in equation (2.8)
and industrial loads. Let R = {1,2,??? ,N} be the set of users. The electric demand of a user is
daily based. Without loss of generality, we assume the one day period is divided into L time slots,
each with length ?. Let pn(t) be the power consumption of user n in time slot t, which is time
varying but remains constant within the time slot. Each user n knows its own total daily demand,
i.e., En =summationtextLt=1pn(t)?, and wishes to schedule the demand over the one day period [91].
28
We assume the total demand En consists of two parts: the priority load and the deferrable
load. The priority load should be strictly guaranteed in a time slot (e.g., for lighting), while the
deferrable load can be served flexibly but with a specific deadline (e.g., charging a household
battery or PHEVs). We define en,p(t) and en,d(t) as the electric energy for priority load in time slot
t, and the deferrable load that must be satisfied by time slot t, respectively. The minimum demand
of user n in time slot t, denoted by eminn (t), is the sum of en,p(t) and en,d(t). Finally, let emaxn (t) be
the maximum possible demand for user n, which is limited by the amount of deferrable loads that
have not been satisfied yet, and the capacity of protective relays and switches of the users.
2.2.2 Cumulative Demand and Supply Curves
At the beginning of a day, the DCC will aggregate the individual demand profiles received
by communicating with the smart meters and smart facilities via the communications network [3].
Let the total minimum electricity demand in time slot t be Emin(t) = summationtextn?Reminn (t). We have
Emin(L) = summationtextn?REn = ?, since the daily aggregated demand of all users, denoted by ?, should
finally be satisfied by the end of the day. We define the cumulative minimum demand curve vectorWmin
as
Wmin(t) =
tsummationdisplay
l=1
Emin(l),1 ?t?L. (2.1)
We define the cumulative maximum demand curve vectorWmax to represent the maximum amount of
electricity demand that can be consumed up to t as
Wmax(t) = min{Wmin(t?1) +
summationdisplay
n?R
[emaxn (t) + ?en(t?1)],?},1 ?t?L
where ?en(t) = emaxn (t)?eminn (t) is the deferrable load that can be served in slot t but with dead-
lines later than t. To incorporate the priority load in the model, Wmax(t) also satisfies Wmax(t) ?
Wmax(t?1) +summationtextn?Ren,p(t).
For given demand curves vectorWmin and vectorWmax, we aim to find a feasible electricity schedule
vectorW, which is the cumulative supply of electricity to the users that satisfies constraints Wmin(t) ?
29
Time
kWh
0
Infeasible supply schedule ?
electricity outage
Infeasible supply schedule 
?electricity wasted
Feasible supply schedule 
W    (t)? W(t) ? W     (t)
Cumulative  min demand 
curve W
Cumulative max demand curve W
Hard guaranteed demand 
Slot interval ?
max
min
maxmin
Deferrable demand
Figure 2.2: Cumulative demand and supply curves.
W(t) ? Wmax(t), for all 1 ? t ? L, and W(L) = ? (i.e., the total demand should be satisfied
by the end of the day). The three cumulative curves are illustrated in Fig. 2.2, which are all
nondecreasing over time.
The proposed demand and supply model is quite general. It does not assume any mathemat-
ical model for either the supply or the demand. It is more practical than the complex statistical
models for supply and demand used in the literature [4]. The cumulative curves represent the
demand/supply status in the power distribution network. In each time slot t, Wmin(t) tracks the
priority load and the deferrable load with deadline t, while Wmax(t) represents an upper bound of
the possible consumption by timet. The gap betweenWmax(t) andWmin(t) may accommodate the
future uncertainty of the electric power usage. The slope of W(t), denoted by P(t), corresponds to
the scheduled electric power. The DCC aims to find an optimal schedule W(t) for every time slot t
to achieve a specific control target. A feasible power supply schedule vectorP = [P(1),P(2),??? ,P(L)]
ensures that vectorW lies between vectorWmin and vectorWmax for all the L time slots, thus preventing both outage
events and energy waste.
30
It can be seen from Fig. 2.2 that the feasible electric power schedule may not be unique.
Among various feasible schedules, we are interested in the one that distributes electricity most
smoothly among the L time slots, i.e., the smoothness optimal schedule. Once the DCC obtains
the smoothness optimal schedule, it can announce the schedule to the smart meters and smart util-
ities at the users? premises via the communication network, and the users can shape their demand
to match the schedule (assuming cooperative users). Therefore, we can achieve smooth electric-
ity generation, transmission and consumption, which is highly preferable for the grid design and
operation [4].
2.2.3 Smooth Power Scheduling Problem
Based on the demand and supply model, we formulate the smooth power scheduling problem
in this section. Let ?P = ?/(L?) be the average power consumption in the power distribution
network through the daily period. The scheduled power for each time slot is P(t) = W(t)/?.
The smoothness optimal schedule minimizes the variations of the supplied power over the entire
period, i.e.,
maximize: S(vectorP) (2.2)
subject to: Wmin(t) ?W(t) ?Wmax(t), for all t
W(L) = Wmin(L) = Wmax(L) = ?
P(t) ?Ep(t)/?, for all t,
where S(vectorP) is the smoothness of a schedule vectorP, Ep(t) = summationtextn?Ren,p(t) is the total priority load in
time slot t.
Generally, smoothness can be measured by different metrics, such as variance, cumulative
absolute difference, etc. Each smoothness measure leads to a different objective function in prob-
lem (2.2), while the solution to the problem will then depend on the specific form of the objective
31
function. In addition, the smoothness measures are generally nonlinear, making the problem non-
trivial to solve. In this chapter, we resort to a mathematic theory of majorization [13], which
explicitly addresses the unique mathematical notion for smoothness. Applying majorization the-
ory, we will see that for an arbitrary smoothness objective function in problem (2.2) that satisfies
the Schur-convex properties [13], the problem can be solved by a universal algorithm in polyno-
mial time. For brevity in the deduction, we minimize variance in the rest of the chapter, while the
solution algorithms developed in Section 2.4 apply to any objective function that is Schur-convex.
We first consider the case where deferrable load is the dominant component [97], i.e. Ep(t) ?
0. Problem (2.2) is then reduced to problem (2.3).
minimize: summationtextLt=1bracketleftbigP(t)? ?Pbracketrightbig2/L (2.3)
subject to: Wmin(t) ?W(t) ?Wmax(t), for all t
W(L) = ?
P(t) ? 0, for all t.
This problem fits well with the majorization theory, since the objective function is Schur-
convex [13]. We briefly review Majorization preliminary in Section 2.3. Applying majorization,
we will design a smooth electric power scheduling algorithm for solving problem (2.3) in Sec-
tion 2.4.1. We will then extend the algorithm for solving problem (2.2) in Section 2.4.3.
2.3 Majorization Preliminaries
Majorization theory [13] describes the ?less spread out? or ?more nearly equal? properties of
the elements of a vector comparing to the elements of another vector. It concerns with the problem
of ordering vectors with nonnegative, real elements, as well as order-preserving functions. For
simplicity, all the vectors in this section are row vectors.
Definition 2.1. For two n-dimensional vectors vectorX = (x1,x2,??? ,xn) and vectorY = (y1,y2,??? ,yn),
with elements sorted in the non-increasing order as x1 ? x2 ? ??? ? xn ? 0 and y1 ? y2 ?
32
??? ? yn ? 0. vectorX is said to be majorized by vectorY, denoted as vectorX ? vectorY, if (i) summationtextti=1xi ? summationtextti=1yi,
t = 1,2,??? ,n?1, and (ii)summationtextni=1xi =summationtextni=1yi [13].
Definition 2.2. A real-valued function ? defined on a set A?Rn is said to be Schur-convex on A
if vectorX ? vectorY on A??(vectorX) ??(vectorY) [13].
Schur-convex functions have the ?order-preserving? property, which bridges majorization to
optimization. Schur-convex functions can be validated with the following fact.
Fact 2.1. If ? is symmetric and convex, then ? is Schur-convex. Consequently, vectorX ? vectorY implies
?(vectorX) ??(vectorY) [13].
This fact provides connection between ordering and its order-preserving functions. By this
fact, we may solve the minimization optimization problem by generating the most possible ?spread
out? vector as the solution. If ? = summationtextg and g is continuous convex, then we have the following
strong fact:
Fact 2.2. summationtextg(xi) ? summationtextg(yi) ? vectorX ? vectorY holds for all continuous convex function g : R ? R
[13].
Lemma 2.1. Let vectorX = (vectorX1,??? , vectorXK), and vectorY = (vectorY1,??? ,vectorYK), where each element has dimension
Ji and satisfying vectorXi ? vectorYi. Then vectorX ? vectorY.
Proof. Let g be the continuous convex function R ? R. By Fact 2.2, vectorXi ? vectorYi ?summationtextJij=1g(xji) ?
summationtextJi
j=1g(y
j
i) ?
summationtextK
i=1
summationtextJi
j=1g(x
j
i) ?
summationtextK
i=1
summationtextJi
j=1g(y
j
i) ? vectorX ? vectorY.
Observation 2.0.1. Let vectorX = (?x,??? ,?x),vectorY = (y1,??? ,yn),vectorZ = (z1,??? ,zn), where summationtextni=1yi =
summationtextn
i=1zi = n?x. If the elements in each vector is non decreasing, we may plot the normalized
points of vectorX/(n?x),vectorY/(n?x) and vectorZ/(n?x) on Fig. 2.3. If we explain the element of vectors as the
income of individual, this is Lorenz Curves, which evaluate the social income inequality [98]. The
curves show the normalized cumulative proportion of the income versus the cumulative percentage
of population. The normalized vectorX forms the straight curve A, which corresponds to the equal
33
0
C (Z)
B (Y)A (X)
B'
C'
Cu
mu
lat
ive
 pr
op
ort
ion
 of
 
inc
om
e
Cumulative proportion 
of population
1
1
Figure 2.3: Lorenz Curves.
distribution. Normalized vectorY and normalized vectorZ represent the unequal distribution and bent in the
middle, and are denoted curve B and C, respectively. We call these bow curves in the convex
shape. By the fact that the Lorenz curves are bent more, concentration increases, the bow curve
inside represents more even distribution [13], which leads to vectorX ? vectorY ? vectorZ. Similarly, we may
mutate the B and C to B? and C? by only changing the order of the elements. We call these
bow curves in the concave shape. Since the order of the vectors plays no role in majorization,
vectorX ? vectorY ? vectorZ still holds for concave shape.
Theorem 2.1. The objective function of problems (2.2) and (2.3) is Schur-convex.
Proof. The proof follows the Fact 2.1 straightforward, due to the symmetric, increasing and convex
of the objective function in problem (2.2) and (2.3).
2.4 Smooth Electric Power Scheduling
2.4.1 SEPS-DL Algorithm
We first develop a smooth electric power scheduling for deferrable load algorithm (SEPS-
DL) based on majorization. With Theorem 2.1, we convert the optimization problem (2.3) into an
ordering problem of vectors, each representing a feasible schedule. Thus, we solve problem (2.3)
34
by finding the most evenly distributed electric power schedule that is feasible for the entire period.
Obviously, the most evenly distributed schedule is vectorPopt = [?/(L?),??? ,?/(L?)], corresponding
to having the average power consumption ?P in each time slot. However, due to time varying user
demands, vectorPopt may not be feasible. In general, each feasible schedule is piece-wise linear with a
set of power changing points, where the scheduled power increases or decreases to prevent outage
events or electric energy waste.
The proposed SEPS-DL algorithm can generate a feasible piece-wise linear schedule, which
keeps each piece as long as possible into the future and keeps the power variation as small as
possible. The algorithm is illustrated in Fig. 2.4. Starting from tstart, SEPS-DL first computes two
probe lines:
? One probe line from tstart to the next corner point of Wmax(t), which can go the furthest
into the future without causing outage events or energy waste (e.g., lines P1P2 in Case 1 and
P5P6 in Case 2 of Fig. 2.4). The power of this probe line is Pmax(t) = Wmax(t)?W(tstart)t?tstart .
? The other probe line fromtstart to the next corner point ofWmin(t), which can go the furthest
into the future without causing outage events or energy waste (e.g., lines P1P3 in Case 1 and
P5P7 in Case 2). The power of this probe line is Pmin(t) = Wmin(t)?W(tstart)t?tstart .
All feasible schedules should reside between the two probe lines in order to go farther into the
future (i.e., to be smooth). Moreover, when the two probe lines are ended, they must hit both on
either Wmax(t) or both on Wmin(t). Otherwise, we can always adjust one of the probe lines to
make it go even further into the future. For example, see lines P1P3 and P1P?4 in Case 1 of Fig. 2.4.
We can use lineP1P2 (which goes farther into the future) to replace lineP1P?4, and both probe lines
hit Wmin(t) eventually. In Case 2 in the figure, both probe lines P5P6 and P5P7 hit Wmax(t).
If both probe lines hit Wmin(t) (i.e., Case 1 in Fig. 2.4), any feasible schedule for this interval
will also hit Wmin(t), since it must lie between the two probe lines. We then trace back the upper
probe line (i.e., line P1P2) to find the latest time when the schedule just satisfies the maximum
35
Time
kWh
Time
Case 1 Case 2
P1
P2
P3
P4
P5
P6
P8
P7kWh
Wmax(t)
Wmin(t)
P'4
P'8
tstart tstop tstart tstop
Wmin(t)
Wmax(t)
Figure 2.4: Illustrate the operation of the SEPS-DL algorithm.
demand (i.e., point P4 at time tstop). Then segment P1P4 will be chosen as the schedule for the
interval [tstart, tstop), with power Wmax(tstop)?W(tstart)tstop?tstart .
If both probe lines hit Wmax(t) (i.e., Case 2 in Fig. 2.4), any feasible schedule for this interval
will also hitWmax(t). We then trace back the lower probe line (i.e., lineP5P7) to find the latest time
when the schedule just satisfies the minimum demand (i.e., point P8 at time tstop). Then segment
P5P8 will be chosen as the schedule for the interval [tstart, tstop), with power Wmin(tstop)?W(tstart)tstop?tstart .
After the schedule for [tstart,tstop) is determined, we set tstart = tstop and repeat the above
procedure to find the schedule for the next time interval.
As shown in Algorithm 1, SEPS-DL probes for the longest feasible power starting from tstart
in Steps 4?10. In Steps 11?14, the power for the interval [tstart,tstop) is determined depending on
which of the two cases it is as illustrated in Fig. 2.4. Steps 16?17 are for the case that the power
does not change in the time slot. Step 19 resets the variables to start the computation for the next
segment of P(t).
2.4.2 Performance of SEPS-DL
The proposed SEPS-DL algorithm is very easy to implement. It can be shown that SEPS-DL
has the following properties.
Theorem 2.2. SEPS-DL is smoothness optimal.
Proof. See Appendix A.1.
36
Algorithm 1: Smooth Electric Power Scheduling for Deferrable Load Demand
1 DCC aggregates the demand from all the users by information networks and calculates
vectorWmin, vectorWmax for the whole scheduling period ;
2 t = 1,tstart = 0,tstop = tc1 = tc2 = 1,Pmin = 0,Pmax = ? ;
3 while some time slots are not scheduled do
4 Calculate Pmax(t) and Pmin(t) over interval [tstart,t] ;
5 if Pmin ?Pmin(t) & Pmin(t) ? min{Pmax,Pmax(t)} then
6 Pmin = Pmin(t) and tc1 = t ;
7 end
8 if Pmax ?Pmax(t) & Pmax(t) ? max{Pmin,Pmin(t)} then
9 Pmax = Pmax(t) and tc2 = t ;
10 end
11 if Pmin > min{Pmax,Pmax(t)} then
12 Select Pmin from tstart to tstop = tc1 ;
13 else if Pmax < max{Pmin,Pmin(t)} then
14 Select Pmax from tstart to tstop = tc2 ;
15 else
16 t+ + ;
17 CONTINUE ;
18 end
19 tstart = tstop,tstop = tc1 = tc2 = tstart + 1,t = tstart + 1,Pmin = 0, Pmax = ? ;
20 end
Corollary 2.2.1. The optimal power schedule is unique.
Proof. Suppose vectorP? is not unique. Then there exists vectorP? ? vectorPk, for all k, and vectorP? negationslash= vectorP?. vectorP? must have
a different set of power changing points from that of vectorP?. According to the proof of Theorem 2.2, we
can construct an auxiliary schedule vectorP1, such that vectorP? ? vectorP1 ? vectorP?, which contradicts the assumption
that vectorP? is optimal.
Theorem 2.3. The complexity of SEPS-DL is O(L2).
Proof. In the worst case, the SEPS-DL algorithm computes the optimal schedule for each time slot
by probing the full length of the remaining power sequence (as in Steps 4?10 in Algorithm 1). The
worst case execution time issummationtext1i=Li= L(L+1)2 ?O(L2).
Theorem 2.4. The smooth electric power schedule computed by SEPS-DL has the smallest peak
power.
37
Proof. From Theorem 2.2, we have vectorP? ? vectorPk, for all k. We may reorder the elements in both vectorP?
and vectorPk to a non-increasing order. According to the definition of Majorization in Definition 2.3, the
first element in the re-ordered vectorP? is not greater than that of vectorPk, which means the largest element in
vectorP? is not greater than that of vectorPk. Thus, the schedule generated by SEPS-DL has the lowest peak
power.
Corollary 2.4.1. The SEPS-DL achieves highest load factor.
Proof. The load factor in electric power grid is defined as in [99]
Load factor(%) = AveragePowerPeakPower ?100%
By Theorem 2.4, the SEPS-DL generates the lowest peak power, which achieves the highest load
factor during the scheduled period.
Theorem 2.4 and Corollary 2.4.1 is highly preferable for grid design and operation. A lower
peak power allows the operator to deploy generators, transformers and power transmission lines
with smaller capacity in the grid, thus reducing the capital investment. In addition, the grid may
be alleviated of the power usage burden during peak hours, and the electric energy usage quality
of users can be improved.
Theorem 2.5. SEPS-DL is generation operating cost optimal.
Proof. See Appendix A.2.
2.4.3 Extension to the General Case
We next extend SEPS-DL to solve the general case problem (2.2). With the priority load,
a feasible power schedule should satisfy P(t) ? Ep(t)/? in every time slot. The presence of
priority load enforces new constraints on the feasibility of the schedules. For example, consider
the aggregated cumulative priority load curves and deferrable load curves shown in Fig. 2.5(a)
38
kWh
0
t1 t3t2
(a) Cumulative priority load curves
kWh
t1 t3t2
(b) Cumulative deferrable load
curves
kWh
t1 t3t2
1
2
3
(c) Aggregated cumulative load
curves
Figure 2.5: Smooth power scheduling with priority load. (Note that although not deferrable, the
cumulative priority load can also be represented with two curves as shown in (a), where the maxi-
mum and minimum curves meet at the corners.)
and Fig. 2.5(b). According to the definition of feasible power supply schedule in SEPS-DL, both
segments 1 and 2 in Fig. 2.5(c) are feasible. However, it can be seen that segment 1 actually cannot
provide enough power to satisfy the priority load in time slot t2, since its slope is smaller than the
required slope (i.e., that of segment 0 in Fig. 2.5(a)).
To solve this problem, we develop the general smooth electric power scheduling algorithm
(GSEPS), which is based on SEPS-DL. Specifically, SEPS-DL assumes no priority load. During
the execution, the generated power segment is compared with the priority load. If the SEPS-DL
generated power segment P?(t) is less than the priority load in time slot t, GSEPS will increase
P?(t) to the priority load (e.g., see segment 3 in Fig. 2.5). Then SEPS-DL will continue to compute
further segments of the schedule by setting the new starting point to t, until the entire schedule is
computed.
2.4.4 Electric Power Allocation Among Individual Users
After the smooth electric power schedule is obtained, the DCC announces the schedule to all
the users and requests them to control their loads to match the supplied electric energy W?(t) =
P?(t)? in each time slot t. To divide the total supply W?(t) among the N users, we assume a
benefit function Un(pn(t)) for each user n, which is a nondecreasing concave function [92] and
39
represents the level of satisfaction of the user when receiving pn(t) in time slot t. We then develop
an algorithm that maximizes the sum of the benefit functions of all users in the power distribu-
tion network. The maximization of the total benefit under the smooth schedule constraint can be
formulated in each time slot t as follows:
maximize: summationtextn?RUn(pn(t)) (2.4)
subject to: pminn (t) ?pn(t) ?pmaxn (t), for all n
summationtext
n?Rpn(t) = W
?(t)/?,
where pmaxn (t) and pminn (t) are the maximum and minimum power consumptions of user n in slot
t, respectively.
Problem (2.4) is a convex optimization problem, which can be solved effectively with a convex
optimization solver. In case that DCC may not know the exact parameters of individual utility
functions in practice, we develop a distributed user benefit maximization load control algorithm
(DUBMLC) based on dual decomposition [100] to solve problem (2.4). For brevity, we omit the
time slot notation t in following equations.
First, we introduce the non-negative Lagrange multiplier ?, and derive the Lagrange function:
L(pn,?) =
summationdisplay
n?R
Un(pn) +?
parenleftBigg
W ?
summationdisplay
n?R
pn
parenrightBigg
(2.5)
=
summationdisplay
n?R
Ln(pn,?) +?W,
whereLn(pn,?) = Un(pn)??pn. Observing thatLn(?) only depends on usern?s local information,
we have the dual decomposition for each user n. Each user solves subproblem (2.6) for given
Lagrange multipliers ??:
pn(??) = argmaxpminn ?pn?pmaxn Ln(pn,??), for all n. (2.6)
40
Subproblem (2.6) can be solved with the subgradient method [14], sinceLn is strictly concave.
User n iteratively updates its power pn until pn converges, as:
pn(l+ 1) =
bracketleftbigg
pn(l) +?(l)? ?Ln(pn)?p
n
bracketrightbigg+
(2.7)
where [?]+ denotes the projection of pn onto the range [pminn ,pmaxn ], and ?(l) is the step size varies
in each step l according to the Armijo Rule [14]. The solution pn can be solved locally by the users
and converges to the optimal solution of ?pn for all n as l??.
For a given optimal solution ?pn, the master dual problem is to solved by DCC:
minimize: L(?pn,?),
subject to: ? ? 0, for all n.
We can also apply the subgradient method to iteratively update the multipliers as:
?(l+ 1) = max
braceleftbigg
?(l)??(l)? ?L(?)?? ,0
bracerightbigg
, (2.8)
where ?(l) is the step size. The Lagrange multipliers converges to the optimal as l ? ?, since
problem (2.4) is a convex problem, the duality gap is zero and the solution of (2.6) is unique. The
primal variable pn will also converge to the optimal solutions [100].
The distributed user benefit maximization load control (DUBMLC) algorithm is presented
in Algorithm 2. With DUBMLC, each user greedily maximizes its own benefit by solving (2.6)
with current ?price? ?, which is controlled by DCC through the master due problem (2.8). Due to
convexity of the problem, the optimization gap is zero and the optimal total maximum benefit is
reached when the algorithm converges.
Combining of GSEPS and DUBMLC, we now present the General Smooth Electric Power
Scheduling Policy. Specifically, at the beginning of each period, which can be daily based or
be an arbitrary period of time, the users send their slotted demand profiles to the DCC through
41
Algorithm 2: Distributed User Benefit Maximization Load Control Algorithm
1 l = 0 and the DCC initializes nonnegative parameter ?(l) ;
2 The DCC announces the parameters to the users via the communications network ;
3 Each user locally solves problem (2.6) as in (2.7) to obtain its requested power ;
4 Each user sends its requested power to the DCC via information networks ;
5 The DCC updates the parameters ?(l) as in (2.8) and announces the new value ?(l+ 1) to
all users ;
6 l = l+ 1 and go to Step 3, until the solution converges ;
the communications network. After aggregating all the demand profiles, the DCC calculates the
deterministic cumulative supply/demand curves for the power distribution networks and executes
GSEPS to compute the smooth power profile. After that, DCC let the users to control their elec-
tricity usage to match the smooth schedule with DUBMLC. The load control does not necessarily
need to be fully executed at the exact beginning of the period. It may operate at some time ahead
of the scheduled time slot. If a user requests a usage exceeding the planned level, the DCC may
allow the distribution substation to temporally fulfill the excess demand but charging a penalty
price based on the electric energy availability of the power distribution network.
2.5 Simulation Evaluation
In this section, we evaluate the proposed algorithms by simulating an electric power distri-
bution network with 250 independent users. We assume a daily period slotted into L = 144
time slots (i.e., ? = 10 min). The demand for each user during the period is randomly dis-
tributed from 35 kWh to 50 kWh. The DCC aggregates the load profiles and generates the
cumulative supply/demand curves at the beginning of the period. We adopt a benefit function
Un(t) = k1qn(t) ? 12k2qn(t)2 [101], where qn(t) ? [0,1] is the normalized value of power sup-
ply pn(t). With this Un(?), problem (2.4) becomes a quadratic programming problem, which can
be effectively solved with the proposed distributed algorithm. Without loss of generality, we set
k1 = k2 = 1 in the simulations.
We first examine the performance of SEPS-DL and GSEPS. For SEPS-DL, all the electric
energy demand is deferrable. For GSEPS, we assume 50% of the demand is deferrable and the
42
0 20 40 60 80 100 120 1400
2000
4000
6000
8000
10000
12000
Time
Cumulative electric energy (kWh)
 
 
Cumulative min demand curve of the grid
Cumulative supply curve of SEPS?DL
Cumulative supply curve of GSEPS
Cumulative max demand curve of the grid
Figure 2.6: Comparison of the power schedules achieved by SEPS-DL an GSEPS.
deadlines are randomly distributed during the daily period. The cumulative demand curves and the
computed schedules are plotted in Fig. 2.6. We find that both electric power schedules lie between
vectorWmin and vectorWmax, meaning they are feasible and satisfying the user demands in the entire period.
In some time slots, e.g., slots [70, 80] and [110, 120], GSEPS requests a larger electric power
than SEPS-DL. This is due to the hard requirement for the priority load, which temporally forces
GSEPS to increase the electric power supply.
After the smooth power schedule is obtained, DUBMLC is executed to divide the power to
individual users in each time slot. For better illustration, we only plot the power convergence
curves for six users in Fig. 2.7. The curves for other users are similar. We find that all the curves
converge to the optimal values very quickly; after one step there is no significant variation in the
electric powers of individual users.
We next compare the proposed algorithms with two alternatives:
? A ?supply until deadline? scheme (SUDP), in which the deferrable load demand is served
until the last minute.
43
0 10 20 30 40 50 60 70 80 90 1000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Steps
Electric power (kW)
 
 
User 1
User 2
User 3
User 4
User 5
User 6
Figure 2.7: Convergence of the individual power allocation achieved by DUMLC.
? The ?utility maximization real-time pricing? scheme (UMRP) that is introduced in [102],
which solves the demand side management problem with a real-time pricing strategy and
has been widely cited in the smart grid research community.
In particular, the UMRP scheme maximizes the social welfare in a smart grid at each time slot
independently, i.e.,
max
summationdisplay
n?R
Un(pn(t))?g(P(t),?(t)), for all t,
subject to the total power generation constraint and user power consumption constraints. This
algorithm can be extended to our simulation scenario. In this work, the operating cost of power
generation is evaluated by g(P(t),?(t)) = (?1 + ?2P(t) + ?3P(t)2)?(t), which is generalized
from [103].We let ?1 = 120.0,?2 = 6,?3 = 0.04 for a generator [103] and ?(t) be uniformly
distributed in [0.5,2.5]. This yields a quadratic programming problem, which can be solved in
either a centralized or distributed manner [102]. In the simulations, we use a centralized interior-
point method to solve the problem.
44
The peak power, load factor and power variation achieved by the above algorithms for net-
works with different numbers of users are presented in Fig. 2.8, Fig. 2.9, and Fig. 2.10, respec-
tively. Each number is the average of 100 simulation runs with different random seeds, with the
95% confidence interval plotted at the top of each bar. We observe that SEPS-DL and GSEPS can
significantly reduce both peak power and power variation in the power distribution network. For
example, for the network with 1000 users, SEPS-DL and GSEPS achieves peak powers 1947 kW
and 2560 kW, respectively, which are only 55% and 73% of the corresponding SUDP and UMRP
peak powers. We also notice that the load factors achieved by SEPS-DL and GSEPS are more
than 100% and 50% larger than those of SUDP and UMRP under all cases. Similiarly, we find in
Figure 2.10 that the SEPS-DL and GSEPS schedules are much more smoother than both SUDP
and UMRP schedules. Therefore, to design the power generation, transmission and distribution
infrastructure for this 1000-user site, we may select components, such as transformers and trans-
mission lines, based on the capacity specifications of 1947 kW and 2560 kW, respectively (with
SEPS-DL and GSEPS), instead of 3513 kW with SUDP and UMRP. As the network size increases,
the performance gap increases as well. For the smallest gap case when network size is 100, the
power reduction is 176 kW for SEPS-DL and 110 kW for GSEPS, which meets the requirement of
100 kW minimum energy reductions for the DR products at ISONE [104].
It is interesting to observe that UMRP and SUDP have almost identical performance in the
simulations. This is largely due to the choice of the object functionsummationtextn?RUn(pn(t))?g(P(t),?(t)).
In the simulation with the above utility functions and cost functions, the effect of the total power
decrement on the cost functions dominates the effect of the individual user power increments in
their utility functions. Thus UMRP attempts to reduce the total power generation, while only main-
taining the minimum user satisfaction level. This strategy indeed degenerates UMRP to SUDP,
both with similar performance. Although UMRP maximizes the welfare of the distribution net-
work, it does not aim to smooth the power schedules. It would be helpful to carefully introduce
some coefficients to balance the contributions of utility and cost to the welfare. This phenomenon
also verifies our motivation of the work that the real-time pricing with utility maximization may
45
100 200 500 1000 15000
1000
2000
3000
4000
5000
6000
Power distribution networks size (# of users)
Average peak electric power (kW)
 
 
SEPS?DL
GSEPS
SUDP
UMRP
Figure 2.8: Peak electric powers achieved by SEPS-DL, GSEPS, SUDP and UMRP.
100 200 500 1000 15000
10
20
30
40
50
60
70
80
90
100
Power distribution network size (# of users)
Load factor (%)
 
 
SEPS?DL
GSEPS
SUDP
UMRP
Figure 2.9: Load factor achieved by SEPS-DL, GSEPS, SUDP and UMRP.
not automatically solve the smooth electric power scheduling and peak power reduction problems.
Compared to UMRP, the proposed algorithms directly target at the smoothness optimization prob-
lem and are robust to various configurations of the distribution network.
46
0 100 200 500 1000 15000
2
4
6
8
10
12
x 105
Power distribution networks size (# of users)
Average electric power variance(KW  
2 )
 
 
SEPS?DL
GSEPS
SUDP
UMRP
Figure 2.10: Variances of the power schedules achieved by SEPS-DL, GSEPS, SUDP and UMRP.
We finally compare the average generator operating cost of SEPS-DL, GSEPS, SUDP and
UMRP. Similar to the previous scenario, we assume a load site with average aggregated demand
100 MW, which is scheduled by SEPS-DL, GSEPS, SUDP and UMRP based on a daily period.
The average generator operating costs are shown in Table 2.2. It can be seen that both SEPS-
DL and GSEPS obtain smaller operating costs than SUDP and UMRP for serving the same load
site. It is not surprising to see that SUDP and UMRP obtain similar results, due to the effect of
the objective function. Also note that UMRP is not operating cost optimal, although it seeks to
maximize the social welfare. This is due to the fact that the operating cost is not independent
from time slot to time slot; greedily optimizing social welfare in each time slot does not guarantee
optimality over the entire period. Suppressing user satisfaction level some time slots, may result
in high power allocations at a later time slot to meet all the delayed demands right before their
deadline, leading to larger peak power and electricity generation cost. On the contrary, SEPS-DL
and GSEPS optimize power scheduling over the entire period, and can flexibly serve the demand
to achieve smooth electric power scheduling and low operating cost.
47
Table 2.2: Average operating cost of power generation
SEPS-DL GSEPS SUDP UMRP
$/hour 1653 1669 1746 1746
2.6 Related Work
SG is regarded as the next generation power grid that exploits a coexisting communications
network for better control and optimization of power generation and distribution. In SG, infor-
mation technologies and computational intelligence are integrated across electricity generation,
transmission, distribution and consumption to achieve green, reliable, efficient and sustainable en-
ergy goals. Comprehensive surveys of SG technologies can be found in [3?5, 105].
The emergence of SG attracts new interest in evolving the next generation of power distri-
bution systems [88]. Demand response is an important power distribution paradigm to reduce the
peak demand and smooth demand profiles in the grid by shaping the user demands. Various imple-
mentation issues of demand response in SG are examined in [104]. Real-time pricing and direct
load control are two important ways to shape user demand profile.
Due to the real-time communications and control through two-way information flows in SG,
new design approaches in demand response are being developed recently [90?93], which are based
on optimization and game theory approaches. In [90], the authors proposed an optimal and au-
tomatic residential energy consumption scheduling framework to achieve the trade-off between
minimization of electricity payment and appliance operation waiting time. In [91], a game the-
ory approach is used to control the power demand at peak hours by dynamic pricing strategies.
In [92], the authors studied demand response for households based on utility maximization, and
showed that there exist time-varying prices that may achieve social optimality. A recent work [93]
introduced the framework for optimal resource allocation under the uncertainty in the two-way
information network and provided a decentralized algorithm that can be implemented in practice.
Although providing some interesting methods to achieve cost efficient electricity usage, these work
do not explicitly address the problem of smooth electric power scheduling.
48
Two recent works [106,107] studied the problem of reducing the peak-to-average ratio (PAR)
of the electric energy consumption in SG. In [106], the authors introduced a game theory frame-
work for a distributed algorithm to minimize the total energy payment and reduce PAR. However,
users need to broadcast control messages to announce their new schedules to the entire network.
The control overhead could be considerable. In [107], energy storage devices were incorporated
in the SG and users? cost and PAR are minimized with a distributed algorithm. Although the dis-
tributed algorithm only needs to exchange information with the energy provider, the achieved Nash
equilibrium cannot be guaranteed to be socially optimal. In addition, this work does not consider
the operating cost of the energy provider.
Majorization is a useful tool for problems involving vectors [13]. It has been used in solving
optimization problems in the communication and networking area [57,95,96]. In [95], majorization
is applied to variable-bit-rate (VBR) video smoothing over a wired CBR link. Ref. [96] presented
an optimal transmission algorithm over a wireless multiple-input single-output (MISO) link based
on majorization. In our recent work [57], we adopted majorization for power efficient VBR video
streaming over a cellular network.
Our work differs from these existing efforts by introducing the mathematic theory of majoriza-
tion to solve the smoothness scheduling problem in electric distribution networks, while explicitly
targeting at the unique mathematical notion of smoothness. To the best of our knowledge, this is
the first work that introduces maojorization into electric energy management in power grid, which
jointly considers smooth power scheduling, electric usage quality provisioning on the user side,
and grid operating cost on the electric energy provider side. The effective electric power smooth-
ing solution provides a highly competitive solution for future SG design and operations.
2.7 Conclusions
In this chapter, we addressed the problem of smooth electric power scheduling in a power dis-
tribution network. We introduced a deterministic model to characterize the complex relationship
49
between demand and supply. A constrained nonlinear optimization problem is formulated aim-
ing to minimize the electric power variation and satisfy user power usage quality. We developed
majorization-based algorithms for deriving smoothness optimal schedules for the network, and a
distributed algorithm for dividing the power supply among the users. Our simulation study shows
that the proposed algorithms can effectively reduce the peak power, minimize the power variation,
and reduce the operating cost of the grid, while satisfying user power usage quality.
50
Chapter 3
Adaptive Electricity Scheduling in Microgrids
3.1 Introduction
In this chapter, we designed a smart energy management systems in Microgrid (MG) by taking
advantage of the plug-and-play interfaces of smart grid. MG is a promising component for future
SG deployment. Due to the increasing deployment of distributed renewable energy resources (DR-
ERs), MG provides a localized cluster of renewable energy generation, storage, distribution and
local demand, to achieve reliable and effective energy supply with simplified implementation of SG
functionalities [4, 108]. We review the typical MG architecture in Fig. 3.1, consisting of DRERs
(such as wind turbines and solar photovoltaic cells), energy storage systems (ESS), a communica-
tion network (e.g., wireless or powerline communications) for information delivery, an MG central
controller (MGCC), and local residents. The MG has centralized control with the MGCC [108],
which exchanges information with local residents, ESS?s, and DRERs via the information net-
work. There is a single common coupling point with the macrogrid. When disconnected, the MG
works in the islanded mode and DRERs and ESS?s provide electricity to local residents. When
connected, the MG may purchase extra electricity from the macrogrid or sell excess energy back
to the market [3].
The balance of electricity demand and supply is one of the most important requirements in
MG management. Instead of matching supply to demand, smart energy management matches the
demand to the available supply using control technology or off-peak pricing to achieve more effi-
cient capacity utilization [4]. In this chapter, we develop a novel access control framework for MG
energy management, exploiting the two-way flows of electricity and information. In particular,
we consider two types of electricity usage: (i) a pre-agreed basic usage that is ?hard?-guaranteed,
such as basic living usage, and (ii) extra elastic quality usage exceeding the pre-agreed level for
51
Macrogrid
Power flow
Inf
orm
ati
on
 flo
w
Microgrid
Power flow
Market 
Broker
Energy
storageMGCC
Figure 3.1: Illustrate the microgrid architecture.
more comfortable life, such as excessive use of air conditioners or entertainment devices. In prac-
tice, residents may set their load priority and preference to obtain the two types of usage [89].
The basic usage should be always satisfied, while the quality usage is controlled by the MGCC
according to the grid status, such as DRER generation, ESS storage levels and utility prices. The
MGCC may block some quality usage demand if necessary. This can be implemented by incor-
porating smart meters, smart loads and appliances that can adjust and control their service level
through communication flows [3]. To quantify residents? satisfaction level, we define the outage
percentage of the quality usage as Quality of Service in Electricity (QoSE), which is specified in
the service contracts. For example, the residents may customize their outage risk of quality usage
in return for paying an insurance premium, which is differentiated according to local residents
preferences [109] [109]. The MGCC adaptively schedules electricity to keep the QoSE below a
target level, and accordingly dynamically balance the load demand to match the available supply.
In this chapter, we investigate the problem of smart energy scheduling by jointly considering
renewable energy distribution, ESS management, residential demand management, and utility mar-
ket participation, aiming to minimize the MG operation cost and guarantee the residents? QoSE.
The MGCC may serve some quality usage with supplies from the DRERs, ESS?s and macrogrid.
On the other hand, the MG can also sell excessive electricity back to the macrogrid to compensate
52
for the energy generation cost. The electricity generated from renewable sources is generally ran-
dom, due to complex weather conditions, while the electricity demand is also random due to the
random consumer behavior, and so do the purchasing and selling prices on the utility market. It is
challenging to model the random supply, demand, and price processes for MG management, and
it may also be costly to have precise, real-time monitoring of the random processes. Therefore,
a simple, low cost, and optimal electricity scheduling scheme that does not rely on any statistical
information of the supply, demand, and price processes would be highly desirable.
We tackle the MG electricity scheduling problem with a Lyapunov optimization approach,
which is a useful technique to solve stochastic optimization and stability problems [12]. We first
introduce two virtual queues: QoSE virtual queues and battery virtual queues to transform the
QoSE control problem and battery management problem to queue stability problems. Second, we
design an adaptive MG electricity scheduling policy based on the Lyapunov optimization method
and prove several deterministic (or, ?hard?) performance bounds for the proposed algorithm. The
algorithm can be implemented online because it only relies on the current system status, without
needing any future knowledge of the energy demand, supply and price processes and any future
information. The proposed algorithm also converges exponentially due to the nice property of
Lyapunov stability design [110]. The algorithm is evaluated with trace-driven simulations and is
shown to achieve significant efficiency on MG operation cost while guaranteeing the residents?
QoSE.
The remainder of this chapter is organized as follows. We present the system model and
problem formulation in Section 3.2. An adaptive MG electricity scheduling algorithm is designed
and analyzed in Section 3.3. Simulation results are presented and discussed in Section 3.4. We
discuss related work in Section 3.5. Section 3.6 concludes the chapter.
The notations used in this chapter are summarized in Table 3.1.
53
3.2 System Model and Problem Formulation
3.2.1 System Model
Overview
We consider the electricity supply and consumption in an MG as shown in Fig. 3.1. We as-
sume that the MG is properly designed such that a portion of the electricity demand related to basic
living usage (e.g., lighting) from the residents, termed basic usage, can be guaranteed by the min-
imum capacity of the MG. There are randomness in both electricity supply (e.g., weather change)
and demand (e.g., entertainment usage in weekends). To cope with the randomness, the MG works
in the grid-connected mode and is equipped with ESS?s, such as electrochemical battery, supercon-
ducting magnetic energy storage, flywheel energy storage, etc. The ESS?s store excess electricity
for future use.
The MGCC collects information about the resident demands, DRER supplies, and ESS levels
through the information network. When a resident demand exceeds the pre-agreed level, a quality
usage request will be triggered and transmitted to the MGCC. The MGCC will then decide the
amount of quality usage to be satisfied with energy from the DRERS, the ESS?s, or by purchasing
electricity from the macrogrid. The MGCC may also decline some quality usage requests. The
excess energy can be stored at the ESS?s or sold back to the macrogrid for compensating the cost
of MG operation.
Without loss of generality, we consider a time-slotted system. The time slot duration is deter-
mined by the timescale of the demand and supply processes, as well as how frequent the MG can
switch on and off to the macrogrid.
Energy Storage System Model
The system model is shown in Fig. 3.2. Consider a battery farm with K independent battery
cells, which can be recharged and discharged. We assume that the batteries are not leaky and do
not consider the power loss in recharging and discharging, since the amount is usually small. It is
54
Resident 1's 
quality usage 
Control outage 
probability at ?N
P(t)
S(t)Q(t)
pN(t)
p1(t)
R1(t) D1(t) Resident N's quality usage 
Served quality 
usage 
Served quality 
usage 
Control outage 
probability at ?1
Battery 1
Battery K
ESSs
MGCC 
Rk(t) Dk(t)
Figure 3.2: The system model considered in this chapter.
easy to relax this assumption by applying a constant percentage on the recharging and discharging
processes. For brevity, we also ignore the aging effect of the battery and the maintenance cost,
since the cost on the utility market dominates the operation cost of MGs.
Let Ek(t) denote the energy level of the kth battery in time slot t. The capacity of the battery
is bounded as
Emink ?Ek(t) ?Emaxk ,?k,t, (3.1)
where Emaxk ? 0 is the maximum capacity. Emink ? 0 is the minimum energy level required for
battery k, which may be set by the battery deep discharge protection settings. The dynamics over
time of Ek(t) can be described as
Ek(t+ 1) = Ek(t)?Dk(t) +Rk(t),?k,t, (3.2)
where Rk(t) and Dk(t) are the recharging and discharging energy for battery k in time slot t,
respectively. The charging and discharging energy in each time slot are bounded as
?
??
??
0 ?Rk(t) ?Rmaxk , ?k,t
0 ?Dk(t) ?Dmaxk , ?k,t.
(3.3)
In each time slot t, Rk(t) and Dk(t) are determined such that (3.1) is satisfied in the next time slot.
55
Usually the recharging and discharging operations cannot be performed simultaneously, which
leads to
??
?
??
Rk(t) > 0 ?Dk(t) = 0, ?k,t
Dk(t) > 0 ?Rk(t) = 0, ?k,t.
(3.4)
Energy Supply and Demand Model
Consider N residents in the MG; each generates basic and quality electricity usage requests,
and each can tolerate a prescribed outage probability ?n for the requested quality usage part. The
MGCC adaptively serves quality usage requests at different levels to maintain the QoSE as well
as the stability of the grid. The service of quality usage can be different for different residents,
depending on individual service agreements.
Let ?n be the average quality usage arrival rate, and ?n a prescribed outage tolerance (i.e., a
percentage) for user n. The average outage rate for the quality usage, ?n, should satisfy
?n ??n ??n. (3.5)
At each timet, the quality usage request from residentnis?n(t) ? [0,?maxn ] units, which is an i.i.d
random variable with a general distribution. The average rate is ?n = limt??(1/t)summationtextt?1?=0?n(?)
by the law of large number.
The DRERs in the MG generate U(t) units of electricity in time slot t. U(t) can offer enough
capacity to support the pre-agreed basic usage in the MG, which is guaranteed by islanded mode
MG planning. due to complex weather conditions. The electric is transmitted over power trans-
mission lines. Without loss of generality, we assume the power transmission line is not subject
to outages and the transmission loss is negligible. Let ?bn(t) be the pre-agreed basic usage for
resident n in time slot t, which can be fully satisfied by U(t), i.e., summationtextNn=1?bn(t) ? U(t), for all t.
In addition, some quality usage request ?n(t) may be satisfied if P(t) = U(t)?summationtextNn=1?bn(t) ? 0.
56
Let pn(t) be the energy allocated for the quality usage of resident n. We have
0 ?pn(t) ??n(t). (3.6)
We define a function In(t) ? 0 to indicate the amount of quality usage outage for resi-
dent n, as In(t) = ?n(t) ? pn(t). Then the average outage rate can be evaluated as ?n =
limt??(1/t)summationtextt?1?=0In(t).
The MGCC may purchase additional energy from the macrogrid or sell some excess energy
back to the macrogrid. Let Q(t) ? [0,Qmax] denote the energy purchased from the macrogrid and
S(t) ? [0,Smax] the energy sold on the market in time slot t, where Qmax and Smax are determined
by the capacity of the transformers and power lines. Since it is not reasonable to purchase and sell
energy on the market at the same time, we have the following constraints
?
??
??
Q(t) > 0 ?S(t) = 0, ?t
S(t) > 0 ?Q(t) = 0, ?t.
(3.7)
To balance the supply and demand in the MG, we have
P(t)+Q(t)+
Ksummationdisplay
k=1
Dk(t)?S(t)?
Ksummationdisplay
k=1
Rk(t)=
Nsummationdisplay
n=1
pn(t),?t. (3.8)
Utility Market Price Model
The price for purchasing electricity from the macrogrid in time slot t is C(t) per unit. The
purchasing price depends on the utility market state, such as peak/off time of the day. We assume
finite C(t) ? [Cmin,Cmax], which is announced by the utility market at the beginning of each time
slot and remains constant during the slot period [111]. Unlike prior work [111], we do not require
any statistic information of the C(t) process, except that it is independent to the amount of energy
to be purchased in that time slot.
57
If the MGCC determines to sell renewable energy on the utility market, the selling price from
the market broker is denoted by W(t) ? [Wmin,Wmax] in time slot t, which is also a stochastic
process with a general distribution and mean ?n. We also assume W(t) is known at the beginning
of each time slot and independent to the amount of energy to be sold on the market. We assume
Cmax ? Wmax, Cmin ? Wmin and C(t) > W(t) for all t. That is, the MG cannot make profit by
greedily purchasing energy from the market and then sell it back to the market at a higher price
simultaneously.
3.2.2 Problem Formulation
Given the above models, a control policyA(t) = {Q(t),S(t),Rk(t),Dk(t),pn(t)}is designed
to minimize the operation cost of the MG and guarantee the QoSE of the residents. We formulate
the electricity scheduling problem as
minimize: limt?? 1t
t?1summationdisplay
?=0
E{Q(?)C(?)?S(?)W(?)} (3.9)
s.t. (3.1), (3.3), (3.4), (3.5), (3.6), (3.7), (3.8)
battery queue stability constraints.
Problem (3.9) is a stochastic programming problem, where the utility prices, utility generation of
DRERs, and utility consumption of residents are all random. The solution also depends on the
evolution of battery states. It is challenging since the supply, demand, and price are all general
processes.
Virtual Queues
We first adopt a battery virtual queue Xk(t) that tracks the charge level of each battery k:
Xk(t) = Ek(t)?Dmaxk ?Emink ?VCmax, ?k,t, (3.10)
58
where 0 <V ?Vmax = mink
braceleftBigEmax
k ?E
min
k ?R
max
k ?D
max
k
Cmax?Wmin
bracerightBig
is a constant for the trade-off between al-
gorithm performance and ensuring the battery constraints. This constant Vmax is carefully selected
to ensure the evolutions of the battery levels always satisfy the battery constraints (3.1), which will
be examined in detail in Section 3.3.3.The virtual queue can be deemed as a shifted version of the
battery dynamics in (3.2) as
Xk(t+ 1) = Xk(t)?Dk(t) +Rk(t), ?k,t. (3.11)
These queues are ?virtual? because they are maintained by the MGCC control algorithm. Unlike
an actual queue, the virtual queue backlog Xk(t) may take negative values.
We next introduce a conceptual QoSE virtual queue Zn(t), whose dynamics are governed by
the system equation as
Zn(t+ 1) = [Zn(t)??n ??n(t)]+ +In(t), ?n,t. (3.12)
where [x]+ = max{0,x}.
Theorem 3.1. If an MGCC control policy stabilizes the QoSE virtual queue Zn(t), the outage
quality usage of resident n will be stabilized at the average QoSE rate ?n ??n ??n.
Proof. See Appendix B.1.
Problem Reformulation
With Theorem 3.1, we can transform the original problem (3.9) into a queue stability problem
with respect to the QoSE virtual queue and the battery virtual queues, which leads to a system
59
stability design from the control theoretic point of view. We have a reformulated stochastic pro-
gramming problem as follows.
minimize: limt?? 1t
t?1summationdisplay
?=0
E{Q(?)C(?)?S(?)W(?)} (3.13)
s.t. (3.3), (3.4), (3.6), (3.7), (3.8)
battery and QoSE virtual queue stability
constraints.
Theorem 3.1 indicates that QoSE provisioning is equivalent to stabilizing the QoSE virtual queue
Zn(t), while stabilizing the virtual queues (3.11) ensures that the battery constraints (3.1) are sat-
isfied. We then apply Lyapunov optimization to develop an adaptive electricity scheduling policy
for problem (3.13), in which the policy greedily minimize Lyapunov drift in every slot t to push
the system toward the stability.
3.2.3 Lyapunov Optimization
We define the Lyapunov function for system state vector?(t) = [vectorX(t),vectorZ(t)]T with dimension (N+
K)?1 as follows, in which vectorX(t) = [X1(t)???XK(t)]T and vectorZ(t) = [Z1(t)???ZN(t)]T.
L(vector?(t)) = 12
Ksummationdisplay
k=1
[Xk(t)]2 + 12
Nsummationdisplay
n=1
[Zn(t)]2, (3.14)
which is positive definite, since L(vector?(t)) > 0 when vector?(t) negationslash= vector0 and L(vector?(t)) = 0 ? vector?(t) = vector0. We
then define the conditional one slot Lyapunov drift as
?(vector?(t)) = E{L(vector?(t+ 1))?L(vector?(t))|vector?(t)}. (3.15)
60
With the drift defined as in (3.15), it can be shown that
?(vector?(t)) = 12E
braceleftBigg Ksummationdisplay
k=1
[(Xk(t+ 1))2 ?(Xk(t))2|Xk(t)]+
Nsummationdisplay
n=1
[(Zn(t+ 1))2 ?(Zn(t))2|Zn(t)]
bracerightBigg
? B +
Nsummationdisplay
n=1
E{Zn(t)(1??n)?n(t)|Zn(t)}+
Ksummationdisplay
k=1
E{Xk(t)(Rk(t)?Dk(t))|Xk(t)}?
Nsummationdisplay
n=1
E{(Zn(t) +?n(t))pn(t)|Zn(t)}, (3.16)
where B = 12summationtextKk=1(max{Dmaxk ,Rmaxk })2 + 12summationtextNn=1(2 + ?2n)(?maxn )2 is a constant. See Ap-
plendix B.2 for the derivation of (3.16).
To minimize the operation cost of the MG, we adopt the drift-plus-penalty method [112].
Specifically, we select the control policy A(t) = {Q(t),S(t),Rk(t),Dk(t),pn(t)} to minimize the
bound on the drift-plus-penalty as:
?(vector?(t)) +VE{Q(t)C(t)?S(t)W(t)|vector?(t)}
? right-hand-side of (3.16) +
VE{Q(t)C(t)?S(t)W(t)|vector?(t)}, (3.17)
where 0 < V ? Vmax is defined in Section 3.2.2 for the trade-off between stability performance
and operation cost minimization. Given the current virtual queue states Xk(t) and Zn(t), market
prices S(t) and W(t), available DRERs energy P(t), and the resident quality usage request ?n(t),
61
the optimal policy is the solution to the following problem.
minimize: B +
Nsummationdisplay
n=1
[Zn(t)(1??n)?n(t)] +
V[Q(t)C(t)?S(t)W(t)] +
Ksummationdisplay
k=1
[Xk(t)(Rk(t)?Dk(t))]?
Nsummationdisplay
n=1
[(Zn(t) +?n(t))pn(t)] (3.18)
s.t. (3.3), (3.4), (3.6), (3.7), (3.8) .
Since the control policy A(t) is only applied to the last three terms of (3.18), we can further
simplify problem (3.18) as
minimize: V[Q(t)C(t)?S(t)W(t)] +
Ksummationdisplay
k=1
[Xk(t)(Rk(t)?
Dk(t))]?
Nsummationdisplay
n=1
[(Zn(t) +?n(t))pn(t)] (3.19)
s.t. (3.3), (3.4), (3.6), (3.7), (3.8) ,
which can be solved based on observations of the current system state{Xk(t),Zn(t),C(t),W(t),P(t),
?n(t)}.
3.3 Optimal Electricity Scheduling
3.3.1 Properties of Optimal Scheduling
With the Lyapunov penalty-and-drift method, we transform problem (3.13) to problem (3.19)
to be solved for each time slot. The solution only depends on the current system state; there is
no need for the statistics of the supply, demand and price processes and no need for any future
information. The solution algorithm to this problem is thus an online algorithm. We have the
following properties for the optimal scheduling.
62
Lemma 3.1. The optimal solution to problem (3.19) has the following properties:
1. If Q(t) > 0, we have S(t) = 0,
(a) If Xk(t) > ?VC(t), the optimal solution always selects Rk(t) = 0; if Xk(t) <
?VC(t), the optimal solution always selects Dk(t) = 0.
(b) If Zn(t) >VC(t)??n(t), the optimal solution always selects pn(t) ? (1??n)?n(t);
if Zn(t) <VC(t)??n(t), the optimal solution always selects pn(t) = 0.
2. When Q(t) = 0, we have S(t) > 0,
(a) If Xk(t) > ?VW(t), the optimal solution always selects Rk(t) = 0; if Xk(t) <
?VW(t), the optimal solution always selects Dk(t) = 0.
(b) If Zn(t) >VW(t)??n(t), the optimal solution always selects pn(t) ? (1??n)?n(t);
if Zn(t) <VW(t)??n(t), the optimal solution always selects pn(t) = 0.
Proof. The proof of Lemma 3.1 is given in Appendix B.3.
Lemma 3.2. The optimal solution to the battery management problem has the following proper-
ties:
1. If Xk(t) >?VWmin, the optimal solution always selects Rk(t) = 0.
2. If Xk(t) <?VCmax, the optimal solution always selects Dk(t) = 0.
Proof. The proof of Lemma 3.2 is given in Appendix B.4.
Lemma 3.3. The optimal solution to the QoSE provisioning problem has the following properties:
1. If Zn(t) >VCmax, the optimal solution always selects pn(t) ? (1??n)?n(t).
2. If Zn(t) <VWmin ??max, the optimal solution always selects pn(t) = 0.
Proof. The proof of Lemma 3.3 is given in Appendix B.5.
63
Lemma 3.1 provides useful insights for simplifying the algorithm design, which will be dis-
cussed in Section 3.3.2. The intuition behind these lemmas is two-fold. On the ESS management
side, if either the purchasing priceC(t) or the selling priceW(t) is low, the MG prefers to recharge
the ESS?s to store excess electricity for future use. On the other hand, if either C(t) or W(t) is
high, the MG is more likely to discharge the ESS?s to reduce the amount of energy to purchase
or sell more stored energy back to the macrogrid. On the QoSE provisioning side, if either C(t)
or W(t) is high and the quality usage ?n(t) is low, the MG is apt to decline the quality usage for
lower operation cost. On the other hand, if eitherC(t) orW(t) is low and?n(t) is high, the quality
usage are more likely to be granted by purchasing more energy or limiting the sell of energy.
3.3.2 MG Optimal Scheduling Algorithm
In this section, we present the MG control policy A(t) to solve problem (3.19). Given the
current virtual queue state {Xk(t),Zn(t)}, market prices C(t) and W(t), quality usage ?n(t) and
available energy P(t) from the DRERS for serving quality usage, problem (3.19) can be decom-
posed into the following two linear programming (LP) sub-problems (since one of S(t) and Q(t)
must be zero, see (3.7)).
minimize:VQ(t)C(t) +
Ksummationdisplay
k=1
[Xk(t)(Rk(t)?Dk(t))]?
Nsummationdisplay
n=1
((Zn(t) +?n(t))pn(t)) (3.20)
s.t.S(t) = 0,(3.3), (3.4), (3.6), (3.8).
minimize:?VS(t)W(t) +
Ksummationdisplay
k=1
[Xk(t)(Rk(t)?Dk(t))]?
Nsummationdisplay
n=1
((Zn(t) +?n(t))pn(t)) (3.21)
s.t. Q(t) = 0,(3.3), (3.4), (3.6), (3.8).
64
Algorithm 3: Adaptive Electricity Scheduling Algorithm
1 MGCC initializes the QoSE target to ?n and the virtual queues backlogs Zn(t) and Xk(t),
for all n and k ;
2 while TRUE do
3 Residents send usage request (with basic and quality usage) to MGCC via the
information network ;
4 MGCC solves LPs (3.20) and (3.21) ;
5 MGCC selects the optimal solution A(t) comparing the solutions to (3.20) and (3.21) ;
6 MGCC updates the virtual queues Xk(t) and Zn(t) according to (3.11) and (3.12), for
all n and k ;
7 end
In sub-problem (3.20), we set Rk(t) = 0 if Xk(t) > ?VC(t), and Dk(t) = 0 if Xk(t) <
?VC(t) according to Lemma 3.1. Also, if Zn(t) <VC(t)??n(t), we set pn(t) = 0; otherwise,
we reset constraint (3.6) to a smaller search space of (1 ??n)?n(t) ? pn(t) ? ?n(t). We take a
similar approach for solving sub-problem (3.21) by replacing C(t) with W(t). Then we compare
the objective values of the two sub-problems and select the more competitive solution as the MG
control policy. The complete algorithm is presented in Algorithm 3.
3.3.3 Performance Analysis
The proposed scheduling algorithm dynamically balances cost minimization and QoSE provi-
sioning. It only requires current system state information (i.e., as an online algorithm) and requires
no statistic information about the random supply, demand, and price processes. The algorithm is
also robust to non-i.i.d. and non-ergodic behaviors of the processes [113, 114].
Theorem 3.2. The constraint on the ESS battery level Ek(t), Emink ? Ek(t) ? Emaxk , is always
satisfied for all k and t.
Proof. The proof of Theorem 3.2 is given in Appendix B.6.
65
Theorem 3.3. The worst-case backlogs of the QoSE virtual queue for each resident n is bounded
by Zn(t) ? Zmaxn = VCmax + ?maxn , for all n, t. Moreover, the worst-case average amount of
outage of quality usage for resident n in a period T is upper bounded by Zmaxn +T?n?maxn .
Proof. The proof of Theorem 3.3 is given in Appendix B.7.
Theorem 3.4. The average MG operation cost under the adaptive electricity scheduling algorithm
in Algorithm3, y?, is bounded as y? ?yopt + ?B/V, where ?B = B +summationtextNn=1Zmaxn (1??n)?maxn .
Proof. The proof of Theorem 3.4 is given in Appendix B.8.
It is worth noting that the choice of V controls the optimality of the proposed algorithm.
Specifically, a larger V leads to a tighter optimality gap. However, from the proof of Theorem 3.2,
V is limited by Vmax, which ensures the feasibility of the battery constraints. This is actually a
similar phenomenon to the so-called performance-congestion trade-off [115]. Through the defi-
nition of Vmax (see Section 3.2.2), it can be seen that if we invest more on the individual storage
components for a larger ESS capacity, the proposed algorithm can achieve a better performance
(i.e., a smaller optimality gap).
It is also worth noting that all the performance bounds of the proposed algorithm are deter-
ministic, which provide ?hard? guarantees for the performance of the proposed adaptive scheduling
policy in every time slot. Unlike probabilistic approaches, the proposed method provides useful
guidelines for the MG design, while guaranteeing the MG operation cost, grid stability, and the
usage quality of residents.
3.4 Simulation Study
We demonstrate the performance of the proposed adaptive MG electricity scheduling algo-
rithm through extensive simulations. We simulated an MG with 500 residents, where the electricity
from DRERs is supplied by a wind turbine plant. We use the renewable energy supply data from the
66
Western Wind Resources Dataset published by the National Renewable Energy Laboratory [116].
The ESS?s consists of 100 PHEV Li-ion battery packs, each of which has a maximum capacity of
16 kWh and the minimum energy level is 0. The battery can be fully charged or discharged within
2 hours [117].
The residents? pre-agreed power demand is uniformly distributed in [2 kW, 25 kW], and the
quality usage power is uniformly distributed in [0, 10 kW]. The MG works in the grid-connected
mode and may purchase/sell electricity from/to the macrogrid. The utility prices in the macrogrid
are obtained from [118] and are time-varying. We assume the sell price by the broker is random
and below the purchasing price in each time slot. The time slot duration is 15 minutes. The MGCC
serves a certain level of quality usage according to the adaptive electricity scheduling policy. The
QoSE target is set to ?n = 0.07 for all residents. The control parameter is V = Vmax, unless
otherwise specified.
3.4.1 Algorithm Performance
We first investigate the average QoSEs and total MG operation cost with default settings for a
five-day period. We use MATLAB LP solver for solving the sub-problems (3.20) and (3.21). For
better illustration, we only show the QoSEs of three randomly chosen users in Fig. 3.3. It can be
seen that all the average QoSEs converge to the neighborhood of 0.08 within 200 slots, which is
close to the MG requested criteria?n = 0.07. In fact the proposed scheme converges exponentially,
due to the inherent exponential convergence property in Lyapunov stability based design [110].
We also plot the MG operation traces from this simulation in Fig. 3.6. The energy for serving
quality usage from the DEREs are plotted in Fig. 3.6(A). It can be seen that the DRERs generate
excessive electricity from slot 150 to 200, which is more than enough for the residents. Thus,
the MGCC sells more electricity back to the macrogrid and obtains significant cost compensation
accordingly. In Fig. 3.6(B), we plot the traces of electricity trading, where the positive values
are the purchased electricity (marked as brown bars), and the negative values represent the sold
electricity (marked as dark blue bars). The MG operation costs are plotted in Fig. 3.6(C). The
67
0 100 200 300 4000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time slot
Av
era
ge
 Q
oS
Es
 
 
Resident 1
Resident 2
Resident 3
Figure 3.3: Average QoSEs of three residents (V = Vmax).
curve rises when the MG purchases electricity and falls when the MG sells electricity. From slot
150 to 200, the operation cost drops significantly due to profits of selling excess electricity from
the DEREs. The operation cost is $418.10 by the end of the period, which means the net spending
of the MG is $418.10 on the utility market.
We then examine the energy levels of the batteries in Fig. 3.4. We only plot the levels of three
batteries in the first 50 time slots for clarity. The proposed control policy charges and discharges
the batteries in the range of 0 to 16 kWh, which falls strictly within the battery capacity limit. It
can be seen that the amount of energy for charging or discharging in one slot is limited by 2 kWh
in the figure, due to the short time slots comparing to the 2-hour fully charge/discharge periods.
For longer time slot durations and batteries with faster charge/discharge speeds, the variation of
the energy level in Fig. 3.4 could be higher. However, Theorem 3.2 indicates that the feasibility of
the battery management constraint is always ensured, if the control parameter V satisfies 0 <V ?
Vmax.
We next evaluate the performance of the proposed adaptive control algorithm under different
values of control parameter V. For different values V = {Vmax,Vmax/2,Vmax/4}, the QoSEs
are stabilized at 0.081, 0.061, and 0.055, and the total operation cost are $418.10, $625.69, and
68
0 10 20 30 40 500
2
4
6
8
10
12
14
16
Time slot
Ba
tte
ry 
lev
els
 (k
W
h)
 
 
Battery 1
Battery 2
Battery 3
Figure 3.4: Energy levels of three Li-ion batteries (V = Vmax).
$717.75, respectively. We find the QoSE decreases from 0.081 to 0.055, while the total oper-
ation cost is increased from $418.10 to $717.75, as Vmax is decreased. This demonstrates the
performance-congestion trade-off as in Theorem 3.4: a larger V leads to a smaller objective value
(i.e., the operating cost), but the system is also penalized by a larger virtual queue backlog, which
corresponds to a higher QoSE. On the contrary, a smaller V favors the resident quality usage, but
increases the total operation cost. In practice, we can select a proper value for this parameter based
on the MG design specifications.
It would be interesting to examine the case where the residents require different QoSEs. We
assume 5 residents with a service contract for lower QoSEs. We plot the average QoSEs of three
residents with V = Vmax/2 in Fig. 3.5. Resident 1 prefers an outage probability ?1 = 0.02, while
residents 2 and 3 require an outage probability ?2 = ?3 = 0.07. It can be seen in Fig. 3.5 that
resident 1?s QoSE converges to 0.015, while the other two residents? QoSEs remains around 0.063.
3.4.2 Comparison with a Benchmark
We compare the performance of the proposed scheme with a heuristic MG electricity control
policy (MECP), which serves as a benchmark. In MECP, the MGCC blocks quality usage requests
69
0 100 200 300 4000
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
Time slot
Av
era
ge
 Q
oS
E
 
 
Resident 1
Resident 2
Resident 3
Figure 3.5: QoSEs for three residents with different service contracts (V = Vmax/2).
simply by tossing a coin with the target probability. We use?n = 0.03 in the following simulations.
If there is sufficient electricity from the DRERs, all the quality usage requests will be granted and
the excess energy will be stored in the ESS?s. If there is still any surplus energy, the MGCC
will sell it to the macrogrid. If there is insufficient electricity from the DRERs, the ESS?s will
be discharged to serve the quality usage requests. The MGCC will purchase electricity from the
macrogrid if even more electricity is required. Finally, with a predefined probability, e.g., 0.5 in
the following simulation, the MG purchases as much energy as possible to charge the ESS?s.
We run 100 simulations with different random seeds for a seven-day period. We assume in the
first five days the resident behavior is the same as previous default settings. In the last two days,
we assume the residents are apt to request more electricity (e.g., more activities in weekends)
We assume in the last two days the resident pre-agreed basic usage power demand is uniformly
distributed from 5 kW to 35 kW. The quality usage power is uniformly distributed from 0 to 20
kW.
We find that the proposed algorithm earns $947.27 from the utility market (with 95% con-
fidence interval [950.65,943.89]). The profit mainly comes from the abundant DRER generation
in the last two days, as shown in Fig. 3.7. MECP only earns $379.74 from the market (with 95%
70
0 50 100 150 200 250 300 350 400 4500
2000
4000
6000
Time slot
DRERs Ele. for QoSE (kWh)
(A) DRERs electricity for QoSE request
0 50 100 150 200 250 300 350 400 450?6000
?4000
?2000
0
2000
Time slot
Bought/Sold Ele. (kWh)
(B) Bought/sold electricity
0 50 100 150 200 250 300 350 400 450?500
0
500
1000
1500
Time slot
MG operation cost ($)
(C) MG operation cost
Figure 3.6: MG operation traces of the proposed algorithm for the 5-day period.
confidence interval [387.96,371.52]), which is 60% lower than that of the proposed control pol-
icy. We also find that the QoSEs under the proposed control policy remains about 0.025, which is
lower than the criteria ?n = 0.03. This is because there are a sudden price jump from $27/MWh
to $356/MWh in the afternoon of the last day. This sharp increment increases Cmax eight times
and decreases the value of Vmax. Due to the performance-congestion trade-off, the QoSEs become
smaller (lower than MECP?s 0.03 level).
3.5 Related Work
SG is regarded as the next generation power grid with two-way flows of electricity and in-
formation. Several comprehensive reviews of SG technologies can be found in [3, 4]. Recently,
71
0 100 200 300 400 500 6000
2000
4000
6000
Time slot
DRERs Ele. for QoSE (kWh)
(A) DRERs electricity for QoSE request
0 100 200 300 400 500 600?6000
?4000
?2000
0
2000
Time slot
Bought/Sold Ele. (kWh)
(B) Bought/sold electricity
0 100 200 300 400 500 600?1000
0
1000
2000
3000
Time slot
MG operation cost ($)
(C) MG operation cost
Figure 3.7: MG operation traces of proposed algorithm for the 7-day period.
SG research is attracting considerable interest from the networking and communications commu-
nities [119?124]. For example, the design of wireless communication systems in SG is studied
in [120]. The authors of [121, 122] explore the important wireless communication security issues
in smart grid. The energy management and power flow control in the grid is investigated in [119]
to reach system-wide reliability under uncertainties. The frequency oscillation in power networks
is studied in [123] by epidemic propagation and a social network based approach. The electric
power management with PHEVs are examined in [124].
Microgrid is a new grid structure to group DRERs and local residents loads, which provides
a promising way for the future SG. In [108], the authors review the MG structure with distributed
energy resources. In [125], the integration of random wind power generation into grids for cost
72
effective operation is investigated. In [126], the authors propose a useful online method to discover
all available DRERs within the islanded mode mircogrid and compute a DRER access strategy. The
problem of optimal residential demand management is studied in [92], aiming to adapt to time-
varying energy generation and prices, and maximize user benefit. In [127], the authors investigate
energy storage management with a dynamic programming approach. The size of the ESS?s for MG
energy storage is explored in [128].
Lyapunov optimization is a useful stochastic optimization method [12]. It integrates the Lya-
punov stability concept of control theory with optimization and provides an efficient framework
for solving schedule and control problems. It has been widely used and extended in the communi-
cations and networking areas [12, 112]. In two recent work [114, 129], the Lyapunov optimization
method is applied to jointly optimize power procurement and dynamic pricing. In [114], the au-
thors investigate the problem of profit maximization for delay tolerant consumers. In [129], the
authors study electricity storage management for data centers, aiming to meet the workload re-
quirement. Both of the work are designed based on a single energy consumption entity model.
In this chapter, we investigate a novel smart energy management system for MGs based on the
concept of QoSE, which is different from above work. By jointly considering multiple residents,
ESS?s and utility market participation, the adaptive electricity scheduling policy is designed with
Lyapunov optimization for ensuring the quality of service of the electricity usage and minimizing
the MG operation cost.
3.6 Conclusion
In this chapter, we developed an online adaptive electricity scheduling algorithm for smart
energy management in MGs by jointly considering renewable energy penetration, ESS manage-
ment, residential demand management, and utility market participation. We introduced a QoSE
model by taking into account minimization of the MG operation cost, while maintaining the out-
age probabilities of resident quality usage. We transformed the QoSE control problem and ESS
management problem into queue stability problems by introducing the QoSE virtual queues and
73
battery virtual queues. The Lyapunov optimization method was applied to solve the problem with
an efficient online electricity scheduling algorithm, which has deterministic performance bounds.
Our simulation study validated the superior performance of the proposed approach.
74
Table 3.1: Notation for Chapter 3
Symbol Description
N total number of residents
K total number of batteries
T total number of slots
Ek(t) energy level for battery k at time slot t
Rk(t) recharging energy for battery k at time slot t
Dk(t) discharging energy for battery k at time slot t
Emaxk maximum battery energy level for battery k
Emink minimum battery energy level for battery k
Rmaxk maximum supported recharging energy for batter k in a slot
Dmaxk maximum supported discharging energy for battery k in a slot
?n average quality usage arrival rate for resident n
?n average outage rate of quality usage for resident n in MG
?n target QoSE for resident n in MG
?n(t) quality usage of residents n in time slot t
?maxn maximum quality usage of resident n in a single slot
?bn(t) basic electricity usage of resident n in time slot t
P(t) available electricity from DRERs to supply quality usage in
time slot t
U(t) electricity generated from DRERs in time slot t
Q(t) electricity purchased from macrogrid in time slot t
S(t) electricity sold on the market in time slot t
pn(t) electricity to the resident n
C(t) purchasing price on the utility market in time slot t
W(t) selling price ob the utility market in time slot t
In(t) indicator function for outage events of quality usage of
resident n in time slot t
Cmin minimum purchasing price of utility from macrogrid
Cmax maximum purchasing price of utility from macrogrid
Wmin minimum selling price of utility to macrogrid
Wmax maximum selling price of utility to macrogrid
Xk(t) battery virtual queue for the battery k
Zn(t) QoSE virtual queue for the resident n
vector?(t) states of the virtual queues Xk(t) and Zn(t)
L(?) Lyapunov function
?(t) Lyapunov one step drift
A(t) proposed scheduling policy including Q(t),S(t),Rk(t),
Dk(t) and pn(t)
y? optimal objective value of problem (3.19)
?A(t) relaxed scheduling policy for problem (B.6)
?y optimal objective value of problem (B.6)
yopt optimal objective value of problem (3.9)
75
Chapter 4
Overview of Green Video Streaming over Celluar Networks and Variable Bit Rate Video
4.1 Green Video Streaming over Celluar Networks
Besides the redesign of the electricity delivery networks, it is equally important to study the
energy efficiency at the demand side. The rapid proliferation of information and communications
technology (ICT) infrastructures continuously contribute to the overall carbon footprint and bring
the intensity of ?green? communications to the research community. Among various green com-
munication technologies, we focus on the energy efficiency of base stations (BS?s) for downlink
video streaming. This is due to the expected surge in wireless video data, as well as the drastic
increase in the deployment of BS?s. It is reported that, in a typical cellular network, more than 50%
of the total power consumption is directly attributed to BS equipment [11]. At every year, 120,000
BS?s are added, catering to the 300 million to 400 million new mobile phone users adopting mobile
services around the world [130]. Furthermore, Many wireless operators have launched femtocell
service recently, such as AT&T, Sprint, Verizon, and Vodafone. The wide adoption of femtocells
will greatly intensify the proliferation of BS?s. Therefore, any small improvement in the energy ef-
ficiency of video coding or wireless video streaming system will be amplified by the huge volume
of wireless video data and number of BS?s deployed, and will result in considerable environmental
impact. Considerable savings on electrical bills could be achieved for wireless operators when the
power of BS?s is minimized for video streaming. The reduced electricity consumption will also
bring about important improvement in the overall carbon footprint of the wireless industry and
achieve the goal of ?green? communications.
In the following chapters, we consider the challenging problem of streaming multiuser vari-
able bit rate (VBR) videos in the downlink of cellular networks. This is motivated by the fact that
VBR video offers stable and superior quality over constant bit rate (CBR) videos. Furthermore,
76
(a) Frame 1, VBR (b) Frame 29, VBR (c) Frame 1, CBR (d) Frame 29, CBR
Figure 4.1: Perceived quality of VBR and CBR videos: Football video coded with an H.264 codec.
many stored video content are coded in the VBR format. It is important to support such stored
VBR video over existing wireless networks without the need for transcoding.
VBR video has stable visual quality for the frames, but at the cost of large variations in the
bit rate, while CBR video maintains a stable bit rate, but the frames have large variations in visual
quality. This is illustrated in Fig. 4.1, where the 15 fps Football sequence is encoded using an
H.264 codec. Both VBR and CBR videos are encoded at the approximately same bit rate (250
kb/s). It can be seen that although the two Frame 1s have similar visual quality, CBR Frame 29
looks worse than VBR Frame 29 when there is high motion. However, we may also observe that
the sizes of frames of VBR video have much larger variation than those of CBR video in Fig. 4.2.
It can be noted that the frame size varies in CBR from frame 42 to 45. That is because that the
content on the video switches from the high motion players to the static field, and this simple scene
is kept from frame 42 to frame 43, which allows the rate control algorithm to choose smaller frame
sizes. Starting from frame 44, the players come back to the scene, thus the larger frame sizes are
selected to compensate the rate decrease in previous frames to keep the average rate constant.
The following chapters distinguish themselves from other energy efficient designs over wire-
less networks in the following aspects: First, instead of power-aware mobile video devices, we
focus on the BS power efficiency when transmitting multiuser videos. As mentioned before, the
BS equipment consumes more than 50% of the total power in a typical cellular network. Thus, it is
important to improve the BS energy efficiency to achieve the goal of green communications. Sec-
ond, we explicitly investigate streaming of multiple VBR videos. VBR videos can offer constant
and better QoE over CBR videos with the same bit budget. However, VBR videos are notori-
ously difficult to schedule and control in wireless networks, due to the high variability and the
77
0 20 40 60 80 100 1200
0.5
1
1.5
2
2.5
3
3.5
4
4.5 x 10
4
Frame
Bits
 
 
VBR(Constant QP)
CBR(Rate Control)
Figure 4.2: Frame size of VBR and CBR videos: Football video coded with an H.264 codec.
complex autocorrelation structure [131?133]. Third, we adopt the control and optimization ap-
proach to optimize the BS energy allocation as well as the quality of experience (QoE) of users,
by jointly considering power control, wireless channel condition, playout buffer constraints, and
playout deadlines.
4.2 VBR Video System Model
It is a challenging problem to support VBR video traffic, which is found to exhibit both strong
long-range and short-range-dependence [131, 132]. It is nontrivial to develop parsimonious traffic
models that can accurately capture the auto-correlation structure. The large frame size variations
may cause frequent playout buffer underflow or overflow [134]. To address this issue, we adopt
a deterministic traffic model for stored VBR video, which considers frame size, frame rate, and
playout buffers [54, 95, 135, 136]. Unlike prior work that is focused on a single video session
over a given CBR or VBR channel, we exploit power control, a unique capability in wireless
networks, to adjust the downlink capacities based on prior knowledge of frame sizes and playout
schedules. Usually large frames are rarely transmitted simultaneously. Thus jointly optimizing
78
Time
Bits
0
Infeasible transmission schedule 
? playout buffer underflow
Infeasible transmission 
schedule ? playout buffer 
overflow
Feasible transmission schedule 
Dn(t) 12Xn(t) 12Bn(t)
Cumulative  consumption 
curve Dn(t)
Cumulative overflow curve Bn(t)
Playout buffer 
size bn
Frame size
Frame interval 3
Figure 4.3: Transmission schedules for VBR video session n.
the BS transmit powers is, in some sense, analogous to statistical multiplexing VBR videos in the
cellular networks.
A stochastic model capturing the auto-correlation structure often requires a large number of
parameters, and is thus hard to be incorporated for scheduling real-time video data. To this end,
we adopt a deterministic model that considers frame sizes, playout buffers, and schedule [95].
Let Di(t) denote the cumulative consumption curve of the i-th user, representing the cumulative
amount of bits consumed by the decoder at time t. The cumulative consumption curve is deter-
mined by video characteristics such as frame sizes and rates, and playout schedule. Assume user i
has a playout buffer of sizebi bits and its video hasLi frames. We can derive a cumulative overflow
curve for user i as
Bi(t) = min{Di(t?1) +bi,Di(Li)}, 0 ?t?Li. (4.1)
Bi(t) is the maximum number of cumulative received bits at time t without overflowing user i?s
playout buffer. Finally we define cumulative transmission curve Xi(t) as the cumulative amount
of bits transmitted to user i at time t. To simplify notation, we assume the video sessions have
79
identical frame rate and the frame intervals are synchronized. Thus a time slot t is equal to the t-th
frame interval, denoted as ?, for 0 ?t? maxi{Li}.1
Since Di(t), Bi(t) and Xi(t) are cumulative curves, they are all nondecreasing functions of
time. The three curves for user n are illustrated in Fig. 4.3. A feasible transmission schedule will
produce a cumulative transmission curveXi(t) that lies withinDi(t) andBi(t), i.e., causing neither
underflow nor overflow at the playout buffer. In practice, Di(t)?s are known for stored videos and
are delivered to the BS?s (or a centralized video scheduler) during the session setup phase, and
Bi(t)?s are then derived as in (4.1).
This deterministic VBR video model will be adopted in following chapters.
1This assumption can be relaxed for more general cases. For example, if the frame rates are different, we can use a time slot duration that is
equal to the greatest common divisor of all the frame intervals (if not too small). If the frame intervals are not synchronized, a time slot can be a
fraction of a frame interval within which the Di(t)?s of all the videos remain constant. In fact, the time slot duration could be arbitrary as in [50]
(i.e., equal to multiple frame intervals). Since the cumulative overflow and consumption curves are known, we can still determine the upper and
lower bounds for the transmission rate in each time slot. The problem formulation and proposed solution procedures to be discussed in the following
sections apply to these cases.
80
Chapter 5
Downlink Power Allocation for Stored Variable-Bit-Rate Video in Cellular Network
5.1 Introduction
In this chapter, we present a downlink power control framework for streaming multiple vari-
able bit rate (VBR) videos in a cellular network with intracell interference. With the determin-
istic VBR video traffic model 4.2, we formulate an optimization problem that jointly considers
donwlink power control, intra-cell interference, VBR video traffic characteristics, playout buffer
underflow and overflow constraints, and base station (BS) peak power constraint. The objective is
to maximize the total throughput, which can achieve high playout buffer utilization. As a result,
playout buffer underflow or overflow events can be minimized. We analyze the convex/concave
regions of the formulated problem and develop a two-step downlink power allocation algorithm
for solving the problem. We also develop a distributed algorithm based on the dual decomposition
technique from convex optimization, in order to reduce the control and computation overhead at
the BS. We evaluate the performance of the proposed distributed algorithm with simulations using
VBR video traces. Our simulation results verify the accuracy of the analysis and demonstrate the
efficacy of the proposed algorithms.
The remainder of this chapter is organized as follows. The deterministic VBR video model is
introduced in Section 4.2. The system model is presented in Section 5.2. We develop a two-step
algorithm to solve the power allocation problem in Section 5.3, and a distributed algorithm based
on dual decomposition in Section 5.4. Simulation results are presented in Section 5.5 and related
work are discussed in Section 5.6. Section 5.7 concludes this chapter.
The notation used in this chapter are summarized in Table 5.1.
81
Table 5.1: Notation for Chapter 5
Symbol Description
N total number of users in a cell
L processing gain
? interference proportion
U set of users sharing the same channel
Tn total number of frames for user n video
bn playout buffer size of user n
Dn(t) cumulative consumption curve at user n
Xn(t) cumulative transmission curve at user n
Bn(t) cumulative overflow curve at user i
vectorP(t) BS transmit power vector in time slot t
vectorPmax max. power allocation vector without overflow
vectorPmin min. power allocation vector without underflow
?P peak power constraint for the BS?s
?Pmin sum of the elements in vectorPmin
vectorP? inflection power vector
vector?P optimal power vector
Gn path gain from BS to user n
Bw channel bandwidth
? duration of a time slot
?n noise power at user n
Cn capacity from the base station to user n
Bw Channel bandwidth
? Constant for the proof of Lemma 5.2
vectorP?(t),vectorP??(t) Auxiliary power allocation in the Lemma 5.2 proof
An Ratio of noise power and channel gain of user n
Pthn Minimum between Pmaxn and P?n
?(l) Stepsize of step l in (5.26)
??(l),??(l),??(l) Stepsize of step l in (5.29)
?n(t) SINR at user unn in time slot t
?minn (t) minimum SINR corresponding to Cminn (t)
?maxn (t) max. SINR for user unn without overflow
?thn receiver sensitivity at user n
F N ?N matrix defined in (5.13)
?,?,? Lagrange Multipliers
L Lagrange function
5.2 System Model and Problem Formation
We consider the downlink of a cellular network. In the cell, a BS streams multiple VBR videos
simultaneously to mobile users in the cell, which share the downlink bandwidth. We assume the
82
last-hop wireless link is the bottleneck, while the wired segment of a session path is reliable with
sufficient bandwidth. Thus the corresponding video data is always available at the BS before
the scheduled transmission time. The deterministic VBR video model is adopted as indicated in
Section 4.2.
We consider N subscribers in the cell and let U denote the set of users. In each time slot t, the
BS transmits to each usernwith powerPn(t) and the power allocation is vectorP(t) = [P1(t),??? ,Pn(t)]T.
We also consider a maximum transmit power constraint ?P, i.e., summationtextn?U Pn(t) ? ?P, for all t. When
the power allocation vectorP(t) is determined, the Signal to Interference-plus-Noise Ratio (SINR) at user
n can be written as [50, 137]
?n(vectorP(t)) = LnGnPn(t)?summationtext
knegationslash=nGnPk(t) +?n
, (5.1)
where Pn is the power allocated to user n, Gn is the path gain between the BS and user n, ?n is
the noise power at user n, Ln is a constant for user n (e.g., processing gain), and ? denotes the
orthogonality factor, with 0 ? ? ? 1. In this chapter, we consider the case ? = 1, where the
SINR of a user not only depends on its own power allocation but also the power allocations of
other users.
We assume slow-fading channels such that the path gains do not change within each time
slot [50]. The downlink capacity Cn(t) depends on the SINR at user n, the channel bandwidth Bw,
and the transceiver design, such as modulation and channel coding. Without loss of generality, we
use the upper bound as predicted by Shannon?s Theorem:
Cn(vectorP(t)) = Bw log
parenleftBig
1 +?n(vectorP(t))
parenrightBig
. (5.2)
In time slot t, Cn(t)? bits of video data will be delivered to user n. The cumulative transmis-
sion curve Xn(t) is
Xn(0) = 0; Xn(t) = Xn(t?1) +Cn(t)?. (5.3)
83
For a feasible power allocation, the cumulative transmission curves should satisfy
Dn(t) ?Xn(t) ?Bn(t), for all n,t, (5.4)
i.e., without causing playout buffer underflow or overflow.
From (5.2)?(5.4), the lower and upper limit on the feasible SINR at user n can be derived as
??
?
??
?minn (t) = max
braceleftBig
exp
braceleftBig
max{0,Dn(t)?Xn(t?1)}
Bw?
bracerightBig
,?thn
bracerightBig
?maxn (t) = exp
braceleftBig
Bn(t)?Xn(t?1)
Bw?
bracerightBig
,
(5.5)
where ?thn is the minimum SINR requirement imposed by the transceiver design. ?minn (t) is the
SINR that the just empties the buffer at the end of time slot t, without causing underflow; ?maxn (t)
is the SINR that just fills up the buffer at the end of time slot t, without causing overflow.
Generally, feasible power allocation vectorP(t) is not unique for a given set of VBR video sessions.
Among the set of feasible solutions, a schedule that transmits more data is more desirable since
it provides more flexibility for optimizing future power allocations. We formulate the problem of
optimal downlink power control for VBR videos, termed problem A, as
(A) maximize
summationdisplay
n?U
log(1 +?n(t)) (5.6)
subject to:
?n(t) = LnGnPn(t)summationtext
knegationslash=nGnPk(t) +?n
, for all n (5.7)
?minn (t) ??n(t) ??maxn (t), for all n (5.8)
summationdisplay
n?U
Pn ? ?P. (5.9)
In problem A, the objective is to achieve the maximum buffer uitilization at the users, under
playout buffer underflow and overflow constraints and BS maximum transmit power constraints.
This is a nonlinear nonconvex problem, to which traditional convex optimization techniques cannot
directly apply. Due to the large variability of VBR traffic, the SINRs may assume values ranging
84
from very low to very high, to avoid playout buffer underflow and overflow. Thus the existing
high SINR approximation [59] and low SINR approximation [138] techniques cannot be directly
applied.
5.3 Two-Step Downlink Power Allocation
In problem A, we consider an interference-limited system, where the capacity of downlink n
depends on the power allocations for all the users. In the following, we first derive conditions for
the optimal solution, and then present a two-step power allocation algorithm for solving problem
A.
Lemma 5.1. If there exists a feasible power allocation vectorP(t) that achieves ?maxn (t) for all n, the
solution is optimal.
Proof. See Appendix C.1.
Lemma 5.2. If the upper limit?maxn (t) cannot be achieved for every usern, then the optimal power
allocation vectorP(t) satisfiessummationtextn?U Pn(t) = ?P.
Proof. See Appendix C.2.
We have the following result for the optimal solution of problem A, which directly follows
Lemmas 5.1 and 5.2.
Theorem 5.1. A solution to problem A is optimal if (i) it achieves the maximum SINR ?maxn (t) for
all n; or (ii) its total transmit power is ?P.
Proof. By Lemma 5.1 and Lemma 5.2, it is straightforward to obtain the result.
Theorem 5.1 implies that we can examine the SINR (or buffer) constraints and the peak power
constraint separately. In the rest of this section, we present a two-step power allocation algorithm
85
for solving problem A. We first examine problem A under condition (i) in Theorem 5.1, to obtain
problem B as
(B) ?maxn (t) = LnGnPn(t)summationtext
knegationslash=nGnPk(t) +?n
, for all n, (5.10)
subject to:
summationdisplay
n?U
Pn ? ?P. (5.11)
In problem B, (5.10) is a system of linear equations of power allocation vectorP(t). Rearranging
the terms, we can rewrite (5.10) in the matrix form as:
(I?F) vectorP(t) = vectoru, for vectorP(t) ?vector0, (5.12)
where I is the identity matrix, F is a N ?N matrix with
Fnm =
??
?
??
0, if n = m
?maxn /Ln, otherwise,
(5.13)
and vectoru = [?1?max1 /LnG1,?2?max2 /LnG2,??? ,?N?maxN /LnGN]T.
Since all the variables are nonnegative, F is a non-negative matrix. According to the Perron-
Frobenius Theorem, we have the following equivalent statements [47]:
Fact 5.1. The following statements are equivalent: (i) there exits a feasible power allocation sat-
isfying (5.12); (ii) the spectrum radius of F is less than 1; (iii) the reciprocal matrix (I?F)?1 =
summationtext?
k=0 (F)
k exists and is component-wise positive.
Based on Theorem 5.1 and Fact 5.1, we derive the first step of the two-step power allocation
algorithm, as given in Algorithm 4. If problem B is solvable, the Step I algorithm in Algorithm 4
produces the optimal solution for problem A according to Theorem 5.1. Otherwise, we derive
86
Algorithm 4: Two-Step Power Allocation Algorithm: Step I
1 BS obtains bn, Dn, and Bn, and computes ?maxn for all user n;
2 BS tests the existence of feasible solutions using (5.12);
3 if (5.12) is solvable then
4 Compute its solution vectorP(t);
5 else
6 Go to Step II of the algorithm, as given in Algorithm 5;
7 end
8 ifsummationtextn?U Pn(t) ? ?P then
9 Stop with the optimal solution vectorP(t);
10 else
11 Go to Step II of the algorithm, as given in Algorithm 5;
12 end
problem C by applying Lemma 5.2, as
(C) maximize
summationdisplay
n?U
log(1 +?n(t)) (5.14)
subject to:
?n(t) = LnPn(t)?P ?P
n(t) +An
, for all n (5.15)
Pminn (t) ?Pn(t) ?Pmaxn (t),for all n (5.16)
summationdisplay
n?U
Pn(t) = ?P, (5.17)
where An = ?n/Gn is the ratio of noise power and channel gain, representing the quality of the
user n downlink channel. Pminn (t) and Pmaxn (t) are solved from (5.8) and (5.15), as
?
??
??
Pminn (t) = ?minn ( ?P +An)/(Ln +?minn )
Pmaxn (t) = ?maxn ( ?P +An)/(Ln +?maxn ).
(5.18)
Since the total transmit power is ?P, the objective value in (5.14) and the SINR in (5.15) for
each user only depends on its own power. Note that all the constraints are now linear. To solve
problem C, we examine the objective function to see if it is convex. We omit time index t in the
following for brevity.
87
0 1 2 3 4 5 6 7 8 9 100
2
4
6
8
10
12
14
16
18
20
P1 / P2 (Watts)
No
rm
ali
ze
d ca
pa
cit
y (
bit
/se
c/H
z)
 
 
Link 1
Link 2
Inflection point for link 2Inflection point for link 1
Figure 5.1: Normalized capacity curves and inflection points for a two-user system, where link 1
has better quality than link 2, i.e. A1 <A2.
Lemma 5.3. The capacity of each user n, Cn, has one inflection point P?n: when Pn <P?n, Cn is
in concave; when Pn >P?n, Cn is convex.
Proof. See Appendix C.3.
The normalized capacities for a two-user system is plotted in Fig. 5.1, with the inflection
points marked. It can been observed that the curves are concave on the left hand side of the
inflection points and convex on the right hand side of the inflection points. The processing gain is
usually large for practical systems (e.g., Ln = 128 in IS-95 CDMA). We assume Ln ? 1 in the
following analysis.
Theorem 5.2. For problem C, there can be at most two links operating in the convex region if
Ln ? (4 ?P + 6An)/( ?P + 3An).
Proof. See Appendix C.4.
For a clean channel where An ? 0, Ln ? 4 will guarantee at most two links operating in the
convex region. The following results are on the impact of channel quality An = ?n/Gn.
88
Theorem 5.3. For a given Ln, the inflection point P?n is an increasing function of An. For two
links i and j with the same transmit power P, if Ai < Aj, we have Ci(P,Ai) > Cj(P,Aj) and
?Ci(Pi,Ai)
?Pi |Pi=P >
?Cj(Pj,Aj)
?Pj |Pj=P > 0.
Proof. See Appendix C.5.
Theorem 5.3 shows that, for two links in the convex region with the same initial power P,
allocating more power to the link with better quality can achieve larger objective value than alter-
native ways of splitting the power between the two links (i.e., achieving the multi-user diversity
gain). Based on the above analysis, we develop the second step of the power allocation algorithm
for solving problem C, as given in Algorithm 5. In Algorithm 5, Lines 3 ? 4 tests the feasibility
of the power allocation. If the sum of the total minimum required power is larger than the BS peak
power, there is no feasible power allocation and there will be buffer underflow. In this case, we
select users with ?good? channels for transmission and suspend the users with ?bad? channels.
The Step II algorithm checks the three possible solution scenarios for problem C depending
on the network status and video parameters:
? All links operate in the convex region;
? One link operates in the convex region and the remaining links operate in the concave region
? Two links operate in the convex region and the remaining links operate in the concave region.
Each of the three phases in Algorithm 5 considers the optimality condition for one of the three
scenarios. In particular, Phase 1 first optimizes the power allocation in the concave region and
then allocates the remaining power to the links that could be moved to the convex region. Phase
2 allocates as much power as possible to the link with the best quality, which could work in the
convex region. Phase 3 attempts to move the second best link to the convex region if the total
power constraint is not violated. Usually when Ln and n are large, Phase 3 will rarely occur due
to the peak power constraint.
89
Algorithm 5: Two-Step Power Allocation Algorithm: Step II
1 Initialization ;
2 BS obtains bn, Dn, and Bn for all user n;
3 BS computes ?maxn , ?minn , and P?n, for all n;
4 BS computes the minimum required sum power ?Pmin =summationtextn?U Pminn and gap ?P = ?P ? ?Pmin;
5 if ?Pmin > ?P then
6 Remove links from U, according to descending order of An, until ?Pmin ? ?P;
7 end
8 Compute Rn = Cn(min{Pmaxn ,Pminn +?P})?Cn(Pminn )min{Pmax
n ,Pminn +?P}?Pminn
, for all Pmaxn > P?n;
9 Phase 1 ;
10 Select all the users satisfying Pminn < P?n as a set U? ?U;
11 Solve problem C under constraints Pminn ? Pn ? min(Pmaxn ,P?n) andsummationtext
n?U? Pn ? ?P
? = ?P ?summationtext
n??U? P
minn , where ?U? is the complementary set of U?, and obtain
solution vectorP1;
12 Calculate Rn by updating Pminn to the solution in Line 11 and assign the remaining power to the
nodes in set U, in descending order of Rn;
13 Obtain the Phase 1 solution, vectorPp1, and objective value fp1;
14 Phase 2 ;
15 Select the link with the maximum Rn, and assign all the available power ?P ? ?Pmin to the link,
until either all the power is assigned or the link attains power Pmaxn ;
16 if there is still power to allocate then
17 Select all the nodes in set U\n and repeat Lines 8 ? 12;
18 end
19 Obtain the Phase 2 solution, vectorPp2, and objective value fp2;
20 Phase 3 ;
21 Select the first 2 links with the largest Rn?s, and assign all the available power ?P ? ?Pmin to the
links, until all the power is assigned or the links attains power Pmaxn , and repeat Lines 16 ? 18;
22 Obtain the Phase 3 solution, vectorPp3, and objective value fp3;
23 Decision ;
24 Choose the largest objective value among fp1, fp2 and fp3, and stop with the corresponding power
assignment;
In Algorithm 5, Line 7 presents a convex optimization component, for which several effective
solution techniques can be applied. In the following section, we describe a distributed algorithm
for Line 7 based on dual decomposition.
5.4 Distributed Algorithm
As discussed in Section 5.3, the core of the Step II algorithm is to solve problem C in the
concave region (see Fig. 5.1). In this section, we present a distributed algorithm for this purpose,
90
where the users are involved in power allocation to reduce the control and computation overhead
on the BS. In the concave region, we have problem D as
(D) maximize
summationdisplay
n?U
log(1 +?n(t)) (5.19)
subject to:
?n(t) = LnPn(t)?P ?P
n(t) +An
, for all n (5.20)
Pminn (t) ?Pn(t) ? min{Pmaxn ,P?n},for all n (5.21)
summationdisplay
n?U
Pn(t) ?Ptot, (5.22)
where Ptot ? ?P is the total power budget for the links in the concave region. For brevity, we define
Pthn = min{Pmaxn ,P?n} and drop the time slot index t in the following analysis.
Introducing non-negative Lagrange multipliers ?n,?n, and? for constraints (5.21) and (5.22),
respectively, we obtain the Lagrange function as
L(vectorP,vector?,vector?,?) (5.23)
=
summationdisplay
n?U
bracketleftbigg
log
parenleftbigg
1 + LnPn?P ?P
n +An
parenrightbigg
+?n(Pn ?Pminn )
bracketrightbigg
+
summationdisplay
n?U
bracketleftbig?
n(Pthn ?Pn)
bracketrightbig+?
parenleftBigg
Ptot ?
summationdisplay
n?U
Pn
parenrightBigg
=
summationdisplay
n?U
bracketleftbigL
n(Pn,?n,?n,?)+(?nPthn ??nPminn )
bracketrightbig+?P
tot,
where
Ln(Pn,?n,?n,?) = log
parenleftbigg
1 + LnPn?P ?P
n +An
parenrightbigg
+ (?n ??n ??)Pn. (5.24)
Since Ln only depends on user n?s own parameters, we have the dual decomposition for each user
n. For given Lagrange multipliers (or, prices) ??n, ??n, and ??, we have the following subproblem for
each user n.
?Pn( ??n, ??n,??) = [Pminn ?Pn ?Pthn ]argmaxLn(Pn, ??n, ??n,??), for all n. (5.25)
91
Subproblem (5.25) has a unique optimal solution due to the strict concavity of Ln. We use the
gradient method [14] to solve (5.25), where user n iteratively updates its power Pn as:
Pn(l+ 1) (5.26)
= [Pn(l) +?(l)?nLn(Pn)]?
=
bracketleftbigg
Pn(l) +?(l) Ln(
?P +An)
( ?P ?Pn +An)( ?P + (Ln ?1)Pn +An) +?(l)(?n ??n ??)
bracketrightbigg?
,
where [?]? denotes the projection onto the range of [Pminn ,Pthn ]. The update stepsize ?(l) varies in
each step l and is determine by the Armijo Rule [14]. Due to the strict concavity of Ln, the series
{Pn(1),Pn(2),???} will converge to the optimal solution ?Pn as l??.
For a given optimal solution for problem (5.25), vector?P = [ ?P1,??? , ?PN]T, the master dual problem
is as follows:
minimize L(vector?P,vector?,vector?,?) (5.27)
subject to: ?n,?n,? ? 0, for all n. (5.28)
Since the objective function (5.27) is differentiable, we also apply the gradient method to solve the
master dual problem [14], where the Lagrange multipliers are iteratively updated as
?
????
?
????
?
?n(l+ 1) = [?n(l)???(l)? ?L(vector?,vector?,?)??n ]+, for all n
?n(l+ 1) = [?n(l)???(l)? ?L(vector?,vector?,?)??n ]+, for all n
?(l+ 1) = [?(l)???(l)? ?L(vector?,vector?,?)?? ]+,
(5.29)
where [?]+ denotes the projection onto the nonnegative axis. The update stepsizes are also deter-
mined by the Armijo Rule [14]. As the dual variablesvector?(l),vector?(l),?(l) converge to their stable values
as l??, the primal variables vector?P will also converge to the optimal solution [100].
92
Algorithm 6: Distributed Power Control Algorithm
1 BS sets l = 0 and prices ?n(l),?n(l),?(l) equal to some nonnegative initial values for all n;
2 BS broadcasts the prices to the selected users;
3 Each user locally solves problem (5.25) as in (5.26) to obtain its requested power;
4 Each user sends its requested power to the BS;
5 BS updates prices ?n(l),?n(l),?(l) as in (5.29) and broadcasts new prices
?n(l + 1),?n(l + 1),?(l + 1) for all n;
6 Set l = l + 1 and go to Step 3, until the solution converges;
The distributed algorithm is given in Algorithm 6, where the above procedures are repeated
iteratively. The BS first broadcasts Lagrange multipliers to the users. Each user updates its re-
quested power as in (5.26), using local information Pminn , Pmaxn , P?n, An, Ln, and BS peak power
?P. Each user then sends its requested power back to the BS, and the BS will updates the Lagrange
multipliers as in (5.29). And so forth, until the optimal solution is obtained.
5.5 Simulation Results
We evaluate the proposed algorithms with MATLAB simulations, where the deterministic
VBR traffic model and the optimization solution algorithms are implemented. We use a cellular
network with 20 users;1 the network topology is illustrated in Fig. 5.2. The downlink bandwidth
is 1 MHz. The path gain averages are Gn = d?4n , where dn is the physical distance from the BS to
user n. The downlink channel is modeled as log-normal block fading with zero mean and variance
8 dB [50]. The processing gains are set to Ln = 128 for all n. The distance dn is uniformly
distributed in [100m, 1000m]. The device temperature is T0 = 290 Kelvin and the equivalent noise
bandwidth is Bw = 1MHz. The BS peak power constraints is set to ?P = 10 Watts. We use three
VBR movies traces, Star Wars, NBC News, and Tokyo Olympics, from the Video Trace Library
maintained at Arizona State University [139]. We plot the sizes of the first 100 frames of the NBC
News video sequence in Fig. 5.3, to illustrate the high variation of VBR video frame sizes, which
makes it very challenging to develop accurate mathematical models. Each playout buffer is set to
1.5 times of the largest frame size in the requested VBR video.
1The number of users/links in the cellular network is chosen according to the resource specified in the simulation:
bandwidth and the total BS power limit.
93
Figure 5.2: Topology of the cellular network.
0 10 20 30 40 50 60 70 80 90 1000
1
2
3
4
5
6
7
8 x 10
4
Frame
Fr
am
e s
ize
 (b
its
)
Figure 5.3: The sizes of the first 100 frames of the NBC News sequence.
In the simulations, we have 7 user streaming NBC news, 7 users streaming Star Wars, and
6 users streaming Tokyo Olympics. The proposed power allocation algorithm is executed at the
beginning of each time slot. In Fig. 5.4, we plot the cumulative consumption, overflow and trans-
mission curves for NBC News transmitted to user 2. The top sub-figure is the overview of 10,000
94
0 2000 4000 6000 8000 100000
0.5
1
1.5
2
2.5 x 10
5
Frame Index
Cu
mu
lat
ive
 bi
ts 
(kb
its)
2620 2625 2630 2635 2640
4.88
4.9
4.92
4.94
4.96
4.98
x 104
Frame Index
Cu
mu
lat
ive
 bi
ts 
(kb
its)
 
 
Cumulative consumption curve
Cumulative overflow curve
Cumulative transmission curve
Figure 5.4: Transmission schedule for video NBC News to user 2.
frames. We also plot the curves from frame 2,620 to 2,640 in the bottom sub-figure. We observe
that the cumulative transmission curve X(t) is very close to the cumulative overflow curve B(t),
indicating that the algorithm always aim to maximize the transmission rate as allowed by the buffer
and power constraints. The playout buffers are almost fully utilized most of the time. There is no
playout buffer overflow and underflow for the entire range of 10,000 frames. Among the NBC
News frames, frame 2,625 is the largest frame. We let seven out of the 20 links playout this largest
frame simultaneously at time slot 2,625 in the simulation. There is no buffer underflow under such
heavy load.
In Fig. 5.5, we plot the power allocation and price updates for all the 20 links in one of the
10,000 time slots. The power and prices converges in around 70 steps. The converged power
vector is vector?P = [0.0022, 1.396, 0.0356, 0.0024, 1.396, 0.0351, 0.0016, 1.396, 0.0356, 0.0026, 1.396,
0.0356, 0.0023, 1.396, 0.0356, 0.0018, 1.396, 0.0356, 0.0034, 1.394] Watts. Note that with the
distributed algorithm, the computation in each iteration only consisting updating power or price as
in (5.26) and (5.29), which takes only a negligible amount of time. The 70-step convergence time
is very small comparing to the power control in cellular standards (e.g., 1500 Hz for UMTS power
95
0 20 40 60 800
1
2
3
Steps
Power (Watts)
(a) Transmit powers
0 20 40 60 800
0.5
1
Steps
?
(b) Lagrange mulipliers ?
0 20 40 60 800
0.5
1
Steps
?
(c) Lagrange mulipliers ?
0 20 40 60 800
0.5
1
1.5
Steps
?
(d) Lagrange muliplier ?
Figure 5.5: Convergence of power allocation and Lagrange multipliers.
control [140]). Since the gradient method is used, the convergence of the algorithm is dependent
on the gradients, which further depend on the system parameters such asLn andAn. Another main
factor for the convergence speed is the choice of the stepsize. As discussed, we use Armijo Rule to
determine step size, in which the stepsize evolves according to the difference of the target values
between steps.
Finally, we compare the proposed algorithm with a diversity-aware power allocation scheme,
where the BS allocates power according to channel quality. With this scheme, the best channel
n will be assigned power to achieve its maximum required power Pmaxn (t). Then the second best
channel will be allocated power until its maximum required power is achieved, and so forth until
all of ?P is allocated. In this simulation, we increase the number of users to 50 to stress the capacity
of the cellular network, such that the system is close to saturate. The purpose is to show the
performance of the algorithms under a nearly congested scenario, which is more interesting in
performance analysis than an under-load scenario.
96
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frame Index
Av
era
ge
 pl
ay
ou
t b
uff
er 
uti
liz
ati
on
 
 
Proposed power control algorithm
Diversity based algorithm
Figure 5.6: Average playout buffer utilization for the entire video sequence (10000 frames).
We compare the algorithms by their average playout buffer utilization. In Fig. 5.6, we plot
the average buffer utilizations achieved by the proposed scheme and the diversity-aware scheme
for the entire video sequence. A zoomed in version is presented in Fig. 5.7 from frames ranging
from 2,000 to 2,500. It can be seen that the proposed algorithm consistently achieves high buffer
utilization, ranging from 60% to 100%. The diversity scheme achieves buffer utilization lower
than 50% for frames from 2,000 to 2,250. Such considerably higher buffer utilization translates to
better video quality: there is no buffer overflow or underflow for proposed algorithm, while there
is buffer underflow in 17% of the playout frames for the diversity scheme.
5.6 Related Work
Most of the prior work on VBR video streaming consider wired networks, which can be clas-
sified according to their traffic models, i.e., statistical or deterministic models. With the former
approach, stochastic models are developed to capture the burstiness in VBR traffic. In [131, 132],
the authors observed the long-range-dependence in VBR video traffic and modeled the autocorre-
lation with self-similar processes. This class of work provides valuable insights on the nature of
97
2000 2050 2100 2150 2200 2250 2300 2350 2400 2450 25000.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frame Index
Av
era
ge
 pl
ay
ou
t b
uff
er 
uti
liz
ati
on
 
 
Proposed power control algorithm
Diversity based algorithm
Figure 5.7: Average playout buffer utilization for frames 2000 to 2500.
VBR video traffic. The stochastic models can be incorporated in quality of service (QoS) mecha-
nisms for VBR videos, and for traffic synthesizing in simulations [133].
With the deterministic approach, the piecewise-constant-rate transmission and transport (PCRTT)
method was used, aiming to optimize one or more objectives while preserving continuous video
playout. In [135], Liew and Chan proposed bandwidth allocation schemes for dynamically shar-
ing a CBR channel among multiple VBR video streams, either i) to minimize the total receiver
buffer size, or ii) to avoid underflow and overflow for a given playout buffer size. In [95], Salehi
et al. considered smoothing VBR video over a CBR link and developed an effective algorithm to
achieve the greatest smoothness in rate. In [141], McManus and Ross introduced a dynamic pro-
gramming framework to set PCRTT rates and intervals to optimize different objective functions.
These techniques do not directly apply to our problem of VBR over wireless networks, due to the
fundamental difference between wireless and wired CBR links.
The downlink power allocation problem was studied in [50, 137], aiming to obtain the power
allocation that maximizes a properly defined system utility. A distributed algorithm based on
dynamic pricing and partial cooperation was proposed. Deng, Webera, and Ahrens [142] studied
98
the achievable maximum sum rate of multi-user interference channels. These papers provide the
theoretical foundation and effective algorithms for utility maximization of downlink traffic, but the
techniques used cannot be directly applied for VBR video over wireless networks with buffer and
delay constraints.
In [54, 143], the authors studied the problem of one VBR stream over a given time-varying
wireless channel. In [143], it was shown that the separation between a delay jitter buffer and
a decoder buffer is in general suboptimal, and several critical system parameters were derived.
In [54], the authors studied the frequency of jitters under both network and video system constraint
and provided a framework for quantifying the trade-offs among several system parameters. In this
chapter, we jointly consider power control in wireless networks, playout buffers, and video frame
information, and address the more challenging problem of streaming multiple VBR videos, and
present a cross-layer optimization approach that does not depend on any specific channel or video
traffic models.
5.7 Conclusions
We developed a downlink power allocation model for streaming multiple VBR videos in a cel-
lular network. The model considers interactions among downlink power control, channel interfer-
ence, playout buffers, and VBR video traffic characteristics. The formulated problem aims at max-
imizing the total transmission rate under both peak power and playout buffer overflow/underflow
constraints. We presented a two-step approach for solving the problem and a distributed algorithm
based on the dual decomposition technique. Our simulation studies validated the efficacy of the
proposed algorithms.
99
Chapter 6
Downlink Power Control for Variable Bit Rate Video over Multicell Wireless Networks
6.1 Introduction
In this chapter, we extend power control in variable bit rate (VBR) video streaming to mul-
ticell wireless networks scenario. We consider video streaming over a multicell wireless network,
a wireless network architecture widely deployed all over the world. We consider the typical case
of downlink video transmissions. For the multicell system, generally intra-cell interference can be
effectively controlled with precise synchronization or the use of guard times. The capacities of the
downlinks are mainly limited by the inter-cell interference due to simultaneous base station (BS)
transmissions using the same channel. Therefore, effective downlink power control is necessary to
support concurrent videos.
In this chapter, we presented a problem formulation that considers downlink power control,
inter-cell interference, VBR video characteristics, and playout buffer requirements. The objective
is to achieve high playout buffer utilization, under playout buffer underflow and overflow con-
straints and peak power constraint. This is a nonlinear nonconvex problem to which traditional
convex optimization techniques [59] and low- or high- Signal to Interference-plus-Noise Ratio
(SINR) approximations [59, 138] do not directly apply.
We first derive the condition of the existence of feasible power assignments, which can achieve
downlink capacities to guarantee no buffer underflow and overflow. We then develop a central-
ized algorithm that can produce solutions with bounded optimality gap. Specifically, we use the
Reformulation-Linearization Technique (RLT) to obtain a linear programming (LP) relaxation of
the original problem. Solving this LP relaxation yields an upper bound to the original problem.
Interestingly, since the constraints are preserved in the relaxation procedure, the upper-bounding
100
solution is also feasible to the original problem; the corresponding objective value with this solu-
tion provides a lower bound to the global optimum. The LP relaxation is then incorporated into
the branch-and-bound framework to obtain a centralized algorithm, which can produce a solution
within the (1-?) range of the global optimal.
To simplify computation and control, we also develop a distributed algorithm based on dis-
tributed constrained power control (DCPC) [48], where each BS iteratively updates transmit power
based on feedback of measured SINR at the target receiver. It is shown that with DCPC, the power
vector converges to a unique power vector that can achieve the goal of maximizing playout buffer
utilization and avoiding playout buffer underflow and overflow. We evaluate the proposed al-
gorithms with simulations using VBR video traces [139] and fading channels. The distributed
algorithm is shown to achieve a performance very close to that of the centralized algorithm. Both
algorithms are demonstrated to be highly effective for streaming VBR videos over multicell wire-
less networks.
In the reminder of this chapter, we present the problem formulation in Section 6.2. We de-
scribe a centralized algorithm in Section 6.3 and a distributed algorithm in Section 6.4. Simulation
results are presented in Section 6.5 and related work is discussed in Section 6.6. Section 6.7 con-
cludes this chapter. The notation used in this chapter are summarized in Table 6.1.
6.2 Problem Statement
6.2.1 Network and Video System Model
We consider the downlinks of an M-cell wireless network as shown in Fig. 6.1. In each cell,
a BS streams video to mobile users in the cell, each allocated with a downlink channel. A channel
is a spectral resource slot, the nature of which depends on the specific multiple access technique
adopted for the multicell network. Without loss of generality, we assume that the downlink chan-
nels within a cell are orthogonal (e.g., due to perfect synchronization of spreading codes or use of
guard times). The main interference at a user stems from the concurrent downlink transmissions
101
Table 6.1: Notation Table for Chapter 6
Symbol Description
M total number of cells (or, BS?s)
U set of users sharing the same channel
Li total number of frames for user i video
bi playout buffer size of user i
Di(t) cumulative consumption curve at user i
Xi(t) cumulative transmission curve at user i
Bi(t) cumulative overflow curve at user i
Pm(t) transmit power of BS m in time slot t
vectorP(t) BS transmit power vector in time slot t
?P peak power constraint for the BS?s
vectorP? optimal power vector to the LP relaxation
Gmk path gain from BS k to user unm
Bw channel bandwidth
? duration of a time slot
?m noise power at user unm
Cm capacity of the cell m downlink
Cminm (t) min. rate for user unm without underflow
?Cminm (t) the largest value of Cminm (t)
?m(t) SINR at user unm
?minm (t) minimum SINR corresponding to Cminm (t)
??minm (t) SINR corresponding to ?Cminm (t)
?maxm (t) max. SINR for user unm without overflow
?thm receiver sensitivity at user unm
A matrix of path gain ratios defined in (D.5)
?min defined as diag{?min1 (t),?min2 (t),??? ,?minM (t)}
??min defined as diag{??min1 (t),??min2 (t),??? ,bar?minM (t)}
? M ?M matrix defined as (??min ??min)
?tarm target user unm SINR for distributed alg.
??tar defined as diag{?tar1 (t),?tar2 (t),??? ,?tarM (t)
?m vector of elements ?m/Gmm
um RLT substitution variable for logarithm terms
vmk RLT substitution variable for quadratic terms
?, ? parameters for the distributed algorithm
in neighboring cells that use the same channel. There is a need for the BS?s to adopt power control
to mitigate such inter-cell interference.
We consider the problem of streaming multiple VBR videos in the multicell network. We
assume the wired segment of a video session path is reliable with sufficient bandwidth, while
the last-hop wireless link is the bottleneck [144]. Thus the corresponding video data is always
102
Figure 6.1: A multicell wireless network with concurrent VBR video sessions. The inter-cell
interference experienced by the central cell user is illustrated.
available at the BS before the scheduled transmission time. We adopt the deterministic VBR video
model in 4.2.
6.2.2 Problem Formation
For the multicell wireless video network, consider a specific channel and let U = {un1,un2,
??? ,unM} denote the set of users sharing the channel, where unm is the user in cell m.1 Let the
BS transmit power vector be vectorP(t) = [P1(t),P2(t),??? ,PM(t)]T in time slot t. The capacity of the
downlink from BS m to user unm, denoted as Cm(t), depends on the SINR at unm, which can be
written as
?m(vectorP(t)) = G
m
mPm(t)summationtext
knegationslash=mG
m
k Pk(t) +?m
, (6.1)
where Gmk is the path gain from BS k to user unm and ?m is the noise power at unm. We assume
slow-fading channels such that the path gains do not change within each time slot [50], but vary
over different time slots following a certain distribution. The downlink capacity Cm(t) also de-
pends on the channel bandwidth Bw and the transceiver design, such as modulation and channel
10-1 index variables can be used to model the case where no user uses the channel in some cells, but are omitted for brevity.
103
coding. Without loss of generality, we use the upper bound as predicted by Shannon theorem:
Cm(vectorP(t)) = Bw log
parenleftBig
1 +?m(vectorP(t))
parenrightBig
. (6.2)
The impact of fading channels is incorporated in the SINR in (6.2). For practical systems, the
achievable capacity may be a fraction of Cm(vectorP(t)), but this part is omitted for brevity.
Once the link capacity is determined, Cm(t)? video bits will be delivered to user unm in that
time slot. The cumulative transmission curve Xm(t) can be written as
Xm(0) = 0; Xm(t) = Xm(t?1) +Cm(t)?. (6.3)
Assume peak power constraint 0 ? Pm ? ?P, for all m. The problem is to determine the transmit
power vector vectorP(t), for 0 <t? maxi{Li}, such that the resulting cumulative transmission curves
satisfy
Dm(t) ?Xm(t) ?Bm(t), for all m,t, (6.4)
i.e., without causing playout buffer underflow or overflow. Since the video frames have variable
sizes and the video sessions have random phases, large frames from different sessions are less
likely to occur in the same time slot. Jointly considering power control for the downlinks is, in
some sense, analogous to statistical multiplexing of VBR video flows.
From (6.2)?(6.4), the feasible SINR range at user unm is
emax{0,Dm(t)?Xm(t?1)}Bw? ?1 ??m ?eBm(t)?Xm(t?1)Bw? ?1. (6.5)
In (6.5), the lower bound is the SINR that just empties the buffer without causing underflow. The
upper bound is the SINR that just fills up the buffer without causing overflow.
Generally, the feasible transmit power vector vectorP(t) is not unique for a given set of VBR video
sessions. Among the set of feasible solutions, a schedule that transmits more data is more desir-
able since it provides a larger search space for optimizing transmit power vectors for future time
104
slots. Omitting the constant Bw, we formulate the optimal power control problem for VBR videos,
termed problem OPT-VBR, as
maximize
summationdisplay
m?U
log(1 +?m(t)) (6.6)
subject to: ?m(t) = G
m
mPm(t)summationtext
knegationslash=mG
m
k Pk(t) +?m
, ?m (6.7)
?minm (t) ??m(t) ??maxm (t), ?m (6.8)
0 ?Pm ? ?P, ?m, (6.9)
where ?maxm (t) is the upper bound in (6.5) and ?minm (t) is the larger one between the lower bound
in (6.5) and ?thm, a minimum SINR requirement imposed by the transceiver design.
In problem OPT-VBR, the total amount of video data delivered in time slot t is maximized,
under playout buffer underflow and overflow constraints and peak transmit power constraints. This
is a nonlinear nonconvex problem, to which traditional convex optimization techniques do not
directly apply. Furthermore, to achieve the objective of avoiding playout buffer underflow and
overflow, the SINRs may assume values ranging from very low to very high. Thus the existing
high SINR approximation [59] and low SINR approximation [138] techniques cannot be used. In
the following, we first prove the existence of feasible solutions. We then derive effective centralized
and distributed algorithms to solve problem OPT-VBR in Sections 6.3 and 6.4.
6.2.3 Existence of Feasible Solutions
Due to the wide range of VBR video frame sizes, the corresponding SINR requirements also
assume a wide range of values. Under conditions where many video sessions coincidently transmit
their large frames in the same time slot, problem OPT-VBR may not have a feasible power assign-
ment to deliver all the frames. In this section, we derive the conditions for the existence of feasible
power assignments. We assume a centralized scheduler in the multicell network, which has prior
knowledge of all the path gains and the cumulative consumption and overflow curves.
105
We define the minimum required rate for user unm in time slot t, denoted as Cminm (t), as the
bit rate such that the playout buffer is just emptied, but without underflow, at the end of time slot t.
We have the following result for Cminm (t).
Lemma 6.1. The largest value for the minimum required rate Cminm (t) is ?Cminm (t) = [Dm(t)?
Dm(t?1)]/?.
Proof. See Appendix D.1.
We have the following condition for the existence of a feasible power assignment for problem
OPT-VBR.
Theorem 6.1. There exits a feasible power assignment for problem OPT-VBR for time slot t, if
there exits a feasible power assignment that can achieve the rate vector bracketleftbig?Cmin1 (t), ?Cmin2 (t), ???,
?CminM (t)bracketrightbig.
Proof. See Appendix D.2.
Theorem 6.1 allows us to evaluate, for a given set of videos, if there is a feasible power
assignment for each time slot. There is no need to consider the transmission schedules and playout
buffer occupancies in previous time slots. At the beginning of time slot t, we obtain ??minm (t) from
the cumulative consumption curve D(t) and channel gains. If the linear system (D.4) is solvable
and the resulting vectorP satisfies constraint (6.9), then there is a feasible power assignment for problem
OPT-VBR for this time slot. The following fact from [51] can be used for the feasibility test.
Fact 6.1. The following statements are equivalent: (i) there exits a feasible power assignment sat-
isfying (D.4); (ii) the maximum modulus eigenvalue ofparenleftbig??minAparenrightbigis less than 1; (iii) the reciprocal
matrix (I???minA)?1 =summationtext?k=0parenleftbig??minAparenrightbigk exists and is positive component-wise.
106
6.2.4 Comparison with a Lazy Scheme
A ?lazy? scheme is proposed in [136] for VBR video transmission over a wired network.
This is an ON-OFF scheme and it transmits a video frame as late as possible before its playout
deadline at the maximum link speed, which minimizes the required client buffer size. In multicell
multi-user wireless VBR video streaming, the maximum link speed varies from time to time due
to interference and channel fading. Thus, the original lazy scheme cannot be applied directly.
We enhance the lazy scheme to support multicell multi-user VBR video streaming, termed W-
Lazy, where every BS transmits a frame that is needed for playout in the next time slot. Then we
can determine the rate vector (and the transmit powers) as given in Theorem 6.1. We use W-Lazy
as a benchmark for comparison and evaluation of the proposed algorithms. We have the following
results for W-Lazy.
Corollary 6.1.1. Problem OPT-VBR has a larger solution space than the W-Lazy scheme.
Proof. This result directly follows Theorem 6.1.
Corollary 6.1.2. If vectorC?(t) = [C?1(t),...,C?n(t)] is the solution to problem VBR-OPT, then any other
vector vectorC(t) that is element-wise smaller than vectorC?(t) has a smaller solution space.
Proof. This result also follows a similar process as in the proof of Theorem 6.1.
6.3 Centralized Algorithm
As discussed, problem OPT-VBR is a nonlinear nonconvex problem, to which traditional
convex optimization techniques do not directly apply. In this section, we present a centralized
algorithm to provide solutions with bounded optimality gap. We first use RLT to obtain a linear
programming (LP) relaxation of problem OPT-VBR [145]. We then incorporate the linear relax-
ation into a branch-and-bound framework, which can produce (1-?)-optimal solutions.
107
6.3.1 Reformulation and Linearization
We first apply polyhedral outer approximation for the logarithm functions in problem OPT-
VBR to obtain a Polynomial Programming Problem OPT-VBR(p) [146]. We then use RLT bound-
factor product constraints to relax the quadratic terms to obtain an LP relaxation OPT-VBR(l).
The time slot index (t) is dropped in the following to simplify notation.
We first process the logarithm functions in the objective function. Letting um = log(1 +?m),
we obtain a linear objective function summationtextm?U um and new constraints um = log(1 +?m). We deal
with the new constraints using polyhedral outer approximation. Since ?minm ? ?m ? ?maxm , we
choose H points, denoted as {?hm}, within this range as
?hm = (1 +?minm )
parenleftbigg1 +?max
m
1 +?minm
parenrightbigg h
H?1
?1,h = 0,??? ,H?1, (6.10)
where ?0m = ?minm and ?H?1m = ?maxm . We can obtain a convex envelop for the logarithm function
in [?minm , ?maxm ], which consists of H tangent lines at the H points given in (6.10) and the line
segment connecting the two end points. We relax the logarithm constraint by using its convex
envelop, represented by the following new linear constraints:
?
??
??
um ? log(1+?minm )?max
m ??minm
(?maxm ??m) + log(1+?maxm )?max
m ??minm
(?m ??minm )
um ? log(1 +?hm) + ?m??hm1+?h
m
, h = 0,1,??? ,H?1.
The first line is for the segment connecting the two end points, and the second line is for the tangent
lines at the H points. A four-point approximation is illustrated in Fig. 6.2.
With the polyhedral outer approximation, we obtain a polynomial programming problem
OPT-VBR(p), as given in (6.11) ? (6.18). We can rewrite the last constraint (6.18) as
summationdisplay
knegationslash=m
Gmk ?mPk ?GmmPm +?m?m = 0,
108
1m
um=log(1+1m)
0
Tangent lines
max 1m1m31m21m11mmin 1m0
Figure 6.2: Four-point polyhedral outer approximation for um = log(1 +?m), 1 <?minm ? ?m ?
?maxm .
maximize
summationdisplay
m?U
um (6.11)
subject to:
GmmPm ?
parenleftBiggsummationdisplay
knegationslash=m
Gmk Pk +?m
parenrightBigg
?minm ? 0, ?m (6.12)
GmmPm ?
parenleftBiggsummationdisplay
knegationslash=m
Gmk Pk +?m
parenrightBigg
?maxm ? 0, ?m (6.13)
0 ?Pm ? ?P, ?m (6.14)
um ? log(1 +?
min
m )
?maxm ??minm (?
max
m ??m) +
log(1 +?maxm )
?maxm ??minm (?m ??
min
m ), ?m (6.15)
um ? log(1 +?hm) + ?m ??
h
m
1 +?hm , ?m,h (6.16)
?hm = (1 +?minm )
parenleftbigg1 +?max
m
1 +?minm
parenrightbigg h
H?1
?1,?m,h (6.17)
?m = G
m
mPmsummationtext
knegationslash=mG
m
k Pk +?m
, ?m. (6.18)
which contains quadratic terms in the form of ?mPk. We next introduce RLT bound-factor product
constraints to remove such terms and to obtain an LP relaxation.
Define substitution variables vmk = ?mPk, for all m, k. Since ?m and Pk are bounded by
their respective lower and upper bounds as ?minm ? ?m ? ?maxm and 0 ? Pk ? ?P, we obtain the
109
following RLT bound-factor product constraints
?
????
????
???
????
?
(?m ??minm )?(Pk ?0) ? 0
(?maxm ??m)?(Pk ?0) ? 0
(?m ??minm )?( ?P ?Pk) ? 0
(?maxm ??m)?( ?P ?Pk) ? 0.
Substituting ?mPk = vmk, we obtain the following four linear constraints for vmk:
?
????
????
???
????
?
vmk ??minm Pk ? 0
?maxm Pk ?vmk ? 0
?m ?P ?vmk ??minm ?P +?minm Pk ? 0
?maxm ?P ??maxm Pk ??m ?P +vmk ? 0.
The quadratic terms Pk?m are thus replaced with vmk with the above linear RLT bound-factor
constraints, and an LP relaxation OPT-VBR(l) is obtained as given in (6.19) ? (6.30).
The LP relaxation OPT-VBR(l) can be effectively solved with an LP solver in polynomial
time. The optimal solution to the LP relaxation consists of {vectorP?,vectoru?,vector??,vectorv?}. It is worth noting that
during the reformulation and linearization procedure, we mainly relax the logarithm function in
the objective function of OPT-VBR. The original constraints of OPT-VBR are preserved in OPT-
VBR(l). Therefore, we have the following theorem regarding the feasibility of the solution, which
greatly simplifies the local search procedure of the branch-and-bound algorithm to be presented in
Section 6.3.2.
Theorem 6.2. The optimal transmit power vector vectorP? to the LP relaxation OPT-VBR(l) is a feasible
solution to the original problem OPT-VBR.
6.3.2 Branch-and-Bound Algorithm
According to Theorem 6.2, we can substitute the optimal power assignment vectorP? for the LP
relaxation into problem OPT-VBR to obtain a lower bound, while the LP solution itself provides
110
maximize
summationdisplay
m?U
um (6.19)
subject to:
GmmPm ?
parenleftBiggsummationdisplay
knegationslash=m
Gmk Pk +?m
parenrightBigg
?minm ? 0, ?m (6.20)
GmmPm ?
parenleftBiggsummationdisplay
knegationslash=m
Gmk Pk +?m
parenrightBigg
?maxm ? 0, ?m (6.21)
0 ?Pm ? ?P, ?m (6.22)
um ? log(1 +?
min
m )
?maxm ??minm (?
max
m ??m) +
log(1 +?maxm )
?maxm ??minm (?m ??
min
m ), ?m (6.23)
um ? log(1 +?hm) + ?m ??
h
m
1 +?hm , ?m,h (6.24)
?hm = (1 +?minm )
parenleftbigg1 +?max
m
1 +?minm
parenrightbigg h
H?1
?1,?m,h (6.25)
vmk ??minm Pk ? 0, ?m,k negationslash= m (6.26)
(?m ??minm ) ?P ?vmk +?minm Pk ? 0, ?m,k negationslash= m (6.27)
?maxm Pk ?vmk ? 0, ?m,k negationslash= m (6.28)
(?maxm ??m) ?P ??maxm Pk +vmk ? 0, ?m,k negationslash= m (6.29)summationdisplay
knegationslash=m
vmkGmk ?GmmPm +?m?m = 0, ?m. (6.30)
an upper bound. We next incorporate the LP relaxation into a branch-and-bound framework to
obtain an algorithm that can produce (1-?)-optimal solutions.
Branch-and-bound is an iterative method for solving optimization problems, especially for
discrete and combinatorial problems. A branch-and-bound procedure has two key components.
The first one, called branching, is to partition a problem into subproblems. The procedure is re-
peated recursively to each of the subproblems and all produced subproblems naturally form a tree
structure, i.e., the branch-and-bound tree. Its nodes are the constructed subproblems. The leaves
of the tree is also call the Problem List. The other component is bounding, which is a fast way of
finding upper and lower bounds for the optimal solution for each subproblem. For a maximization
problem, an infeasible upper bound (UB) can be found by solving a relaxed problem. A local
111
search algorithm is then used to explore the neighborhood, to find a feasible lower-bounding solu-
tion (LB). As discussed, we can easily derive upper and lower bounds by solving the LP relaxation
(no need for local search). The core of the approach is an observation that, for a maximization
task, if the upper bound for a subproblem l1 is smaller than the lower bound for any other sub-
problem l2, then l1 and the branch rooted at l1 can be safely discarded from the tree, such that the
computational complexity can be reduced. This procedure is called pruning.
The algorithm terminates when the upper bound reaches (1 +?) of the lower bound. Let the
optimal object value be O ?UB, we have LB ? 11+?UB ? 11+?O = (1??+?2 ??3 +???)O ?
(1 ? ?)O, for 0 ? ? ? 1. The pseudo code for the branch-and-bound algorithm is given in
Algorithm 7.
6.3.3 Enhancement
In this section, we further introduce a heuristic to accelerate the convergence of the branch-
and-bound algorithm. At the beginning of time slot t, if the playout buffer occupancy is above a
certain threshold, say, 80%, and Xm(t?1) ?Dm(t) at user m, we set Pm(t) = 0 and remove the
link from the optimization process.
Generally the playout buffer size should at least be greater than the largest frame size. Given
the large variations in VBR frame sizes, there could be multiple frames stored when the buffer is
close to full. When the above conditions are satisfied, there is little chance of buffer underflow
at the end of time slot t even if we do not transmit anything to user m. On the other hand, if we
schedule a non-zero power Pm(t) for this link, only a small amount of bits can be transmitted due
to the buffer overflow constraint, but at the cost of reduced SINRs at all other links. Excluding
such links from transmission not only greatly speeds up the convergence of the branch-and-bound
algorithm, but also increases the SINR and capacity of other active links.
112
Algorithm 7: Branch-and-Bound Algorithm
1 Initialization ;
2 Obtain LP relaxation OPT-VBR(l) as Prob 1 ;
3 Set optimal solution sol = ?, Problem list S = {Prob 1},UB = ?, and LB = 0 ;
4 Solve Prob 1 for solution {vectorP?,vectoru?,vector??,vectorv?} and upper bound UB1 ;
5 Use vectorP?, (6.6), and (6.7) to get lower bound LB1 ;
6 Set UB = UB1 and LB = LB1 ;
7 Iteration & pruning ;
8 Select Prob l with the largest UBl in S and set UB = UBl;
9 if LBl >LB then
10 Set sol = vectorP?l and LB = LBl ;
11 if UB ? (1 +?)LB then
12 stop with solution sol ;
13 else
14 remove all probs k in S with UBk ? (1 +?)LB ;
15 end
16 end
17 Partition ;
18 For Prob l, find the maximum relaxation error among all RLT variables, e.g.,
maxm,k{|?mPk ?vmk|} ;
19 Evaluate the following condition:
(?maxm ??minm )?min{??m??minm ,?maxm ???m}? (Pmaxm ?Pminm )?min{P?m?Pminm ,Pmaxm ?P?m};
20 if true then
21 partition [?minm ,?maxm ] into [?minm ,??m] and [??m,?maxm ] ;
22 else
23 partition [Pminm ,Pmaxm ] into [Pminm ,P?m] and [P?m,Pmaxm ] ;
24 end
25 Bounding ;
26 Solve the partitioned probs l1 and l2 to get solutions soll1, soll2 and bounds UBl1, UBl2,
LBl1, LBl2 ;
27 Remove Prob l from S ;
28 if (1 +?)LB <UBl1 then
29 add Prob l1 into S ;
30 end
31 if (1 +?)LB <UBl2 then
32 add Prob l2 into S ;
33 end
34 if S = ? then
35 stop ;
36 else
37 go to Step 8 ;
38 end
113
6.4 Distributed Algorithm
Although the RLT-based branch-and-bound algorithm can provide a (1??)-optimal solution,
it requires a centralized implementation. A centralized controller is needed to collect network, link
and video related information, and to update transmit power for each downlink. In this section, we
develop a distributed algorithm for problem OPT-VBR that can be implemented in each BS and
operate with local information.
We assume each BS obtains video cumulative consumption curves and playout buffer sizes
for its users during the video session initiation phase. At the beginning of time slot t, each BS
m computes for user unm the minimum rate as [Dm(t) ?Xm(t? 1)]/?, i.e., the data rate that
empties the playout buffer at the end of time slot t but without underflow, and the maximum rate
as [Bm(t)?Xm(t?1)]/?, i.e., the data rate that makes the playout buffer full at the end of time
slot t but without overflow. BS m then translates the minimum and maximum rates to minimum
and maximum SINRs, i.e., ?minm (t) and ?maxm (t) as given in (6.5). In the following, we again drop
the time slot index (t) to simplify notation.
To maximize objective function (6.6), BS m sets a target SINR as ?tarm = ?maxm , and tries to
achieve the target SINR by adjusting its transmit power. The problem then becomes a Distributed
Constrained Power Control (DCPC) problem [48]. BS m first randomly sets its initial transmit
power as 0 < P0m ? ?P. Let ?im be the i-th SINR measurement at user unm, which is fed back to
BS m. BS m then uses the following DCPC algorithm to update its power after receiving the i-th
SINR feedback:
Pim = min
braceleftbigg
?P,?tarm
?im P
i?1
m
bracerightbigg
, i = 1,2,??? . (6.31)
If the ?tarm ?s are feasible (see Section 6.2.3), the power vector series {vectorP0,vectorP1,??? ,vectorPi,???} is
proved to converge to a unique positive power vector satisfying the following equation [48]
vectorP = minbraceleftBigvector?P,?tar(AvectorP +vector?)bracerightBig, (6.32)
114
Algorithm 8: DCPC Algorithm
1 Initialization ;
2 BS m obtains bm, Dm, and Bm for user unm ;
3 BS m computes SINR bounds ?maxm and ?minm ;
4 BS m sets ?tarm = ?maxm and Pm(0) ? (0, ?P] ;
5 Iteration ;
6 BS m receives SINR feedback ?im and updates its power as:
Pim = minbraceleftbig?P,(?tarm /?im)Pi?1m bracerightbig;
7 if (Pim = ?P for ? iterations) & (?im negationslash= ?tarm ) then
8 reset the target SINR as: ?tarm = ?minm +??(?tarm ??minm ) ;
9 end
10 i = i+ 1 and go to Step 6 ;
where ?tar = diag{vector?tar} = diag{?tar1 ,?tar2 ,??? ,?tarM }. Furthermore, the converged power vector
vectorP?(t) also achieves the target SINR ?tarm (t) for each BS m. The convergence result is summarized
as the following fact from [48].
Fact 6.2. With the DCPC algorithm (6.31), the transmit power vector converges to a unique pos-
itive power vector vectorP? satisfying (6.32). After convergence, either vectorP? achieves vector?tar or at least one
of the components in vectorP? is equal to ?P.
The pseudo code for the distributed DCPC algorithm is given in Algorithm 8, where ? is a
fraction in (0,1) and ? is a positive integer. If BS m?s transmit power remains at the maximum
power ?P for ? iterations, while the target SINR ?tarm is still not achieved, we reset the target SINR
as ?tarm = ?minm +??(?tarm ??minm ) and restart the iterative update process. We choose ? = 0.618,
the reciprocal of the golden ratio, and ? from 2 to 5 in our simulations.
In practice, the path gains vary over time due to channel fading. It is possible that during some
time slot, the transmission is not feasible even for the minimum required rate. It is nontrivial to test
the feasibility of the target SINR vector vector?tar in a distributed manner with only local information.
In fact, if the target SINR vector is infeasible, the problem of finding the largest set of links that
can be supported at the given SINRs is proved to be NP-Complete [147]. Therefore, we adopt the
following heuristic strategies to handle the case when the target SINR vector cannot be achieved
by a feasible power assignment due to deep fading channels.
115
i) In the first time slot, if the DCPC algorithm does not converge in a certain number of steps,
suspend the transmission of the video with the largest frame size for sometime and retry the
algorithm.
ii) Adopt the acceleration enhancement as in the centralized algorithm, which is described in
Section 6.3.3.
iii) If the DCPC algorithm does not converge for the reduced ?tarm (see Line 5 in Algorithm 8),
further reduce the target SINR as ?tarm = ?minm +?? (?tarm ??minm ). If still no convergence
when ?tarm = (1 +?)??minm , for 0 < ? ? 1, all the links whose buffer will not be empty in
the next time slot will pause their transmissions. Since the algorithm always tries to transmit
as more data as possible (i.e., by setting a high target SINR whenever possible), it is highly
likely that such links won?t have buffer underflow in the following time slots.
iv) If all the above steps fail, the BS suspends its transmission and the user freezes the playout
precess until the next time slot.
6.5 Simulation Results
To evaluate the performance of the proposed algorithms, we simulate streaming VBR videos
in a 7-cell wireless network. We assume the channels within a cell are orthogonal and inter-cell
interference is the major limiting factor. The channel bandwidth is Bw = 1 MHz. The path
gain averages are set to ?Gmk = d?4km, where dkm is the physical distance from BS k to user unm.
We assume Rayleigh fading channels in all the simulations, where the normalized path gain is
exponentially distributed as f(Gmk ) = exp{?Gmk /?Gmk } for Gmk ? 0. The distance from a user to
its corresponding BS is uniformly distributed from 100 m to 1000 m and the inter-cell BS distance
is from 1600 m to 2000 m. The temperature isT0 = 290 Kelvin and the equivalent noise bandwidth
is also 1 MHz. The peak power constraint is ?P = 1 Watt.
In each cell, the channel is dedicated to one mobile user for VBR video streaming. We assume
BS?s 1, 4 and 7 are streaming movie Star Wars, BS?s 2 and 5 are streaming NBC News, and the
116
remaining links 3 and 6 are transmitting Tokyo Olympics. We use the VBR traces for these videos
from the Video Trace Library hosted at Arizona State University [139] in all the simulations. The
playout buffer size is set to be 1.5 times of the largest frame size in the requested VBR video.
6.5.1 Centralized Algorithm
We implement the branch-and-bound centralized algorithm using MATLAB. We choose ? =
10% for the simulations. From the VBR video traces, we derive the cumulative consumption and
overflow curves. The centralized algorithm computes the optimized power assignment for the BS?s
at beginning of each time slot. In Fig. 6.5.1, we plot the cumulative consumption, overflow and
transmission curves for Star Wars transmitted on link 1. The top subfigure is for 10,000 frames. We
also plot the curves from frame 1,960 to frame 1,980 in the bottom subfigure, while frame 1,969
has the largest size among the 10,000 frames. We observe that the cumulative transmission curve
X1(t) is very close to the cumulative overflow curveB1(t), indicating that the centralized algorithm
always aims to maximize the transmission rate as allowed by the buffer and power constraints, and
the playout buffer is fully utilized for most of the time. There is no playout buffer overflow or
underflow for the entire range of the movies.
In Fig. 6.6, we plot the upper and lower bounds for objective function (6.6) for time slot 1.
This is the hardest time slot with respect to power control, since all the sessions are transmitting
I-frames and all the playout buffers are empty in this time slot in our simulations. We observe
the optimality gap between UB and LB is continuously decreased until the ? = 0.1 threshold is
reached. In other time slots where the frame sizes are not consistently large and the playout buffers
are close to full, it usually takes only a few (e.g., 5 or 6) iterations to reach the optimality gap
threshold.
We also evaluate the accelerated scheme under the same video and network conditions. The
curves for link 1 are plotted in Fig. 6.5.1. It can be seen that during time slots 1,963, 1,967,
and 1,971, there is no transmission on link 1 since the playout buffer is over 80% full. Pausing
transmission in these time slots makes it easier for other links to transmit large frames and speeds
117
0 2000 4000 6000 8000 100000
5
10
15 x 10
4
Frame-time
Cu
mu
lat
ive
 bi
ts 
(kb
its)
1960 1965 1970 1975 1980
2.34
2.36
2.38
2.4
2.42
x 104
Frame-time
Cu
mu
lat
ive
 bi
ts 
(kb
its)
 
 
Cumulative consumption curve
Cumulative overflow curve
Cumulative transmission curve
Figure 6.3: The cumulative overflow, transmission, and consumption curves when transmitting
Star Wars at link 1 with centralized algorithm in the seven-cell network.
0 2000 4000 6000 8000 100000
5
10
15 x 10
4
Frame-time
Cu
mu
lat
ive
 bi
ts 
(kb
its)
1960 1965 1970 1975 1980
2.34
2.36
2.38
2.4
2.42
x 104
Frame-time
Cu
mu
lat
ive
 bi
ts 
(kb
its)
 
 
Cumulative consumption curve
Cumulative overflow curve
Cumulative transmission curve
Figure 6.4: The cumulative overflow, transmission, and consumption curves when transmitting
Star Wars at link 1 with Accelerated centralized algorithm in the seven-cell network.
118
0 2000 4000 6000 8000 100000
5
10
15 x 10
4
Frame-time
Cu
mu
lat
ive
 bi
ts 
(kb
its)
1960 1965 1970 1975 1980
2.34
2.36
2.38
2.4
2.42
x 104
Frame-time
Cu
mu
lat
ive
 bi
ts 
(kb
its)
 
 
Cumulative consumption curve
Cumulative overflow curve
Cumulative transmission curve
Figure 6.5: The cumulative overflow, transmission, and consumption curves when transmitting
Star Wars at link 1 with DCPC in the seven-cell network.
up the convergence of the algorithm, while causing no buffer underflow at link 1. Since usually
large frames rarely occur in the same time slot (except for time slot 1), this is analogous to statistical
multiplexing of VBR videos. We find in the simulation, a link can pause in over 60% of the time
slots with the acceleration heuristic, resulting in significant reduction in computation time.
6.5.2 Distributed Algorithm
We next examine the performance of DCPC. The network and video setups are the same
as those in the centralized algorithm simulations. The cumulative overflow, transmission, and
consumption curves obtained by DCPC are plotted in Fig. 6.5.1 for Star Wars transmitted on link
1. We observe very similar performance as in the case of the centralized algorithm shown in
Fig. 6.5.1. The cumulative transmission curve is again very close to Bm(t), and there is neither
buffer overflow nor underflow during the transmission of 10,000 frames.
To compare the distributed and centralized algorithms, we compute the sum of the bit rates
of all the links in each time slot. The acceleration scheme is not used for both algorithms in this
119
50 100 150 200 250 3000
5
10
15
20
25
30
35
40
Number of Iterations
Sum Throughput (bits/sec/Hz)
 
 
UB
LB
Figure 6.6: Convergence of the branch-and-bound algorithm in time slot 1 when all the I-frames
are transmitted and all the buffers are empty (i.e., the worst case scenario).
6800 6810 6820 6830 68400
5000
10000
15000
Frame?time
Sum of bit rates (kbps)
 
 
Centralized algorithm
Distributed algorithm
Figure 6.7: Rate sums with the algorithms with a seven-link network.
simulation. The rate sums are plotted in Fig. 6.5.2 from time slot 6,800 to 6,840. We observe
that the sum rates achieved by the centralized algorithm and that by the distributed algorithm are
identical for most part of this interval. Examining the rate sums for the entire 10,000 time slots,
we find that the rate sum achieved by the DCPC algorithm is within 99% of the corresponding rate
sum achieved by the centralized algorithm in over 97% of the time slots.
120
5 10 15 20 25 30 35 400
0.2
0.4
0.6
0.8
1
Steps
Power (Watts)
 
 
Link1
Link2
Link3
Link4
Link5
Link6
Link7
Figure 6.8: Convergence of transmit powers of DCPC with a seven-link network.
10 20 30 400
5000
10000
15000
Steps
Bit rate (kbps)
 
 
Link1
Link2
Link3
Link4
Link5
Link6
Link7
Figure 6.9: Convergence of bit rates of DCPC with a seven-link network.
The convergence of the distributed DCPC algorithm is plotted in Figs. 6.5.2 and 6.5.2 for one
of the time slots. The accelerated scheme is incorporated with DCPC, such that a linkmmay pause
its transmission if its buffer is over 80% full and Xm(t? 1) > Dm(t). The evolution of the BS
transmit powers are plotted in Fig. 6.5.2, where after 23 steps, all the transmit powers converges
to a value between 0 and ?P = 1 Watt. The converged power vector is vectorP? = [0.0023, 0.208, 0.185,
0.0013, 0.163, 7.1?10?04, 0.188] Watt. The evolution of the bit rates are plotted in Fig. 6.5.2. It
is interesting to see the data rates converge faster than the transmit powers in this case. All the data
rates reach stable values after a few steps.
121
6.5.3 Empirical Performance Evaluation
We evaluate the performance of the proposed schemes by comparing them with the following
two schemes.
? A round-robin scheme where the BS allocates power in a quality of service (QoS) based
round-robin fashion, which favors the session that would suffer buffer starvation if no trans-
mission is scheduled. When a specific BS is selected for transmission, it transmits the video
with maximum power without overflowing the client buffer, and all its neighbors remain
silent in the same frame-time slot.
? W-Lazy, as described in Section 6.2.3.
First, we investigate the average buffer utilization at end of each time slot. When underflow
happens, the missing frame is discarded, and the next frame will be scheduled for the transmission
in the next time slot. We observe that the proposed RLT and DCPC schemes achieve higher average
buffer utilization than the other two schemes. Fig. 6.10 shows the average buffer utilization from
frame-time slot 1,600 to 1,700. We find that the buffer utilization of RLT and DCPC fluctuate
around 90% mostly, while the utilization of the Round-robin scheme is in the range of 50% to
80%. We also find that the W-Lazy scheme always achieves a zero buffer utilization,since it only
transmits each frame as late as possible in each time slot. At the end of a time slot, all the data will
be comsumed by the user and the buffer is left empty.
We then compare the average number of underflow events in Table 6.2. we find RLT achieves
underflow free transmission, while the number of underflow events for DCPC is negligible in the
simulations. This is because both schemes aim to transmit as much video data as possible under the
feasible condition in each frame-time slot. The extra video data transmitted will be in the playout
buffer to provide a cushion to future large frames or network dynamics. On the other hand, both
Round-robin and W-Lazy suffers a large number of underflow events. We also illustrate the buffer
underflow events in the period from 1,680 to 1,700 in Fig. 6.11. The red dot circles indicate the
buffer underflow. It can be seen that the cumulative transmission curve lies below the cumulative
122
1600 1620 1640 1660 1680 17000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frame-Time
Ave
rag
e b
uff
er 
Uti
liza
tio
nu
 
 
RLT
DCPC
Round-Robin
W-Lazy
Figure 6.10: Average buffer utilization of the four schemes.
Table 6.2: Number of underflow events
RLT DCPC Round-robin W-Lazy
Mean 0 0.2 516 1076
Conf. Int. [0,0] [?0.355,0.755] [442,591] [514,1637]
consumption curve when buffer underflow events occur. This results in an infeasible transmission
schedule, which causes frozen playout.
The average power consumption of the schemes are shown in Fig 6.12. W-Lazy has the lowest
power consumption. Due to the variation of frame size and network condition, the transmission
of W-Lazy are infeasible in many time slots. To prevent the divergence of power allocation, some
video sessions should be paused and the power savings of W-Lazy are achieved by pausing video
transmissions. However, this is at the cost of significantly more buffer underflow events, which are
undesirable for user experience. The Round-robin scheme tries to transmit as much video data as
possible. However, it chooses a session greedily and pauses other unselected video sessions. This
also causes many underflow events for the unselected sessions. Also, due to the round robin fashion
and limited buffer size, when the unselected session become selected, its low buffer utilization will
lead to a larger power consumption in order to fill the buffer, especially when it misses the previous
good channel condition and the channel condition is worse at the current time slot. Thus, the
123
1680 1685 1690 1695 1700
2.7
2.72
2.74
2.76
2.78
x 104
Frame-time
Cu
mu
lat
ive
 bi
ts 
(kb
its)
(a) RLT - NBC News
 
 
Cumulative service curve
Cumulative overflow curve
Cumulative transmission curve
1680 1685 1690 1695 1700
2.7
2.72
2.74
2.76
2.78
x 104
Frame-time
Cu
mu
lat
ive
 bi
ts 
(kb
its)
(b) DCPC - NBC News
 
 
Cumulative service curve
Cumulative overflow curve
Cumulative transmission curve
1680 1685 1690 1695 1700
2.7
2.72
2.74
2.76
2.78
x 104
Frame-time
Cu
mu
lat
ive
 bi
ts 
(kb
its)
(c) Round-Robin - NBC News
 
 
Cumulative service curve
Cumulative overflow curve
Cumulative transmission curve
1680 1685 1690 1695 1700
2.7
2.72
2.74
2.76
2.78
x 104
Frame-time
Cu
mu
lat
ive
 bi
ts 
(kb
its)
(d) W-Lazy - NBC News
 
 
Cumulative service curve
Cumulative overflow curve
Cumulative transmission curve
Figure 6.11: Illustration of underflow events.
proposed algorithms achieve the better balance between the energy consumption and the quality of
experience of the video streaming.
6.6 Related Work
A thorough review in VBR video model and VBR video streaming over wired networks has
been explored in 5.6. Due to the fundamental difference between wireless and wired CBR links,
these techniques do not directly apply to our problem of VBR over the multicellular wireless
networks. In this chapter, we take advantage of power control in wireless networks to adjust the
capacity of wireless links based on video frame size information, such that we can jointly optimize
the transmission of multiple VBR video sessions over multiple VBR channels. Our approach does
not depend on any channel or video traffic models, and can be adopted for CBR video as well.
Power control is an important problem for interference-limited wireless networks. Most prior
work focuses on maximizing network utility in the forms of SINR or bit rate [48,59,138]. In [48],
Grandhi, Zander, and Yates presented centralized and distributed power control algorithms for
124
1 2 3 40
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
Schemes (1: RLT 2: DCPC 3: Round-Robin 4: W- Lazy)
Ave
rag
e p
ow
er 
co
nsu
mp
tio
n (
W)
Figure 6.12: Average power consumption.
achieving target SINRs in a cellular network. In [59], Chiang studied the problem of joint power
control and congestion control, aming to maximize the throughput of TCP-Vegas over an ad hoc
network. Gjendemsj et al. [138] presented centralized binary power control algorithms for max-
imizing the sum rate over multiple interfering links. Although laid out the theoretical foundation
and developed effective algorithms, these techniques cannot be directly applied for VBR video
over multicell wireless networks with buffer and delay constraints.
6.7 Conclusions
We studied downlink power control for VBR video streaming in multicell wireless networks.
The problem formulation considers downlink power control, inter-cell interference, VBR video
characteristics, and playout buffer requirements. We developed a centralized algorithm that can
provide (1-?)-optimal solutions, and a fast distributed algorithm that only needs local informa-
tion. The algorithms are evaluated with extensive simulations with VBR video traces and fading
channels, and are demonstrated to be effective for streaming VBR videos over multicell wireless
networks.
125
Chapter 7
Energy Efficiency on Downlink Multiuser VBR Video Streaming
7.1 Introduction
In this chapter, we present a power efficient downlink power control framework in wireless
system with orthogonal channels for variable bit rate (VBR) streaming with focus on minimizing
the power consumption for the total streaming period. We consider the problem of optimal power
allocation for multiuser VBR video streaming in the downlink of a cellular network with orthog-
onal channels. We assume the wireline segment of a video session path is reliable with sufficient
bandwidth, while the last-hop wireless link is the bottleneck. Thus the corresponding video data
is always available at the BS before the scheduled transmission time. We adopt a deterministic
model for VBR video traffic that incorporates video frame and playout buffer characteristics. The
BS allocates a transmit power to each user in each time slot. The problem is to find the optimal
power control schedule to stream the requested VBR video data to users, such that the total trans-
mit power consumption can be minimized, while minimizing the buffer underflow and overflow
events.
The problem is formulated as a constrained stochastic optimization problem. We show that the
problem fits well with majorization theory, which concerns with partial ordering of real vectors and
order-preserving functions. It answers the question of how to order vectors with nonnegative real
components and its order-preserving functions [13]. A majorization-based solution framework is
developed to tackle the problem. First, we prove that the objective function of the formulated prob-
lem is Schur-convex with the order-preserving property [13]. Second, we investigate the case of a
single VBR video session with relaxed peak power constraint. We develop a majorization-based
power optimal algorithm with low complexity, and prove the power optimality of the proposed
126
algorithm and the uniqueness of the global optimum. We also demonstrate that the proposed algo-
rithm is smoothness optimal as well. Third, we investigate the case of multiuser VBR streaming,
where power allocations for the users are coupled with the BS peak power constraint. We develop
a heuristic algorithm that selectively suspends some video sessions, which will not incur underflow
in the next time slot, when the peak power constraint is violated. Finally, the proposed algorithms
are evaluated with trace-driven simulations [139], and are shown to achieve considerable power
savings and improved video quality over a conventional ?lazy? scheme [136]. The rest of this
chapter is organized as follows. The system model and problem statement are presented in Sec-
tion 7.2. We transform the problem into a majorization problem in Section 7.3. The proposed
algorithms are described in Section 7.4 and simulation results are presented in Section 7.5. We
review related work in Section 7.6. Section 7.7 concludes this chapter. The notation used in this
chapter is summarized in Table 7.1.
7.2 System Model
7.2.1 Network and Video Source Model
We consider the downlink in a cellular network, as shown in Fig. 7.1. There are N active
mobile users in a set U = {1,2,??? ,N} in the cell that subscribe to the video service. A BS
transmits multiple VBR videos to the mobile users. Each user occupies a downlink channel, which
is a spectral/time resource slot, the nature of which depends on the specific multiple access tech-
nique adopted. We assume that the downlink channels within a cell are orthogonal, due to perfect
synchronization of the spreading codes or the use of guard times or frequencies. We further as-
sume the wireline segment of a video session path is reliable with sufficient bandwidth, while the
last-hop wireless link is the bottleneck. Thus the corresponding video data is always available at
the BS before the scheduled transmission time. We adopt a deterministic VBR video model, which
was presented in the Section 4.2.
The BS allocates a transmit power to each user in each time slot. Let vectorP(t) = [P1(t),??? ,PN(t)]
be the power allocation in time slot t. The Signal to Interference-plus-Noise Ratio (SINR) at user
127
Table 7.1: Notation Table for Chapter 7
Symbol Description
N number of mobile users in the cell
U set of users
bn playout buffer size at user n
Tn total number of frames of the user n video
?n total number of bits of the use n video
Dn(t) cumulative consumption curve of user n
Bn(t) cumulative overflow curve of user n
Xn(t) cumulative transmission curve of user n
Pn(t) transmit power of user n in time slot t
vectorP(t) power allocation in time slot t
?n(t) SINR at user n in time slot t
Gn(t) path gain from BS to user n in time slot t
?n(t) noise power at user n in time slot t
cn(t) downlink data rate of user n in time slot t
Bw channel bandwdith
? a transceiver dependent constant
?P peak power constraint
vectorCin the i-th feasible transmission schedule
vectorC?n the optimal solution to (7.9)
vectorCoptn an evenly distributed rate vector
vectorC1n an auxiliary schedule used in Theorem 7.2 proof
?(?) a mapping function RTn ?R defined in (7.12)
vectorX,vectorY,vectorZ n-dimensional nonnegative vectors
Cmax(t),Cmin(t) rate of probe lines
U(vectorC) smoothness utility function
n in time slot t can be written as
?n(t) = Gn(t)Pn(t)/?n(t), (7.1)
where Gn(t) is the path gain from BS to user n and ?n(t) is the noise power at user n in time
slot t. We assume block fading channels, where the Gn(t)?s are i.i.d. random variables with a
certain distribution, for t = 1,??? ,Tn [50]. The downlink data rate can be written as cn(t) =
Bw log(1 +??n(Pn(t))), where Bw is the channel bandwidth and ? depends on the transceiver
design, such as modulation and channel coding. Without loss of generality, we use the Shannon
128
Mobile user
BS
Video server
High speed 
wired link
Figure 7.1: Cellular network and video streaming system model.
capacity as an upper bounding approximation:
cn(t) = Bw log(1 +?n(Pn(t))). (7.2)
Once the link capacity is determined, cn(t)? bits of video data will be delivered to user n in
that time slot. The cumulative transmission curve Xn(t) can be written as
Xn(0) = 0; Xn(t) = Xn(t?1) +cn(t)?. (7.3)
A feasible transmission schedule should cause neither playout buffer underflow nor overflow, i.e.,
satisfying
Dn(t) ?Xn(t) ?Bn(t), for all t,n. (7.4)
7.2.2 Power-Aware Transmission Scheduling
As discussed, we jointly consider the traffic source model in the application layer and power
allocation in the physical layer. We adopt cross-layer design to compute the optimal feasible
transmission schedule {Xn(t), 0 <t?Tn}, for all users n?U, such that the total transmit power
consumption can be minimized. From (7.2), the required transmit power for user n is
Pn(t) = (2cn(t)/Bw ?1)?n(t)/Gn(t). (7.5)
129
A peak power constraint may be applied at the BS, i.e.,summationtextn?U Pn(t) ? ?P, for allt. We then formu-
late the following constrained stochastic optimization problem, aiming to minimize the expected
total transmit power.
minimize
summationdisplay
n?U
Tnsummationdisplay
t=1
E[Pn(t)] (7.6)
subject to: Dn(t) ?Xn(t) ?Bn(t),for all n,t (7.7)
summationdisplay
n?U
Pn(t) ? ?P,for all t. (7.8)
Due to orthogonal channels, transmission in one channel does not interfere with those in other
channels. We first relax the peak power constraint (7.8) (i.e., the case when ?P is large). Then,
problem (7.6) can be decomposed into N sub-problems, each minimizing the transmit power of a
video session.
minimize
Tnsummationdisplay
t=1
E[Pn(t)],for all n?U (7.9)
subject to: Dn(t) ?Xn(t) ?Bn(t), for all n,t. (7.10)
For given Bn(t) and Dn(t), the feasible transmission schedule satisfying (7.10) is not unique.
The i-th feasible transmission schedule is a piece-wise linear curve that can be represented as a
vector vectorCin = [cin(1),??? ,cin(Tn)], where cin(t) ? 0 is the data rate in time slot t, for all t. Let
vectorC?n = [c?n(1),??? ,c?n(Tn)] be the optimal solution to (7.9). For a given VBR video, all the feasible
transmission schedules transmit the same amount of video data, i.e.,
Tnsummationdisplay
t=1
cin(t) =
Tnsummationdisplay
t=1
c?n(t) = ?n, for all i,n. (7.11)
Furthermore, the total transmit power for a feasible schedule can be viewed as a mapping function
? : RTn ?R with
?(vectorCin) =
Tnsummationdisplay
t=1
(2cin(t)/Bw ?1)?n(t)/Gn(t). (7.12)
130
Given such an interpretation of the relaxed problem (7.9), the objective is to find an optimal
feasible vector vectorC?n, such that its total powerP?n, obtained through the mapping ?(?), is the minimum
among all feasible vectors vectorCin. This interpretation fits well with the majorization theory introduced
in Chapter 2, which provides useful order preserving results for inequality problems. Applying
these results, we design an optimal algorithm for solving the decomposed sub-problem (7.9) in
Section 7.4.1. Then we will examine the case of multiuser VBR video streaming coupled with the
peak power constraint in Section 7.4.3.
7.3 Problem Reformulation
7.3.1 Majorization Preliminaries
The preliminaries of majorization have been given in Section 2.3 of Chapter 2.
7.3.2 Schur-convexity of Problem (7.9)
As discussed, problem (7.9) fits well with majorization theory with a mapping function (7.12).
To solve the problem, we need to find the optimal rate vector vectorC?n that is majorized by all other
feasible transmission rate vectors vectorCin, as vectorC?n ? vectorCin, for all i. If the mapping (7.12) is Schur-convex,
then the total transmit power to achieve vectorC?n will also be dominated by those of other feasible
transmission rate vectors. That is, the minimum power is found for problem (7.9). Due to random
path gains and noise powers, stochastic majorization (rather than ordinary majorization) should be
used, which investigates the inequality properties related to random variables [13]. We have the
following theorem for the mapping (7.12) in problem (7.9).
Theorem 7.1. The objective function of (7.9) is an increasing Schur-convex function.
Proof. See Appendix E.1.
With Theorem 7.1, solving problem (7.9) is equivalent to finding the optimal rate vector vectorC?n,
such that vectorC?n ? vectorCin, for all i. Then the total power associated with vectorC?n is the minimum since the
131
mapping (7.12) is order-conserving. The feasible rate vector that is closest to equal distribution
will be majorized by all other feasible rate vectors. Therefore, we transform problem (7.9) to
finding a transmission schedule with the most evenly distributed rates for all the time slots.
7.4 Power Allocation Algorithms
Based on the stochastic majorization interpretation of problem (7.9) and the Schur-convex
property of its objective function, we first develop a power minimization algorithm (PMA) for the
case of relaxed peak power constraint. We prove its optimality and the uniqueness of the global
optimal, as well as the equivalence of power optimal and smoothness optimal. We then describe a
heuristic algorithm for the case of multiple videos coupled with the peak power constraint.
7.4.1 Power Minimization Algorithm
From Section 7.3, an evenly distributed rate vector vectorCoptn = [?n/(Tn?),??? ,?n/(Tn?)] is
majorized by all feasible schedules, i.e., vectorCoptn ? vectorCin, for all i. However, due to the high variability
of VBR video frames, limited playout buffer size, random path gains and noise powers, vectorCoptn may
not always be feasible. In general, each feasible schedule is piece-wise linear with a set of rate
change points, where the rate is increased or decreased to prevent buffer underflow or overflow.
vectorCoptn is a special case with no such rate change points.
The algorithm in Algorithm 9, termed PMA, can generate a piece-wise linear schedule, while
keeping each piece as long as possible and rate variation as small as possible. The operation of the
algorithm is illustrated in Fig. 7.2. Starting from tstart (e.g., h1 in Fig. 7.2), PMA first computes
two probe lines:
? One through the starting point and one of the future corner points of Bn(t), which can go the
furthest into the future without causing buffer underflow or overflow (e.g., linesh1h2 in Case
1 and h5h6 in Case 2 of Fig. 7.2). The rate of this probe line is Cmax(t) = Bn(t)?Xn(tstart)t?tstart .
132
Time
Bits
Time
Case 1 Case 2
h1
h2
h3
h4
h5
h6
h8
h7Bits
Bn(t) Bn(t)Dn(t)
Dn(t)
h'4
h'8
tstart tstop tstart tstop
Figure 7.2: Two cases for determine the next transmission rate.
? The other through the starting point and one of the future corner points of Dn(t), which
can go the furthest into the future without causing buffer underflow or overflow (e.g., lines
h1h3 in Case 1 and h5h7 in Case 2 of Fig. 7.2). The rate of this probe line is Cmin(t) =
Dn(t)?Xn(tstart)
t?tstart .
All feasible transmission curves should lie in between these two probe lines in order to go that far.
Furthermore, when the probe lines end, they hit both on either Bn(t) or Dn(t). Otherwise, we can
always adjust one of the probe lines to make them go even further into the future. For example,
see lines h1h3 and h1h?4 in Case 1 of Fig. 7.2. We can use line h1h2, which goes further into the
future, to replace line h1h?4, and both probe lines hit Dn(t) eventually (also see lines h5h6 and h5h?8
in Case 2).
If both probe lines hit Dn(t) (i.e., Case 1 in Fig. 7.2), any feasible schedule for this interval
will also hit Dn(t), since they must lie in between the two probe lines. We then trace back the
upper probe line (i.e., line h1h2) to find the latest time when the buffer is full (i.e., point h4 at time
tstop). Then segment h1h4 will be chosen as the transmission schedule for this interval, with rate
Bn(tstop)?Xn(tstart)
tstop?tstart .
If both probe lines hit Bn(t) (i.e., Case 2 in Fig. 7.2), any feasible schedule for this interval
will also hit Bn(t). We then trace back the lower probe line (i.e., line h5h7) to find the latest time
when the buffer is empty (i.e., point h8 at time tstop). Then segment h5h8 will be chosen as the
transmission schedule for this interval, with rate Dn(tstop)?Xn(tstart)tstop?tstart .
133
Algorithm 9: Power Minimization Algorithm (PMA-1)
1 BS obtains bn, Dn, Bn, and Bw for all user n ;
2 Set t = 1,tstart = 0,tstop = tc1 = tc2 = 1,Cmin = 0,Cmax = ? ;
3 while some time slots are not assigned a rate do
4 Calculate Cmax(t) and Cmin(t) over interval [tstart,t] ;
5 if Cmin ?Cmin(t) & Cmin(t) ? min{Cmax,Cmax(t)} then
6 Cmin = Cmin(t) and tc1 = t ;
7 end
8 if Cmax ?Cmax(t) & Cmax(t) ? max{Cmin,Cmin(t)} then
9 Cmax = Cmax(t) and tc2 = t ;
10 end
11 if Cmin > min{Cmax,Cmax(t)} then
12 Select Cmin from tstart to tstop = tc1 ;
13 else if Cmax < max{Cmin,Cmin(t)} then
14 Select Cmax from tstart to tstop = tc2 ;
15 else
16 t+ + ;
17 CONTINUE ;
18 end
19 tstart = tstop,tstop = tc1 = tc2 = tstart + 1,t = tstart + 1,Cmin = 0,Cmax = ? ;
20 end
21 while more video frames to transmit do
22 Measure the channel gain of the time slot, calculate power using (7.5), and transmit
the video data ;
23 end
After the transmission schedule for [tstart,tstop) is determined, we set tstart = tstop and repeat
the above procedure to find the schedule for the next time interval. In Algorithm 9, the algorithm
probes for the longest feasible rate starting from tstart in Steps 4?10. In Steps 11?14, the transmis-
sion rate for the interval [tstart,tstop) is determined depending on which of the two cases it is as
illustrated in Fig. 7.2. Steps 16?17 are for the case that the rate does not change in the time slot.
Step 19 resets the variables to start the computation for the next segment of Xn(t). Finally, Steps
21?23 transmit the frames following the computed schedule.
134
7.4.2 Optimality Proof
We next show that the algorithm given in Algorithm 9 computes the optimal solution to prob-
lem (7.9).
Theorem 7.2. The power minimization algorithm PMA is optimal to problem (7.9).
Corollary 7.2.1. The power optimal transmission scheme vectorC?n is unique for given Bn(t) and Dn(t).
Corollary 7.2.2. The computational complexity of Algorithm PMA is O(T2n).
The proofs of above conclusions are similar to the proofs of in Chapter 2 and thus omitted for
brevity.
Note that Algorithm PMA is executed during the session setup time. It only incurs a small
initialization delay. In our simulations with VBR video traces, we find the execution time is usually
negligible. When the channel statistics are changed (i.e., due to handoff), the schedule will be
recomputed for the remaining video frames.
Corollary 7.2.3. The power optimal transmission schedule vectorC?n is also the smoothest one among
all feasible schedules.
Proof. The proof of Corollary 7.2.3 is given in Appendix E.2.
7.4.3 Multiuser Video Transmissions
We now consider problem (7.6) to compute transmission schedules forN VBR video sessions,
which are coupled by the peak power constraint (7.8). Due to the peak power constraint and random
channel gains, the individually calculated transmit powers may violate (7.8) in some time slots.
The problem is further complicated because of the random channel gains, which is not available a
priori (except for the statistics of the channels).
135
Algorithm 10: Power Minimization Algorithm for Multiuser Videos (PMA-m)
1 Execute power minimization algorithm PMA to compute transmission schedules for all
active users ;
2 while there are more video frames to transmit do
3 Measure channel gains of the current time slot and calculate the transmit powers using
(7.5) ;
4 if peak power constraint is violated then
5 Select the users who won?t have underflow even without transmission in this time
slot ;
6 Sort the selected users in decreasing order of powers ;
7 while peak power constraint is not satisfied do
8 Decrease the power of the selected users by the order ;
9 end
10 end
11 Transmit the videos and recalculate the optimal transmission scheme for the paused
mobile users for the next time interval ;
12 end
To solve problem (7.6), we develop a heuristic algorithm, termed PMA-m, as presented in
Algorithm 10. The PMA-m algorithm uses PMA to compute transmission schedules for all active
users. Then based on current channel state information, it computes the power needed to achieved
the rate for each user, and checks the peak power constraint summationtextn?Un Pn(t) ? ?P. If the constraint
is not violated, each user?s video data will be transmitted at the computed power. Otherwise, as in
Steps 4?10, PMA-m selects those users who will not have buffer underflow if their transmissions
are suspended in the following time slot, and sort them in the deceasing order of their required
powers. Starting with the first user, PMA-m decreases the powers of the users in the list; if the first
user?s power reaches 0 W but the peak power constraint is till not satisfied, PMA-mstarts to reduce
the power of the second user in the list; and so forth until the peak power constraint is satisfied.
In some extremely severe channel conditions, the total power ?P cannot even support the
minimum required bit rate for all the users. Some users have to be paused and the current frames
be discarded. The corresponding playout of such a user will be frozen until the next time slot.
Finally, the transmission schedules for the suspended users will be recomputed using PMA as in
Step 11 and the above procedure is repeated.
136
7.4.4 Application to Interactive Video Streaming
The ubiquitous spread of mobile devices and trend of multimedia applications require the
interactive service be supported [148?150]. Thus, it is necessary to investigate how to apply the
proposed schemes to the case of interactive video streaming. Interactive video is a relatively new
and still evolving technology with a broad scope. We focus on the three interactive video streaming
related typical scenarios in the following, and show that the proposed schemes are applicable for
these scenarios for improved performance.
First, for quick response to user inputs, many interactive video streaming applications have
tight delay requirements. Such stringent delay requirements have two implications: (i) unlike
stored video, not all the future frame sizes are known now; only the frames sizes for a short look-
ahead period (LAP) are known. (ii) the playout buffer sizes cannot be large, since a large buffer
usually introduces large delay. Clearly, the proposed schemes can be applied to the look-ahead
time period for which the future frame sizes are known, to compute a schedule for the near future.
Furthermore, given the small playout buffer size, usually a chosen transmission rate won?t last
very long into the future before it hits either the cumulative overflow curve or the cumulative
consumption curve (see Fig.4). Therefore the impact of limited look-ahead period would be small
or moderate at best.
Second, many interactive video applications support VCR controls [151]. For example, a
user may slide the progress bar of the video player to replay or skip a part of the video. This
case is equivalent to a change in the cumulative overflow and consumption curves. The proposed
algorithms will seek to the new start frame that the user requires and recalculate the transmission
schedules for the following frames.
Third, in both ?exploratory? online interactive videos (where a user can move through dif-
ferent locations in a space or view an object from different angles) and video click throughs [152]
(where a user can click objects in the video that are linked to other contents), new data will be
transmitted after each user input. These are equivalent to the case of VCR controls. A new set
137
Table 7.2: VBR Video Trace Statistics
Video Trace Average Rate Frame Rate Average PSNR
Star Wars 331,681 b/s 30 f/s 44.62 dB
NBC News 784,840 b/s 30 f/s 38.80 dB
Tokyo Olympics 509,191 b/s 30 f/s 41.46 dB
Terminator 2 5,085,453 b/s 30 f/s 43.92 dB
From Mars to China 4,849,711 b/s 30 f/s 39.26 dB
Sony Demo 5,803,650 b/s 30 f/s 44.07 dB
of cumulative overflow and consumption curves will be delivered (derived from the new data re-
quested) and new schedules computed.
In Section 7.5.1, we evaluate the performance of the proposed schemes under the above in-
teractive video streaming scenarios. Our simulation results show that the proposed schemes still
achieve considerable power savings and better video quality than a conventional scheme for inter-
active videos.
7.5 Performance Evaluation
We demonstrate the performance of the proposed optimal power control algorithm through
trace-driven simulations. We simulate the downlink of a cell with 1 mile radius. The channels are
assumed to be orthogonal, each with Bw = 1 MHz bandwidth. We assume that bit errors can be
corrected by error correction codes. The path gain averages are ?Gn = d?4n , wheredn is the physical
distance from the BS to user n. We assume log-normal fading with zero mean and 8 dB standard
deviation. The device temperature is 290 Kelvin and the equivalent noise bandwidth is Bw = 1
MHz. The BS streams three movies Star Wars, NBC News, and Tokyo Olympics to active users.
The video traces are obtained from the Video Trace Library at Arizona State University [139]. The
statistics of the three video traces are summarized in Table 7.2.
We first investigate the performance of the power optimal algorithm. In the simulation, the BS
streams 3,000 frames of a video sequence to each mobile user located at different distances to the
BS. The cumulative consumption, overflow and transmission curves of the Star wars video session
138
0 500 1000 1500 2000 2500 30000
1
2
3
4 x 10
4
Frame-time
Cu
mu
lat
ive
 bi
ts 
(kb
its
)
50 100 150 200 250 300
200
400
600
800
1000
1200
1400
Frame-time
Cu
mu
lat
ive
 bi
ts 
(kb
its
)
 
 
Cumulative consumption curve
Cumulative overflow curve
Cumulative transmission curve
Figure 7.3: Simulation results: transmission curves of Star wars.
are plotted in Fig. 7.3. It can be seen that the transmission schedule always lie in between the
cumulative consumption and overflow curves, indicating that there is no playout buffer underflow
or overflow events in this simulation.
We next compare the optimal power algorithm with a conventional transmission scheme with
respect to the average power consumption at the BS. In each time slot, the conventional scheme
only transmits the video data that is needed by the decoder at the end of the time slot. It achieves
a cumulative transmission curve that connects all the corner points of Dn(t) (also called the ?lazy?
scheme). Intuitively, such lazy approach should be energy efficient since it always transmits the
minimal amount of data as needed. However, we will see that the proposed algorithm outperform
this lazy approach in the simulations.
In Fig. 7.4, we plot the average power consumption achieved by the two schemes for increased
distance to the BS. Each point in the figure is the average of 10,000 simulation runs. The 95%
confidence intervals are plotted as vertical bars in the figure, which are all very small.
It can be seen that the proposed algorithm outperforms the conventional scheme for the entire
range of distances examined. When the distance to BS is small, both schemes use small transmit
139
100 200 400 500 800 1000 1200 16000
0.01
0.02
0.03
0.04
0.05
0.06
Distance (meters)
Av
era
ge
 po
we
r c
om
su
mp
tio
n (
W
att
s)
 
 
Optimal
Conventional
Figure 7.4: Simulation results: average power consumption.
powers and the power savings are not very big. However, when the distance is increased, channel
fading has a bigger impact on interference and channel capacity. The proposed algorithm achieves
considerable power savings than the conventional scheme. When the distance is 1,600 m, the total
power of the proposed scheme is only 46.62% of that of the conventional approach, corresponding
to a 54.34% normalized improvement.
We further investigate in more detail the difference of the transmissions between the two
schemes. The position of a mobile node is set to 1,000 m from the BS. The first 3,000 frames
of the Star wars movie are transmitted to the node using the PMA-1 scheme and conventional
scheme, respectively. Fig. 7.5 shows the cumulative power consumption for the first 1,000 frames,
while the energy consumption for each video frame is plotted in Fig. 7.6 for frames in [200, 250].
We observe that at the beginning, the ?lazy? scheme archives smaller power consumption than
the PMA scheme, due to the fact that it only transmits the minimum amount of required frames
in each time slot. However, the transmission of frame 241 of the conventional scheme generates
a sharp power increase, because it encounters a large frame as well as bad channel condition, as
indicated in Fig. 7.6. The transmission curves of the two schemes are plotted in Fig. 7.7 for the
140
0 200 400 600 800 10000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Frame-time
Cu
mu
lat
ive
 P
ow
er 
(W
att
s)
 
 
PMA
Conventional
Figure 7.5: Cumulative power consumptions achieved by PMA-1 and the conventional scheme.
first 250 frames for the two schemes. Although the conventional scheme archives smaller power
consumption frame by frame for about 80% of the 3,000 frames, the average power consumption
of the proposed scheme during the entire period (0.0055 W) is much smaller than that of the
conventioanal scheme (0.0141 W). In summary, the ?lazy? scheme only uses the current video
and channel status, and transmits only the minimum amount of required video data, It does not
effectively utilize the playout buffer capacity. Thus during the entire transmission period, the
cumulative power may increased due to some large frames and bad channel conditions. On the
contrary, the proposed scheme aims at minimizing the total average transmission power during the
entire period. Thus, it achieves considerable power savings comparing to the conventional scheme.
We also obtain the average execution time of the proposed algorithm, under the same setting
but for 20,000 Star Wars frames. We find that the average execution time is about 0.06 s on an
IBM Laptop with Intel T2400 1.83 GHz processor and 2 GB RAM.
Finally, we examine the buffer underflow events. The following scenarios are simulated:
141
200 210 220 230 240 2500
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Frame-time
Po
we
r (
W
att
s)
 
 
Optimal
Conventional
Figure 7.6: Power consumption for each frame achieved by PMA-1 and the conventional scheme.
? Scenario 1: ?P = 1 W; the movies are Star Wars, NBC News, and Tokyo Olympics; 50 mobile
users; Bw = 1 MHz;
? Scenario 2: The same setting as i) except that Bw = 125 KHz;
? Scenario 3: ?P = 10 W; the HD movies are Terminator 2, From Mars to China, and Sony
Demo; 20 mobile users; Bw = 1 MHz.
The HD movies have larger frame sizes and higher variability in frame sizes. The buffer underflow
rates are presented in Table 7.3, each being the ratio of the number of underflow frames over the
total number of frames. PMA-m achieves considerably lower underflow rates for all the three sce-
narios. The PMA-m underflow rates are 0.056%, 13.60%, and 14.26% of that of the conventional
scheme. Therefore, PMA-m achieves not only considerable energy savings, but also much better
video quality for the mobile users.
142
0 50 100 150 200 2500
500
1000
1500
Frame-time
Cu
mu
lat
ive
 bi
ts 
(kb
its
)
 
 
Cumulative consumption curve
Cumulative overflow curve
Cumulative transmission curve
0 50 100 150 200 2500
500
1000
1500
Frame-time
Cu
mu
lat
ive
 bi
ts 
(kb
its
)
 
 
Cumulative consumption curve
Cumulative overflow curve
Cumulative transmission curve
(A) Conventional
(B) PMA
Figure 7.7: Transmission curves achieved by PMA-1 and the conventional scheme.
Table 7.3: Simulation Results: Playout Buffer Underflow Rates
Scenario 1 Scenario 2 Scenario 3
PMA-m 0.0005% 1.8% 1.66%
Conventional 0.89% 13.24% 11.64%
7.5.1 Simulation Results for Interactive Video Streaming
In this section, we study the performance of the proposed algorithms for interactive video
streaming. First, we simulate the interactive real-time video streaming with stringent delay require-
ments and small playout buffer sizes. The same simulation settings are used. All the positions of
mobile nodes are randomly generated in the cell. We apply the proposed PMA-m, but only assume
only the frame sizes in a small LAP are known.
The transmission curves of VBR movie NBC News are plotted in Fig. 7.8 for the first 100
frames, where the length of the LAP is 16 frames. We may observe that the proposed algorithm
PMA-m is executed piecewisely for each block of LAP frames, where LAP=16. For this range
of frames, there is neither playout buffer overflow nor playout buffer underflow occurs. We then
143
0 20 40 60 80 1000
500
1000
1500
2000
2500
3000
Frame-time
Cu
mu
lat
ive
 bi
ts 
(kb
its
)
 
 
Cumulative consumption curve
Cumulative overflow curve
Cumulative transmission curve
Figure 7.8: Transmission curves for real-time interactive video streaming: NBC News and LAP =
16.
Table 7.4: Playout Buffer Underflow Rates for Interactive VBR Videos with Different LAPs
Scenario I Scenario II Scenario III
LAP=16
PMA-m 0.387% 5.969% 4.766%
Conventional 0.711% 12.446% 12.495%
LAP=8
PMA-m 0.377% 5.690% 4.894%
Conventional 0.620% 11.409% 9.645%
compute the number of the underflow frames over the total number of frames, as presented in
Table 7.4. The PMA-m still archive the underflow rate that are 54.56%, 47.96%, 38.15% of those
of the conventional scheme in this case.
To illustrate the impact of the length of LAP, we further decrease it to 8 frames and then
run the simulations with random deployed mobile nodes. The playout buffer underflow rates are
presented in Table 7.4. For the halved delay requirement, naturally the proposed scheme?s perfor-
mance is slightly degraded due to limited information about the video frame sizes. However, the
PMA-m scheme still archives underflow rates that are 60.90%, 49.88%, 50.74% of those of the
144
200 250 300 350 400 450 500500
1000
1500
2000
2500
3000
Frame-time
Cu
mu
lat
ive
 bi
ts 
(kb
its
)
 
 
Cumulative consumption curve
Cumulative overflow curve
Cumulative transmission curve
Figure 7.9: Transmission curves after the user skips the frames in [10s, 20s]: Star Wars.
conventional scheme, indicating considerably superior viewer performance over the conventional
?lazy? approach.
Finally, we demonstrate the application of the proposed schemes to VCR control in interactive
video streaming. We assume that after 10 seconds of streaming the VBR video (i.e, corresponding
to 300 frames), the user skips the next 10 seconds of the video, and then resumes the video playout
from 20 second. We plot the dynamics in the transmission/schedule curves of the Star Wars video
in Fig. 7.9. Comparing the curves with the original non-skipped transmissions in Fig. 7.10, we
observe that after playing out the 300th frame, the frame from 301 to 600 are skipped by user?s
operation. Then the frame 601 is moved to the time-slot 301 and a new transmission schedule is
computed for the following frames.
7.6 Related Work
A thorough review in VBR video model and VBR video streaming over wired networks has
been explored in 5.6. In an interesting work [95], Salehi, et al. applied majorization to VBR video
smoothing and developed a smoothness optimal algorithm. The proof of Theorem 7.2 follows a
145
200 300 400 500 600 700 800500
1000
1500
2000
2500
3000
3500
4000
Frame-time
Cu
mu
lat
ive
 bi
ts 
(kb
its
)
 
 
Cumulative consumption curve
Cumulative overflow curve
Cumulative transmission curve
Figure 7.10: Transmission curves for the original non-skipped video streaming: Star Wars.
similar approach as in [95]. These prior work are based on the assumptions of a single video session
and constant rate channels, which may not be directly applied to the case of wireless networks.
In this chapter, we consider multiuser VBR video streaming within a cellular network with or-
thogonal channels. We jointly consider power control, video traffic, and video palyout information
for power minimization. Our stochastic majorization theory based approach is quite different from
the prior works [54,143], which allow us to develop effective algorithms with low complexity and
proven optimality.
7.7 Conclusion
In this chapter, we studied the problem of downlink multiuser VBR video streaming in cel-
lular networks. Our formulation takes into account the interactions among power control, fading
channels, VBR video traffic and playout characteristics. We formulate a constrained stochastic
optimization problem aiming to minimize the BS power consumption and to avoid playout buffer
146
overflow or underflow. We developed majorization-based algorithms to solve the formulated prob-
lem. For the case of large peak power constraints, we prove the optimality of the proposed algo-
rithm and the uniqueness of the global optimal, as well as the equivalence of power optimal and
smoothness optimal. For the case of multiple videos coupled with the peak power constraint, we
develop an effective heuristic algorithm that selectively suspends some video sessions when the
peak power constraint is violated. The superior performance of the proposed algorithms over a
conventional scheme is validated with trace-driven simulations.
147
Chapter 8
Summary and Future Work
8.1 Summary
In the previous chapters, we proposed frameworks for energy efficient designs in Cyber-
physical systems. We investigated the problems by a control and optimization approach, which
contains Lyapunov optimization [12], majorization [13], nonlinear and convex optimization [14].
The synergy of these advanced mathematical tools produces new visions for the energy efficient
solutions to alleviate energy resource depletion, decrease greenhouse gases emission and air pol-
lution, which evolves a green world in the future.
In Chapter 2, we presented the electric power scheduling policies for smoothing the demand
profile in power distribution networks. We introduced a deterministic electricity supply/demand
model that takes into account time-varying demands and their deadlines. We formulated a con-
straint nonlinear optimization problem and incorporated the theory of majorization to develop
algorithms that can compute smooth optimal electric power schedules. After the smooth power
schedule is obtained, a distributed user benefit maximization load control scheme is used to al-
locate the scheduled power to individual users, while maximizing their level of satisfaction. The
proposed algorithms are highly desirable for grid design and operation, which provide smooth
electric power scheduling, minimum peak power and operating cost. The simulation showed that
the proposed algorithms can alleviate the peak power up to 45%. This means we can deploy trans-
formers, transmission lines and other electrical devices with much smaller capacity, which save the
captital investment in the power grid construction. The generation cost saving is about 5% by the
proposed algorithms. Due to the tremendous amount of energy of bulk generation, 5% cost savings
could indeed contribute to billions of dollars.
148
In Chapter 3, we investigated the problem of balancing supply and demand of electric en-
ergy in microgrid (MG). We presented a novel framework for smart energy management based
on the concept of quality-of-service in electricity (QoSE). Specifically, the resident electricity de-
mand is classified into basic usage and quality usage. The basic usage is always guaranteed by
the MG, while the quality usage is controlled based on the MG status. The microgrid control
center (MGCC) aims to minimize the MG operation cost and maintain the outage probability of
quality usage, i.e., QoSE, below a target value, by scheduling electricity among DRERs, ESS?s,
and macrogrid. The problem is formulated as a constrained stochastic programming problem. The
Lyapunov optimization technique is then applied to derive an adaptive electricity scheduling algo-
rithm by introducing the QoSE virtual queues and energy storage virtual queues. The proposed
algorithm is an online algorithm since it does not require any statistics and future knowledge of the
electricity supply, demand and price processes. We derive several ?hard? performance bounds for
the proposed algorithm. and evaluate its performance with trace-driven simulations. The proposed
electricity scheduling algorithm enables an efficient integration of DRERs, ESS?s, and residential
power quality management into the smart grid by plug-and-play interfaces, and provides a promis-
ing paradigm for smart energy management systems in smart grid. The simulation showed that the
MG operation cost can be saved up to 60%, while keeping the power quality of the users.
In Chapter 5, we studied the problem of power allocation of base station (BS) for streaming
multiple variable bit rate (VBR) videos in the downlink of a wireless cellular network with intracell
interference. We considered a deterministic model for VBR video traffic and finite playout buffer
at the mobile users. The objective is to derive the optimal downlink power allocation for the VBR
video sessions, such that the video data can be delivered in a timely fashion without causing playout
buffer overflow and underflow. The formulated problem is a nonlinear nonconvex optimization
problem. We analyzed the convexity conditions for the formulated problem and proposed a two-
step greedy approach to solve the problem. We also developed a distributed algorithm based on the
dual decomposition technique. The proposed algorithms effectively allocated the power in BS?s
149
to stream the VBR video in cellular networks, while preserving the quality of experience (QoE)
requirement.
In Chapter 6 we further extended the problem of downlink power control for streaming mul-
tiple VBR videos in a multicell wireless networks, where downlink capacities are limited by inter-
cell interference. We adopted a deterministic model for VBR traffic that considers video frame
sizes and playout buffers at the mobile users. The problem is to find the optimal transmit powers
for the BS?s, such that VBR video data can be delivered to mobile users without causing play-
out buffer underflow or overflow. We formulated a nonlinear nonconvex optimization problem
and proved the condition for the existence of feasible solutions. We then developed a centralized
branch-and-bound algorithm incorporating the Reformulation-Linearization Technique, which can
produce (1??)?optimal solutions. We also proposed a low-complexity distributed algorithm with
fast convergence. Numerical results showed that the proposed algorithms is QoE performance
bounded and achieve effective usage of the BS?s power in streaming VBR videos.
In Chapter 7, we addressed the energy saving for BS in wireless cellular networks with or-
thogonal channels to achieve energy efficient VBR video streaming. We took into account the
interactions among power control, fading channels, VBR video traffic, and playout characteristics.
A constrained stochastic optimization problem was formulated aiming to minimize the BS power
consumption and to avoid playout buffer overflow or underflow. We then developed majorization-
based algorithms to achieve energy efficiency, while preserving the QoE demands, for BS downlink
VBR streaming in cellular networks. The simulation showed that the average power consumption
can achieve 54% improvement and the proposed algorithms are also compativle to interactive video
streaming in wireless cellular networks.
8.2 Future Work
Although there has been considerable advances in energy efficient Cyber-physical systems,
many problems still remain open in this interesting area. We briefly extend our discussion on
150
the control and optimization in distributed smart electric energy management systems and low
complexity energy efficient wireless multimedia networks.
Distributed Electric Energy Management Systems
We investigated the efficient electric energy scheduling in the power delivery in previous
chapters. The proposed policies are mainly based on the centralized control in the electric power
delivery networks and Microgrids, which were largely inherited from the control infrastructure of
legacy grid. In the evolution of smart grid, all the components, such as DRERs, ESS?s, MGs, and
users, will be deployed in an ad-hoc mode and accessed in a plug-and-play fashion. Accordingly,
such distributed electric power delivery networks prefer distributed intelligent electric energy man-
agement systems.
Compared to the centralized approaches, distributed management systems make decisions
based on local information with very limited timely information exchange, which reduce the
amount of real time information delivery in the communication networks. The distributed manage-
ment systems are less susceptible to the information loss due to the impairment of the communi-
cation links and also have faster response to the grid status. The security design and cryptographic
systems would also be simplified due to reduced amount of information exchange. However, due to
the limitation on the accessible information, the distributed management systems usually are not
capable of computing a globally optimal decision. Further, analytical models and mathematical
tools would be necessary to provide the performance bounded optimal decisions locally.
Low Complexity Energy Efficient Wireless Multimedia Networks
Due to the intrinsic complexity of video sources and the dynamics and uncertainty of wireless
systems, we conjecture that a holistic approach that encompasses the parameter space would be
necessary and the trade-off between complexity and efficiency of a solution algorithm should be
carefully investigated. In particular, we list some interesting problems that may be worth of further
investigation in the following.
151
? Complexity-distortion analysis and the design of energy-efficient video codecs require ac-
curate models of video codecs. However, a video codec is a combination of complex func-
tional blocks, which makes accurate mathematic modeling extremely difficult. In addition,
the quality of compressed video and the power consumption of the video codec depend on
a large number of parameters. A content-based power-aware design may encounter a large
search space for optimal solutions. It would be useful to develop an accurate and effective
model for video codec that can be incorporated into the mathematical optimization frame-
works for both energy efficient codec design and wireless multimedia system design.
? Cross-layer design has been widely adopted for video networking problems. It has been
shown that an adaptive strategy with cooperation of several layers can achieve optimal power
efficiency for video streaming. Most prior work assume that the wireless channel informa-
tion and network status are known apriori (e.g., by accurate estimation, measurement, and
timely feedback). In practice, this assumption may not be true, because of channel/network
uncertainty and dynamics, and delay and congestion in the network. Thus, balancing the
achievable performance and the control overhead of the design is still an open problem.
Effective schemes that are robust to the channel/network uncertainties would be highly ap-
pealing.
152
Appendices
153
Appendix A
Proofs in Chapter 2
A.1 Proof of Theorem 2.2
The schedule computed by SEPS-DL, vectorP?, could be a straight line or in the general case,
consist of one or more convex and concave segments. If vectorP? is a straight line, it is obvious that
vectorP? ? vectorPk for any other vectorPk (see Fig. 2.3) and it is smooth optimal. In the general case, we need to
show vectorP? ? vectorPk, for all k in every convex or concave segment. Then according to Lemma 2.1, we
have vectorP? ? vectorPk for all k and it is optimal.
Let vectorPk denote an arbitrary feasible schedule. We introduce an auxiliary schedule vectorP1, which
intersects with vectorP? at all its power changing points in every convex segment, and with vectorPk at all its
power changing points in every concave segment, as shown in Fig. A.1.
First, we prove that vectorP? ? vectorP1. For a convex segment of vectorP?, because vectorP1 intersects with vectorP? at
all the power changing points of vectorP?, we have vectorP? = vectorP1 in all the convex segments. For a concave
segment of vectorP?, the endpoints of the concave segment should be the last (first) power changing
point of the previous (next) convex segment, where vectorP? intersects with vectorP1. The power changing
points within the concave segment are all on Wmin(t), as in SEPS-DL. Therefore, vectorP1 is an outer
concave curve above vectorP? (or, it is farther away from the straight line A in Fig. 2.3) in this segment.
From the discussion of Fig. 2.3, we have vectorP? ? vectorP1 for all the concave segments. It follows that
vectorP? ? vectorP1 according to Lemma 2.1.
We next prove that vectorP1 ? vectorPk. For a convex segment of vectorP?, the endpoints of the convex segment
should be the last (first) power changing point of the previous (next) concave segment, where vectorPk
intersects with vectorP1. The power changing points of vectorP1 (or, of vectorP?) in the convex segment are all on
Wmax(t). Therefore, vectorPk is an outer convex curve below vectorP1 in this segment. From the discussion of
Fig. 2.3, it follows that vectorP1 ? vectorPk in all the convex segments. In a concave segment, we have either
154
Time
kWh
0
Wmax(t)
Wmin(t)
Pk
P *
P1
Cross point of Pk and P1
in a concave interval
Cross point of P and P1 in 
a convex interval
*
Figure A.1: Illustration for the optimality proof of SEPS-DL algorithm
vectorPk = vectorP1 or vectorPk ? vectorP1, because vectorP1 intersects with vectorPk at each power changing point. Thus, we obtain
vectorP1 ? vectorPk for all the concave segments, and vectorP1 ? vectorPk according to Lemma 2.1.
Finally we have vectorP? ? vectorP1 ? vectorPk. Proposition 2.1 states that problem (2.3) is Schur-convex and
order preserving. It follows from Fact 2.1 that vectorP? is optimal to problem (2.3).
A.2 Proof of Theorem 2.5
Without loss of generality, we assume the fuel cost at time slot t is C(t) = g(P(t),?(t)),
which a nondecreasing convex function of the supplied power P(t) [92]. This assumption is gen-
erally practical, e.g. classically, the fuel cost for the electric energy generation is usually considered
as a quadratic function of its power generation [103]. We also assume the cost C(t) is affected by
the random factors ?(t), which correspond to the cost uncertainty during the period, such as the
fuel market price disturbance, and etc. We assume each element in ?(t) is i.i.d over slots. Thus,
155
the minimization of the expectation of the total cost over the period L is:
Minimize:
Lsummationdisplay
t=1
E{g(P(t),vector?(t))}
s.t. Wmin(t) ?W(t) ?Wmax(t),?t
P(t) = W(t)/?
Lsummationdisplay
t=1
W(t) = ? (A.1)
Due to the convexity of the cost function g(?) respective to P(t), problem (A.1) is similar to
problem (2.3), except the random variable ?(t). Thus, we resort to stochastic majorization (rather
than ordinary majorization) to solve this constraint nonlinear stochastic optimization problem.
Lemma A.1. The objective function of problem (A.1) is an increasing Schur-convex function.
Proof. The i.i.d. random variables ?(t) are exchangeable for all t. The objective function (A.1)
can be rewritten as
G(vectorP) = E
braceleftBigg Lsummationdisplay
t=1
g(P(t),?(t))
bracerightBigg
,
where g(P(t),?(t)) is convex and increasing with respective to P(t) for each fixed ?(t), and
summationtextL
t=1g(P(t),?(t)) is a symmetric, convex and increasing function w.r.t. P(t). According to
Proposition 11.B.5 in [13], the expectation G(vectorP) is symmetric, convex and increasing. Following
Fact 2.1, the objective function (A.1) is Schur-convex and increasing.
By Lemma A.1, the solution of problem (A.1) is equivalent to finding the optimal power vector
vectorP?, which is majorized by any other feasible power vectors. Thus, the smooth optimal solution vectorP?
in Table 1 is also the solution for problem (A.1). Thus, the proposed SEPS-DL achieves fuel cost
optimal for energy generation.
156
Appendix B
Proofs in Chapter 3
B.1 Proof of Theorem 3.1
According to the system equation (3.12), we have
?
????
?
???
??
Zn(1) ?Zn(0)??n ??n(0) +In(0)
???
Zn(t) ?Zn(t?1)??n ??n(t?1) +In(t?1).
(B.1)
Summing up the inequalities in (B.1), we have
Zn(t) ?Zn(0)??n ?
t?1summationdisplay
?=0
?n(?) +
t?1summationdisplay
?=0
In(?). (B.2)
Dividing both sides by t and letting t go to infinity, we have
limt?? Zn(t)?Zn(0)t ? limt?? 1t
bracketleftBigg
??n
t?1summationdisplay
?=0
?n(?)+
t?1summationdisplay
?=0
In(?)
bracketrightBigg
.
Zn(0) is finite. If Zn(t) is rate stable by a control policy In(t), it is finite for all t. We have
limt?? Zn(t)?Zn(0)t = 0, which yields ?n ??n ??n due to the definitions of ?n and In(t).
157
B.2 Derivation of Equation (3.16)
With the drift defined as in (3.15), we have
?(vector?(t)) = 12E
braceleftBigg Ksummationdisplay
k=1
[(Xk(t+ 1))2 ?(Xk(t))2|Xk(t)]+
Nsummationdisplay
n=1
[(Zn(t+ 1))2 ?(Zn(t))2|Zn(t)]
bracerightBigg
? 12
Ksummationdisplay
k=1
Ebraceleftbigbracketleftbig(Dk(t))2 + (Rk(t))2 + 2Xk(t)(Rk(t)?
Dk(t))|Xk(t)]}+ 12
Nsummationdisplay
n=1
EbraceleftbigbracketleftbigIn(t)2+
(?n?n(t))2 + 2Zn(t)(In(t)??n?n(t))|Zn(t)}
= 12
Ksummationdisplay
k=1
E{[(Dk(t))2 + (Rk(t))2]}+
Ksummationdisplay
k=1
E{Xk(t)(Rk(t)?Dk(t))|Xk(t)}+
1
2
Nsummationdisplay
n=1
E{[(1 + (?n)2)(?n(t))2 + (pn(t))2]}+
1
2
Nsummationdisplay
n=1
E{2Zn(t)(1??n(t))?n(t)?
(Zn(t) +?n(t))pn(t)|Zn(t)}
?B +
Nsummationdisplay
n=1
E{Zn(t)(1??n)?n(t)|Zn(t)}+
Ksummationdisplay
k=1
E{Xk(t)(Rk(t)?Dk(t))|Xk(t)}?
Nsummationdisplay
n=1
E{(Zn(t) +?n(t))pn(t)|Zn(t)}.
where B = 12summationtextKk=1(max{Dmaxk ,Rmaxk })2 + 12summationtextNn=1(2 +?2n)(?maxn )2 is a constant.
158
B.3 Proof of Lemma 3.1
In part 1) of Lemma 3.1, if Q(t) > 0, we have S(t) = 0 according to (3.7). The objective
function of problem (3.19) becomes
VQ(t)C(t) +
Ksummationdisplay
k=1
Xk(t)(Rk(t)?Dk(t))?
Nsummationdisplay
n=1
(Zn(t) +?n(t))pn(t). (B.3)
We first prove Lemma 3.1-1a). If Xk(t) > ?VC(t), we assume Rk(t) > 0. Then we have
Dk(t) = 0 according to (3.4). Accordingly, the object function (B.3) is transformed to
VQ(t)C(t) +
summationdisplay
inegationslash=k
Xi(t)(Ri(t)?Di(t))?
Nsummationdisplay
n=1
(Zn(t) +?n(t))pn(t) +Xk(t)Rk(t)
> VQ(t)C(t) +
summationdisplay
inegationslash=k
Xi(t)(Ri(t)?Di(t))?
Nsummationdisplay
n=1
(Zn(t) +?n(t))pn(t)?VC(t)(P(t) +Q(t)?
summationdisplay
inegationslash=k
(Ri(t)?Di(t))?
Nsummationdisplay
n=1
pn(t)
= V
bracketleftBiggsummationdisplay
inegationslash=k
(Ri(t)?Di(t)) +
Nsummationdisplay
n=1
pn(t)?P(t)
bracketrightBigg
C(t) +
summationdisplay
inegationslash=k
Xi(t)(Ri(t)?Di(t))?
Nsummationdisplay
n=1
(Zn(t) +?n(t))pn(t).
The above inequality is due toXk(t) >?VC(t) andRk(t) = P(t)+Q(t)?summationtextinegationslash=k(Ri(t)?Di(t))?
summationtextN
n=1pn(t) ? 0. The last expression shows, given the assumptionRk(t) > 0, we may find another
feasible electricity allocation scheme ?Q(t) = summationtextinegationslash=k(Ri(t)?Di(t)) +summationtextNn=1pn(t)?P(t), which
can achieve a smaller objective value by choosing Rk(t) = 0 and Dk(t) = 0. This contradicts with
159
the assumption Rk(t) > 0. Thus, we prove that Rk(t) = 0 when Xk(t) > ?VC(t), under the
situation Q(t) > 0,S(t) = 0.
We then prove the second part of Lemma 3.1-1a). It follows (3.4) thatRk(t) = 0 ifDk(t) > 0.
Then (B.3) becomes
VQ(t)C(t) +
summationdisplay
inegationslash=k
Xi(t)(Ri(t)?Di(t))?
Nsummationdisplay
n=1
(Zn(t) +?n(t))pn(t)?Xk(t)Dk(t)
> VQ(t)C(t) +
summationdisplay
inegationslash=k
Xi(t)(Ri(t)?Di(t))?
Nsummationdisplay
n=1
(Zn(t) +?n(t))pn(t) +VC(t)(
Nsummationdisplay
n=1
pn(t)?P(t)?
Q(t) +
summationdisplay
inegationslash=k
(Ri(t)?Di(t)))
= V
bracketleftBiggsummationdisplay
inegationslash=k
(Ri(t)?Di(t)) +
Nsummationdisplay
n=1
pn(t)?P(t)
bracketrightBigg
C(t) +
summationdisplay
inegationslash=k
Xi(t)(Ri(t)?Di(t))?
Nsummationdisplay
n=1
(Zn(t) +?n(t))pn(t).
The above inequality is due to Xk(t) <?VC(t) < 0 and Di(t) = ?P(t)?Q(t)+summationtextinegationslash=k(Ri(t)+
Dk(t)) +summationtextNn=1pn(t) > 0. The last expression shows, given the assumption Dk(t) > 0, we may
find another electricity allocation scheme with ?Q(t) = summationtextinegationslash=k(Ri(t) ? Di(t)) + summationtextNn=1pn(t) ?
P(t), which can achieve a smaller objective value by choosing Rk(t) = 0 and Dk(t) = 0. This
contradicts with the assumptionDk(t) > 0. We thus prove thatDk(t) = 0 whenXk(t) <?VC(t),
under the situation Q(t) > 0,S(t) = 0, which completes the proof of Lemma 3.1-1a).
160
We next prove Lemma 3.1-1b). For the first part, if Zn(t) > VC(t) ??n(t), we assume
0 ?pn(t) < (1??n)?n(t). Following (3.18) and S(t) = 0, we have
B +VQ(t)C(t) +
Ksummationdisplay
k=1
Xk(t)(Rk(t)?Dk(t)) +
summationdisplay
jnegationslash=n
(Zj(t)(1??j)?j(t)?(Zj(t) +?j(t))pj(t)) +
Zn(t)(1??n)?n(t)?(Zn(t) +?n(t))pn(t)
=B +VQ(t)C(t) +
Ksummationdisplay
k=1
Xk(t)(Rk(t)?Dk(t)) +
summationdisplay
jnegationslash=n
(Zj(t)(1??j)?j(t)?(Zj(t) +?j(t))pj(t)) +
Zn(t)[(1??n)?n(t)?pn(t)]??n(t)pn(t)
>B +VQ(t)C(t) +
Ksummationdisplay
k=1
Xk(t)(Rk(t)?Dk(t)) +
summationdisplay
jnegationslash=n
(Zj(t)(1??j)?j(t)?(Zj(t) +?j(t))pj(t)) +
(VC(t)??n(t))[(1??n)?n(t)?pn(t)]??n(t)pn(t)
=B +V[
Ksummationdisplay
k=1
(Rk(t)?Dk(t)) +
summationdisplay
jnegationslash=n
pj(t)?P(t) +
(1??n)?n(t)]C(t) +
Ksummationdisplay
k=1
Xk(t)(Rk(t)?Dk(t)) +
summationdisplay
jnegationslash=n
(Zj(t)(1??j)?j(t)?(Zj(t) +?j(t))pj(t)) +
Zn(t)(1??n)?n(t)?(Zn(t) +?n(t))(1??n)?n(t).
The above inequality is due to Zn(t) >VC(t)??n(t) and the assumption pn(t) < (1??n)?n(t).
The last equality shows, given the assumptionpn(t) < (1??n)?n(t), we may find another electric-
ity allocation scheme withpn(t) = (1??n)?n(t) and ?Q(t) =summationtextKk=1(Rk(t)?Dk(t))+summationtextjnegationslash=npj(t)?
P(t) + (1??n)?n(t), which can achieve a smaller objective value. This contradicts with the pre-
vious assumption. Thus, we have pn(t) ? (1??n)?n(t).
161
For the second part of Lemma 3.1-1b), assume pn(t) > 0 for 0 ? Zn(t) <VC(t)??n(t).It
follows (3.7) that S(t) = 0. The objective function (3.18) can be written as
B +VQ(t)C(t) +
Ksummationdisplay
k=1
Xk(t)(Rk(t)?Dk(t)) +
summationdisplay
jnegationslash=n
(Zj(t)(1??j)?j(t)?(Zj(t) +?j(t))pj(t)) +
Zn(t)(1??n)?n(t)?(Zn(t) +?n(t))pn(t)
> B +VQ(t)C(t) +
Ksummationdisplay
k=1
Xk(t)(Rk(t)?Dk(t)) +
summationdisplay
jnegationslash=n
(Zj(t)(1??j)?j(t)?(Zj(t) +?j(t))pj(t)) +
Zn(t)(1??n)?n(t)?VC(t)pn(t)
? B +V[?P(t) +
Ksummationdisplay
k=1
(Rk(t)?Dk(t)) +
summationdisplay
jnegationslash=n
pj(t)]C(t) +
Ksummationdisplay
k=1
Xk(t)(Rk(t)?Dk(t)) +
summationdisplay
jnegationslash=n
(Zj(t)(1??j)?j(t))?
summationdisplay
jnegationslash=n
((Zj(t) +?j(t))pj(t)).
The first inequality is due to 0 ? Zn(t) < VC(t) ??n(t) and the assumption pn(t) > 0. The
second inequality is due to the non-negativity of Zn(t) and ?n(t). The last equation shows, given
the assumption pn(t) > 0, we may find another electricity allocation scheme with pn(t) = 0 and
?Q(t) = ?P(t) + summationtextKk=1(Rk(t) ? Dk(t)) + summationtextjnegationslash=npj(t), which can achieve a smaller objective
value. This contradicts with the previous assumption. Thus, we have pn(t) = 0, which completes
the proof of Lemma 3.1-1b).
In part 2) of Lemma 3.1, if S(t) > 0, we have Q(t) = 0 according to (3.7). The objective
function (3.19) becomes
?VS(t)W(t) +
Ksummationdisplay
k=1
Xk(t)(Rk(t)?Dk(t))?
Nsummationdisplay
n=1
(Zn(t) +?n(t))pn(t). (B.4)
162
We can prove part 2) with a similar approach as in the case of part 3.1. The detailed proof is
omitted for brevity.
B.4 Proof of Lemma 3.2
Since 0 ? Cmin ? C(t) ? Cmax and V > 0, we have Rk(t) = 0 when Xk(t) < ?VCmax,
and Dk(t) = 0 when Xk(t) > ?VCmin according to Lemma 3.1-3.1). Similarly, since 0 ?
Wmin ?W(t) ?Wmax andV > 0, we obtainRk(t) = 0 whenXk(t) <?VWmax, andDk(t) = 0
whenXk(t) >?VWmin according to Lemma 3.1-2)
Since Cmax >Wmax and Cmin >Wmin, we conclude that if Xk(t) > ?VWmin, the optimal
solution always selectRk(t) = 0. IfXk(t) <?VCmax, the optimal solution always selectDk(t) =
0. The proof is completed.
B.5 Proof of Lemma 3.3
The proof directly follows Lemma 3.1 and is similar to the proof of Lemma 3.2. We omit the
details for brevity.
B.6 Proof of Theorem 3.2
From the battery virtual queue definition (3.10), the constraint Emink ? Ek(t) ? Emaxk is
equivalent to
?VCmax ?Dmaxk ?Xk(t)?Emaxk ?VCmax ?Dmaxk ?Emink .
We assume all the batteries satisfy the battery capacity constraint at the initial time t = 0, i.e.,
Emink ? Ek(0) ? Emaxk , for all k. Supposing the inequalities hold true for time t, we then show
the inequalities still hold true for time t+ 1.
First, we show Xk(t + 1) ? Emaxk ?VCmax ?Dmaxk ?Emink . If ?VWmin < Xk(t) ?
Emaxk ?VCmax?Dmaxk ?Emink , then with Xk(t) >?VWmin ?Rk(t) = 0 from Lemma 3.2, we
163
haveXk(t+1) = Xk(t)?Dk(t) ?Xk(t) ?Emaxk ?VCmax?Dmaxk ?Emink . IfXk(t) ??VWmin,
then the largest value is Xk(t+ 1) = ?VWmin +Rmaxk . For any 0 <V ?Vmax, we have
Emaxk ?VCmax ?Dmaxk ?Emink
?Emaxk ?min
k
braceleftbiggEmax
k ?E
min
k ?R
max
k ?D
max
k
Cmax ?Wmin
bracerightbigg
Cmax
?Dmaxk ?Emink ?Rmaxk ?Xk(t+ 1).
It follows that Xk(t+ 1) ?Emaxk ?VCmax ?Dmaxk ?Emink .
Next, we show Xk(t + 1) ? ?VCmax ?Dmaxk . Assuming ?VCmax ?Dmaxk ? Xk(t) ?
?VCmax, then from Lemma 3.2, we have Xk(t) ??VCmax ?Dk(t) = 0. It follows that
Xk(t+ 1) = Xk(t) +Rk(t) ?Xk(t) ??VCmax ?Dmaxk .
If Xk(t) ??VCmax, following (3.10), we have
Xk(t+ 1) = Xk(t)?Dk(t) +Rk(t) ? Xk(t)?Dmaxk
? ?VCmax ?Dmaxk .
Therefore, we have Xk(t+ 1) ? ?VCmax ?Dmaxk . Thus the inequalities also hold true for time
t+ 1.
It follows that Emink ?Ek(t) ?Emaxk is satisfied under the optimal scheduling algorithm for
all k, t.
B.7 Proof of Theorem 3.3
(i) We first prove the upper bound Zmaxn . Initially, we have Zn(0) = 0 ? VCmax + ?maxn .
Assume that in time slot t the backlog of the QoSE virtual queue of resident n satisfies Zn(t) ?
Zmaxn = VCmax +?maxn . We then check the backlog at time t+ 1 and show the bound still holds
true.
164
If Zn(t) > VCmax, following Lemma 3.3, the optimal scheduling for the quality usage of
resident n satisfies pn(t) ? (1??n)?n(t). From the virtual queue dynamics (3.12), we have
Zn(t+ 1) ? [Zn(t)??n?n(t)]+ +?n?n(t).
If Zn(t) ? ?n?n(t), we have Zn(t + 1) ? Zn(t) ? VCmax + ?maxn ; otherwise, it follows that
Zn(t+ 1) ??n?n(t) <VCmax +?maxn .
If Zn(t) ?VCmax, we have Zn(t+1) ? [Zn(t)??n?n(t)]+ +?maxn . If Zn(t) ??n?n(t), we
have Zn(t+ 1) ? Zn(t) ??n?n(t) +?maxn ? VCmax +?maxn ; otherwise, we have Zn(t+ 1) ?
?maxn ?VCmax +?maxn .
Thus we have Zn(t + 1) ? Zmaxn = VCmax + ?maxn . The proof of the QoSE virtual queue
backlog bound is completed.
(ii) Consider an interval [t1,t2] with length of T = t2 ?t1. Summing (3.12) from t1 to t2, we
have Zn(t2 +1) ?Zn(t1)??nsummationtextt2?=t1 ?n(?)+summationtextt2?=t1[?n(?)?pn(?)] ?summationtextt2?=t1[?n(?)?pn(?)]?
T?n?maxn . It follows that
t2summationdisplay
?=t1
[?n(?)?pn(?)] ?Zmaxn +T?n?maxn .
B.8 Proof of Theorem 3.4
From Theorem 3.2, the battery capacity constraints is met in each time slot with the adaptive
control policy. Take expectation on (3.2) and sum it over the period [0,t?1]:
E{Ek(t)}?E{Ek(0)} =
t?1summationdisplay
?=0
[E{Rk(t)}?E{Dk(t)}], ?k.
Since Emink ?Ek(t) ?Emaxk , we divide both sides by t and let t go to infinity, to obtain
limt?? 1t
t?1summationdisplay
?=0
E{Rk(t)} = limt?? 1t
t?1summationdisplay
?=0
E{Dk(t)}, ?k. (B.5)
165
Consider the following relaxed version of problem (3.9).
minimize: limt?? 1t
t?1summationdisplay
?=0
E{Q(?)C(?)?S(?)W(?)} (B.6)
s.t. (3.3), (3.4), (3.5), (3.6), (3.7), (3.8), and (B.5).
Since the constraints in problem (B.6) are relaxed from that in problem (3.9), the optimal solution
to problem (3.9) is also feasible for problem (B.6). The solution of (B.6) is a stationary and
randomized policy does not depend on battery energy levels [115, 129]. Let the optimal solution
for problem (B.6) be ?A(t) = {?Q(t), ?S(t), ?Rk(t), ?Dk(t),?pn(t)} and the corresponding object value
is ?y ?yopt. According to the properties of optimality of stationary and randomized policies [115],
the optimal solution ?A(t) satisfies E{?Rk(t)? ?Dk(t)} = 0 and ?y = E{?Q(?)C(?)? ?S(?)W(?)}.
We substitute solution ?A(t) into the right-hand-side of the drift-and-penalty (3.17). Since our
proposed policy minimizes the right-hand-side of (3.17), we have
?(vector?(t)) +VE{Q(t)C(t)?S(t)W(t)|vector?(t)}
? B +
Nsummationdisplay
n=1
E{Zn(t)(1??n)?n(t)|Zn(t)}+
Ksummationdisplay
k=1
Xk(t)E{?Rk(t)? ?Dk(t)|Xk(t)}?
Nsummationdisplay
n=1
(Zn(t) +?n(t))E{?pn(t)|Zn(t)}+
VE{?Q(t)C(t)? ?S(t)W(t)|vector?(t)}
? B +
Nsummationdisplay
n=1
Zmaxn (1??n)?maxn +V ?yopt.
166
The second inequality is due to the properties of stationary and randomized policy and ?y ? yopt.
Taking expectation and sum up from 0 to T ?1, we obtain
T?1summationdisplay
t=0
VE{Q(t)C(t)?S(t)W(t)}
? T ? ?B +T ?V ?yopt ?E{L(vector?(T))}+E{L(vector?(0))}
? T ? ?B +T ?V ?yopt +E{L(vector?(0))}.
The second inequality is due to the nonnegative property of Lyapunove functions. Divide both
sides by V ? T and let T go to infinity. Since the initial system state vector?(0) is finite, we have
limT?? 1T summationtextT?1t=0 VE{Q(t)C(t)?S(t)W(t)} = y? ?yopt + ?BV .
167
Appendix C
Proofs in Chapter 5
C.1 Proof of Lemma 5.1
If the feasible power allocation vectorP(t) achieves ?maxn (t) for all n, then all the user buffers are
full at the end of the time slot, according to (5.5). The objective value (5.6) cannot be further
improved without causing buffer overflow. Thus the solution is optimal.
C.2 Proof of Lemma 5.2
Consider a feasible power allocation vectorP?(t) = [P?1(t),P?2(t),??? ,P?N(t)]T and summationtextn?U P?n(t) <
?P. We can construct another feasible power allocation vectorP??(t) = [P??1 (t),P??2 (t),??? ,P??N(t)]T, such
that P??n(t) = ??P?n(t), for all n, and ??summationtextn?U P?n(t) = summationtextn?U P??n(t) ? ?P, where ? > 1. For the
SINR at user n, we have
?n(vectorP??(t)) = LnGnP
??
n(t)summationtext
knegationslash=nGnP
??
k (t) +?n
= ?LnGnP
?
n(t)summationtext
knegationslash=n?GnP
?
k(t) +?n
> ?LnGnP
?
n(t)summationtext
knegationslash=n?GnP
?
k(t) +??n
= ?n(vectorP?(t)).
It follows that summationtextn?U log(1 + ?n(vectorP??(t))) > summationtextn?U log(1 + ?n(vectorP?(t))), since log(1 + x) is an
increasing function of x.
Choosing ? = ?P/summationtextn?U P?n(t), we can construct a feasible solution vectorP???(t) = ?? vectorP?(t), such
that summationtextn?U P???n (t) = ?P. Then we have ?n(vectorP???(t)) > ?n(vectorP?(t)) and summationtextn?U log(1 +?n(vectorP???(t))) >
summationtext
n?U log(1 +?n(vectorP
?(t))). That is, any feasible solution withsummationtext
n?U P
?
n(t) < ?P will be dominated
168
by feasible solutions with summationtextn?U P???n (t) = ?P. We conclude that the optimal solution vectorP(t) must
satisfysummationtextn?U Pn(t) = ?P.
C.3 Proof of Lemma 5.3
Taking the first and second derivatives of the objective function (5.14) with respect to Pn, we
have
?Cn(Pn)
?Pn =
Ln( ?P +An)
( ?P ?Pn +An)[ ?P + (Ln ?1)Pn +An] (C.1)
?2Cn(Pn)
?Pn2 =
?Ln[(Ln ?2)( ?P +An) + 2(1?Ln)Pn]( ?P +An)
[( ?P ?Pn +An)2 +LnPn( ?P ?Pn +An)]2 . (C.2)
Since Pn ? ?P and An > 0, both the first and second derivatives exist. Letting ?2Cn(Pn)?P
n2
= 0, we
derive the unique inflection point
P?n = Ln ?22(L
n ?1)
( ?P +An). (C.3)
WhenPn <P?n, it can be shown that ?2Cn(Pn)?P
n2
< 0; whenPn >P?n, it can be shown that ?2Cn(Pn)?P
n2
>
0.
C.4 Proof of Theorem 5.2
The reflection point is P?n = Ln?22(Ln?1)( ?P +An). As Ln ? ?, we have P?n = 0.5( ?P +An).
Only one link can operate in the convex region due to constraint (5.17). Since ?P?n?Ln > 0, P?n is an
increasing function of Ln. When 1 ?Ln <?, we have P?n < 0.5?( ?P +An). Letting 3P?n = ?P,
we have Ln = (4 ?P + 6An)/( ?P + 3An).
169
C.5 Proof of Theorem 5.3
The first part can be easily shown by the first derivative of P?n with respect to An, which is
?P?n
?An =
Ln?2
2(Ln?1) > 0, for Ln > 2. The second part can be easily shown by evaluating (5.14), (5.15),
and (C.1).
170
Appendix D
Proofs in Chapter 6
D.1 Proof of Lemma 6.1
According to the definition of Xm(t) in (6.3), we have Cm(t) = [Xm(t)?Xm(t?1)]/?.
From the definition ofCminm (t), the playout buffer is emptied at the end of time slott, i.e., Xm(t) =
Dm(t). Therefore, we can derive the minimum required rate as
Cminm (t) = max{0,Dm(t)?Xm(t?1)}/?. (D.1)
From the feasibility condition (6.4), we have Xm(t?1) ? Dm(t?1). Substituting it into (D.1),
we have
Cminm (t) ? [Dm(t)?Dm(t?1)]/? ? ?Cminm (t). (D.2)
Rate ?Cminm (t) occurs when the playout buffer is empty at both the beginning and end of time slot t,
but without buffer overflow during the entire time slot.
D.2 Proof of Theorem 6.1
Recall that?minm is the SINR corresponding to the minimum required rateCminm (t). Let ??minm (t)
be the SINR corresponding to ?Cminm (t). Since (6.2) is a monotonically increasing function, we have
0 ??minm (t) ? ??minm (t).
We now consider the power assignment that achieves rates ?Cminm (t), or, the corresponding
SINRs ??minm (t). From (6.7) and (6.8), the minimum SINR constraint is:
?m(t) = G
m
mPm(t)summationtext
knegationslash=mG
m
k Pk(t) +?m
? ??minm (t), ?m. (D.3)
171
Eqn. (D.3) is a system of linear equations of the power vector vectorP(t), which can be written in the
matrix form as:
parenleftbigI???minAparenrightbigvectorP(t) followsequal ??minvector?, (D.4)
where I is the identity matrix, A is an M ?M matrix with
Amk =
??
?
??
0, m = k
Gmk /Gmm, mnegationslash= k,
(D.5)
??min = diag{??min1 (t),??min2 (t),??? ,??minM (t)} is a diagonal matrix, and vector? = [?1/G11,?2/G22,??? ,
?M/GMM]T.
Define ?min = diag{?min1 (t),?min2 (t),??? ,?minM (t)} and ? = ??min ??min followsequal 0. Assume vectorP
is a power assignment that achieves ??minm (t) for all m, which satisfies (D.4). Substituting ??min =
?+?min into (D.4), we have
parenleftbigI??minAparenrightbigvectorP followsequal?minvector? +?parenleftBigvector? +AvectorPparenrightBig.
Since ?, vector?, A and vectorP all have non-negative elements, we have ?
parenleftBig
vector? +AvectorP
parenrightBig
followsequal0, and therefore,
parenleftbigI??minAparenrightbigvectorP followsequal?minvector?.
That is, vectorP can also achieve ?minm (t) for all m and it satisfies the minimum SINR constraint in (6.8).
Once the minimum SINR constraint in (6.8) (i.e., no buffer underflow) is satisfied, the max-
imum SINR constraint in (6.8) (i.e., no buffer overflow) can be satisfied since BS m can stop
transmission when the playout buffer at user unm is full.
172
Appendix E
Proofs in Chapter 7
E.1 Proof of Theorem 7.1
Due to i.i.d. channel gains and noise powers, the random variables ?n(t)/Gn(t)?s are ex-
changeable, for all t. Define w(g,?,c) = (2c/Bw ?1)?/g, which is convex and increasing with c,
for all g ? 0 and ? ? 0. Let ?(vectorC) = E[?(vectorC)] = E[summationtextTt=1w(g(t),?(t),c(t))]. ?(vectorC) is a symmet-
ric, convex and increasing function in vectorC for each fixed vectorG and vector?. According to Proposition 11.B.5
in [13], ?(vectorC) is symmetric, convex and increasing. Following Fact 2.1, the objective function (7.9)
is Schur-convex and increasing.
E.2 Proof of Corollary 7.2.3
To evaluate the smoothness of a transmission schedule vectorC, the following smoothness utility
function can be used:
U(vectorC) =
Tnsummationdisplay
t=1
([c(t)??c]/Tn), (E.1)
where ?c = summationtextTnt=1c(t)/Tn is the average rate. This is a continuous symmetric convex function
U : RTn ? R. From Fact 2.1, U is Schur-convex and order preserving. The optimal power
transmission schedule vectorC?n satisfies vectorC?n ? vectorCin for all i. Therefore, it also achieves the minimum
value for U(?).
173
Appendix F
Acronyms
AMI Automated Meter Infrastructure
BS Base Station
CBR Constant Bit Rate
CDMA Code Division Multiple Access
CPS Cyber-Physical System
CSMA Carrier Sense Multiple Access
DCC Distribution Control Center
DCPC Distributed Constrained Power Control
DCT Discrete Cosine Transform
DR Demand Response
DRER Distributed Renewable Energy Resource
DUBMLC Distributed User Benefit Maximization Load Control
DVS Dynamic Voltage Scaling
ESS Energy Storage System
GSEPS General Smooth Electric Power Scheduling
FDMA Frequency-Division Multiple Access
HAN Home Area Network
ICT Information and Communications Technology
IDCT Inverse DCT
LAN Local Area Network
LB Lower Bound
LP Linear Programming
174
MB Macro Block
MG Microgrid
MGCC MG Central Controller
PMA Power Minimization Algorithm
PMU Phasor Measurement Unit
PHEV Plug-in Hybrid Electric Vehicles
PLC Power Line Communication
QoE Quality of Experience
QoS Quality of Service
QoSE Quality of Service in Electricity
RLT Reformulation-Linearization Technique
RTP Real-time Transport Protocol
SEPS-DL Smooth Electric Power Scheduling for Deferrable Load
SINR Signal to Interference-plus-Noise Ratio
SG Smart Grid
SST Solid State Transformer
SUDP Supply Until Deadline Policy
TCP Transmission Control Protocol
TDMA Time Division Multiple Access
UB Upper Bound
UDP User Datagram Protocol
UMRP Utility Maximization Real-time Pricing
V2G Vehicle-to Gird
VBR Variable Bit Rate
VPP Virtual Power Plant
VSN Visual Sensor Networks
WAN Wide Area Network
175
Bibliography
[1] N. Wu and X. Li, RFID Applications in Cyber-Physical Sys-
tem. InTech, 2011, ch. 16, [online] Available: http://www.
intechopen.com/books/deploying-rfid-challenges-solutions-and-open-issues/
rfid-applications-in-cyber-physical-system.
[2] K. Kim and P. R. Kumar, ?Cyber-physical systems: A perspective at the centennial,? Pro-
ceedings of the IEEE, vol. 100, no. 3, pp. 1287?1308, May 2012.
[3] H. Farhangi, ?The path of the smart grid,? IEEE Power and Energy Mag., vol. 8, no. 1, pp.
18?28, Jan.-Feb. 2010.
[4] X. Fang, S. Misra, G. Xue, and D. Yang, ?Smart grid - the new and improved power grid: A
survey,? IEEE Commun. Surveys & Tutorials, vol. PP, no. 99, pp. 1?37, Dec. 2011.
[5] F. Li, W. Qiao, H. Sun, H. Wan, J. Wang, Y. Xia, Z. Xu, and P. Zhang, ?Smart transmission
grid: Vision and framework,? IEEE Trans. Smart Grid, vol. 1, no. 2, pp. 168?177, Sept.
2010.
[6] C. W. Clark, The Smart Grid: Enabling Energy Efficiency and Demand Response. Lilburn,
GA: Fairmont Press, 2009.
[7] Energy.gov, ?How the smart grid promotes a greener future,? [online] Available: http://
energy.gov/oe/downloads/how-smart-grid-promotes-greener-future.
[8] G. Fettweis and E. Zimmermann, ?Ict energy consumption - trends and challenges,? in Proc.
WPMC?08, Lapland, Finland, Sep. 2008, pp. 1?4.
[9] Cisco, ?Cisco visual networking index: Global mobile data traffic forecast update, 2010-
2015,? Feb. 2011, [online] Available: http://www.cisco.com/en/US/solutions/collateral/
ns341/ns525/ns537/\\ns705/ns827/white\ paper\ c11-520862.html.
[10] C. Schaefer, C. Weber, and A. Voss, ?Energy usage of mobile telephone services in Ger-
many,? Elsevier Energy, vol. 28, no. 5, pp. 411?420, Apr. 2003.
[11] S. Vadgama, ?Trends in green wireless access,? FUJITSU Scientific Technical Journal,
vol. 45, no. 4, 2009.
[12] L. Tassiulas and A. Ephremides, ?Stability properties of constrained queueing systems and
scheduling policies for maximum throughput in multihop radio networks,? IEEE Trans.
Autom. Control, vol. 37, no. 12, pp. 1936?1948, Dec. 1992.
176
[13] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications.
New York, NY: Academic Press, 1979.
[14] D. Bertsekas, Nonlinear Programming. Belmont, MA: Athena Scientific, 1995.
[15] Nationalacademies.org, ?Greatest engineering achievements of the 20th century,? [online]
Available: http://www.nationalacademies.org/greatachievements/.
[16] Whitehouse.gov, ?Battery and electric vehicle report,? Jul. 2010, [online] Available: http:
//www.whitehouse.gov/files/documents/Battery-and-Electric-Vehicle-Report-FINAL.pdf.
[17] G. Iyer, P. Agrawal, E. Monnerie, and R. S. Cardozo, ?Performance analysis of wireless
mesh routing protocols for smart utility networks,? in IEEE SmartGridComm?11, Oct. 2011,
pp. 114?119.
[18] Z. Zhong, C. Xu, B. J. Billian, L. Zhang, S.-J. Tsai, R. W. Conners, V. A. Centeno, A. G.
Phadke, and Y. Liu, ?Power system frequency monitoring network (FNET) implementa-
tion,? IEEE Trans. Power Systems, vol. 20, no. 4, pp. 1914?1921, Nov. 2005.
[19] S. Xu, A. Q. Huang, S. Lukic, and M. E. Baran, ?On integration of solid-state transformer
with zonal DC microgrid,? IEEE Trans. Smart Grid, vol. 3, no. 2, pp. 975?985, Jun. 2012.
[20] J. A. P. Lopes, F. J. Soares, and P. M. R. Almeida, ?Integration of electric vehicles in the
electric power system,? Proceedings of the IEEE, vol. 99, no. 1, pp. 168?183, Jan. 2011.
[21] J. Kumagai, ?Virtual power plants, real power,? Spectrum IEEE, vol. 49, no. 3, pp. 13?14,
Mar. 2012.
[22] A. Q. Huang, M. L. Crow, G. T. Heydt, J. P. Zheng, and S. J. Dale, ?The future renew-
able electric energy delivery and management (FREEDM) system: the energy internet,?
Proceedings of IEEE, vol. 99, no. 1, pp. 133?148, Jan. 2011.
[23] J. Huang, Z. Li, M. Chiang, and A. Katsaggelos, ?Joint source adaptation and resource allo-
cation for multi-user wireless video streaming,? IEEE Trans. Circuits Syst. Video Technol.,
vol. 18, no. 5, pp. 582?595, May. 2008.
[24] A. Brooks, E. Lu, D. Reicher, C. Spirakis, and B. Weihl, ?Demand dispatch,? IEEE Power
and Energy Mag., vol. 8, no. 3, pp. 20?29, May/Jun. 2010.
[25] A. N. Netravali and J. O. Limb, ?On the performance of distributed polling service-based
medium access control,? Proceedings of the IEEE, vol. 68, no. 3, pp. 366?406, Mar. 1980.
[26] A. N. Netravali and B. G. Haskell, Digital pictures: representation and compression. New
York, NY: Plenum Press, 1988.
[27] C. J. Sreenan, J. Chen, P. Agrawal, and B. Narendran, ?Delay reduction techniques for
playout buffering,? IEEE Trans. Multimedia, vol. 2, no. 2, pp. 88?100, Jun. 2000.
177
[28] T. Zhang, E. van den Berg, J. Chennikara, P. Agrawal, J. Chen, and T. Kodama, ?Local
predictive resource reservation for handoff in multimedia wireless ip networks,? IEEE J.
Sel. Areas Commun., vol. 19, no. 10, pp. 1931?1941, Oct. 2001.
[29] P. Ramanathan, K. M. Sivalingam, P. Agrawal, and S. Kishore, ?Dynamic resource alloca-
tion schemes during handoff for mobile multimedia wireless networks,? IEEE J. Sel. Areas
Commun., vol. 17, no. 7, pp. 1270?1283, Jul. 1999.
[30] P. Agrawal, E. Hyden, P. Krzyzanowski, P. Mishra, M. B. Srivastava, and J. A. Trotter,
?SWAN: a mobile multimedia wireless network,? IEEE Personal Communications, vol. 3,
no. 2, pp. 18?33, Apr. 1996.
[31] S. Misra, M. Reisslein, and X. Guoliang, ?A survey of multimedia streaming in wireless
sensor networks,? IEEE COMMUNICATION Surveys & Tutorials, vol. 10, pp. 18?39, 2008.
[32] I. F. Akyildiz, T. Melodia, and K. R. Chowdury, ?Wireless multimedia sensor networks: A
survey,? IEEE Trans. Wireless Commun., vol. 14, no. 6, pp. 32?39, Dec. 2007.
[33] P. Agrawal, J. Yeh, J. Chen, and T. Zhang, ?Ip multimedia subsystems in 3GPP and 3GPP2:
overview and scalability issues,? IEEE Commun. Mag., vol. 46, no. 1, pp. 138?145, Jan.
2008.
[34] A. K. Katsaggelos, F. Zhai, Y. Eisenberg, and R. Berry, ?Energy-efficient wireless video
coding and delivery,? IEEE Trans. Wireless Commun., vol. 12, no. 4, pp. 24?30, Aug. 2005.
[35] P. Agrawal, J. Chen, S. Kishore, P. Ramanathan, and K. Sivalingam, ?Battery power sen-
sitive video processing in wireless networks,? in Proc. IEEE Personal, Indoor and Mobile
Radio Communications?98, Sep. 1998, pp. 116?120.
[36] Z. He, Y. Liang, L. Chen, I. Ahmad, and D. Wu, ?Power-rate-distortion analysis for wireless
video communication under energy constraints,? IEEE Trans. Circuits Syst. Video Technol.,
vol. 15, no. 5, pp. 645?658, May 2005.
[37] D. N. Kwon, P. F. Driessen, A. Basso, and P. Agathoklis, ?Performance and computational
complexity optimization in configurable hybrid video coding system,? IEEE Trans. Circuits
Syst. Video Technol., vol. 16, no. 1, pp. 31?42, Jan. 2006.
[38] H. Cheng and L. Dung, ?A content-based methodology for power-aware motion estimation
architecture,? IEEE Trans. Circuits Syst. Video Technol., vol. 52, no. 10, pp. 631?635, Oct.
2005.
[39] N. J. August and D. S. Ha, ?Low power design of dct and idct for low bit rate video codecs,?
IEEE Trans. Circuits Syst. Video Technol., vol. 6, no. 3, pp. 414?422, Jun. 2004.
[40] Z. Cao, B. Foo, L. He, and M. van der Schaar, ?Optimality and improvement of dynamic
voltage scaling algorithms for multimedia applications,? IEEE Trans. Circuits Syst. Video
Technol., vol. 57, no. 3, pp. 681?690, Mar. 2010.
178
[41] H. M. Wang, H. S. Choi, and J. T. Kim, ?Workload-based dynamic voltage scaling with the
qos for streaming video,? in Proc. IEEE Electronic Design, Test and Applications?08, Jan.
2008, pp. 236?239.
[42] M. Li, Z. Guo, R. Y. Yao, and W. Zhu, ?A novel penalty controllable dynamic voltage scaling
scheme for mobile multimedia applications,? IEEE Trans. Mobile Computing, vol. 5, no. 12,
pp. 1719?1733, Dec. 2006.
[43] A. Goldsmith, Wireless Communications. New York, NY: Cambridge University Press,
2005.
[44] IEEE, ?Part 11, Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY)
Specifications: Medium Access Control (MAC) Enhancements for Quality of Service
(QoS),? July 2003, ANSI/IEEE Std 802.11e, Draft 5.0.
[45] Y. Huang, P. Walsh, Y. Li, and S. Mao, ?A gnu radio testbed for distributed polling service-
based medium access control,? in Proc. IEEE MILCOM?11, Baltimore, MD, Nov. 2011.
[46] G. J. Foschini and Z. Miljanic, ?A simple distributed autonomous power control algorithm
and its convergence,? IEEE Trans. Veh. Technol., vol. 42, no. 4, pp. 641?646, Nov. 1993.
[47] D. Mitra, ?An asynchronous distributed algorithm for power control in cellular radio sys-
tem,? in Proc. 4th Winlab Workshop on Third Generation Wireless Information Networks,
New Brunswick, NJ, Oct. 1993, pp. 249?257.
[48] S. Grandhi, J. Zander, and R. Yates, ?Constrained power control,? Int. J. Wireless Personal
Commun., vol. 1, no. 4, pp. 257?270, Apr. 1995.
[49] P. Hande, S. Rangan, M. Chiang, and X. Wu, ?Distributed uplink power control for optimal
sir assignment in cellular data networks,? IEEE/ACM Trans. Networking., vol. 16, no. 6, pp.
1420?1433, Dec. 2008.
[50] J. Lee, R. Mazumdar, and N. Shroff, ?Downlink power allocation for multi-class wireless
systems,? IEEE/ACM Trans. Networking, vol. 13, no. 4, pp. 854?867, Aug. 2005.
[51] N. Bambos, S. C. Chen, and G. J. Pottie, ?Radio link admission algorithm for wireless net-
works with power control and active link quality protection,? in Proc. IEEE INFOCOM?95,
Boston, MA, Apr. 1995, pp. 97?104.
[52] M. Xiao, N. B. Shroff, and E. K. P. Chong, ?Distributed admission control for power-
controlled cellular wireless systems,? IEEE/ACM Trans. Networking, vol. 9, no. 6, pp. 790?
800, Dec. 2001.
[53] C. Comaniciu and H. V. Poor, ?Jointly optimal power and admission control for delay sensi-
tive traffic in cdma networks with lmmse receivers,? IEEE Trans. Signal Processing, vol. 51,
no. 8, pp. 2031?2042, Aug. 2003.
[54] G. Liang and B. Liang, ?Balancing interruption frequency and buffering penalties in VBR
video streaming,? in Proc. IEEE INFOCOM?07, Anchorage, AK, May 2007, pp. 1406?
1414.
179
[55] Y. Huang, S. Mao, and Y. Li, ?Downlink power allocation for stored variable-bit-rate
videos,? in Proc. ICST QShine?10, Houston, TX, Nov. 2010, pp. 423?438.
[56] Y. Huang and S. Mao, ?Downlink power control for variable bit rate video over multicell
wireless networks,? in Proc. IEEE INFOCOM?11, Shanghai, China, Apr. 2011, pp. 2561?
2569.
[57] Y. Huang, S. Mao, and Y. Li, ?Downlink power control for VBR video streaming in cellular
networks: A majorization approach,? in Proc. IEEE Globalcom?11, Houston, TX, Dec.
2011, pp. 1?6.
[58] ??, ?On downlink power allocation for multiuser variable-bit-rate video streaming,? Wiley
Security and Communication Networks, to appear.
[59] M. Chiang, ?Balancing transport and physical layers in wireless multihop networks: jointly
optimal congestion control and power control,? IEEE J. Sel. Areas Commun., vol. 23, no. 1,
pp. 104?116, Jan. 2005.
[60] S. Mao, S. Lin, Y. Wang, S. S. Panwar, and Y. Li, ?Multipath video transport over wireless
ad hoc networks,? IEEE Wireless Commun., vol. 12, no. 4, pp. 42?49, Aug. 2005.
[61] S. Mao, Y. T. Hou, H. D. Sherali, and S. F. Midkiff, ?Multimedia-centric routing for multiple
description video in wireless mesh networks,? IEEE Network, vol. 22, no. 1, pp. 19?24,
Jan./Feb. 2008.
[62] H. Y. Shutoy, D. Gunduz, E. Erkip, and Y. Wang, ?Cooperative source and channel coding
for wireless multimedia communications,? IEEE J. Sel. Signal Process., vol. 1, no. 2, pp.
295?307, Aug. 2007.
[63] T. Koponen, T. Koponen, B. Chun, A. Ermolinskiy, K. H. Kim, and S. Shenker, ?A data-
oriented (and beyond) network architecture,? in ACM SIGCOMM?07, Kyoto, Japan, Agu.
2007, pp. 181?192.
[64] X. Lu, E. Erkip, Y. Wang, and D. Goodman, ?Power efficient multimedia communication
over wireless channels,? IEEE J. Sel. Areas Commun., vol. 21, no. 10, pp. 1738?1751, Dec.
2003.
[65] Y. Eisenberg, C. E. Luna, T. N. Pappas, R. Berry, and A. K. Katsaggelos, ?Joint source
coding and transmission power management for energy efficient wireless video communi-
cations,? IEEE Trans. Circuits Syst. Video Technol., vol. 12, no. 6, pp. 411?424, Jun. 2002.
[66] Y. Li, M. Reisslein, and C. Chakrabarti, ?Energy-efficient video transmission over a wireless
link,? IEEE Trans. Veh. Technol., vol. 58, no. 3, pp. 1229?1244, Mar. 2009.
[67] Z. Li, F. Zhai, and A. K. Katsaggelos, ?Joint video summarization and transmission adap-
tation for energy-efficient wireless video streaming,? EURASIP J. on Advances in Signal
Processing, vol. 2008, pp. 1?11, Jan. 2008.
180
[68] D. Wu, S. Ci, and H. Wang, ?Cross-layer optimization for video summary transmission over
wireless networks,? IEEE J. Sel. Areas Commun., vol. 25, no. 4, pp. 841?850, May 2007.
[69] S. Soro and W. Heinzelman, ?A survey of visual sensor networks,? Advances in Multimedia,
vol. 2009, 2009, article ID 640386, 21 pages, doi:10.1155/2009/640386.
[70] A. Seema and M. Reisslein, ?Towards efficient wireless video sensor networks: A survey of
existing node architectures and proposal for a flexi-wvsnp design,? IEEE Commun. Surveys
Tutorials, vol. 13, no. 3, pp. 462?486, Quarter 2011.
[71] M. Chen, T. Kwon, S. Mao, Y. Yuan, and V. C. M. Leung, ?Reliable and energy-efficient
routing protocol in dense wireless sensor networks,? International Journal of Sensor Net-
works, vol. 4, no. 1/2, pp. 104?117, 2008.
[72] M. Chen, V. C. M. Leung, and S. Mao, ?Directional controlled fusion in wireless sensor
networks,? ACM/Springer MONET, vol. 14, no. 2, pp. 220?229, Apr. 2009.
[73] ??, ?Directional controlled fusion in wireless sensor networks,? in Proc. QShine?08, Hong
Kong, P.R. China, Jul. 2008, pp. 1?7.
[74] J. Liu and S. Singh, ?Atcp: Tcp for mobile ad hoc networks,? IEEE J. Sel. Areas Commun.,
vol. 19, no. 7, pp. 1300?1315, Jul. 2001.
[75] V. Tsaoussidis and H. Badr, ?Tcp-probing: towards an error control schema with energy and
throughput performance gains,? in Proc. Network Protocols?00, 2000, pp. 12?21.
[76] D. Hu, S. Mao, Y. T. Hou, and J. H. Reed, ?Fine grained scalability video multicast in
cognitive radio networks,? IEEE J. Sel. Areas Commun., vol. 28, no. 3, Apr. 2010.
[77] D. Hu and S. Mao, ?Streaming scalable videos over multi-hop cognitive radio networks,?
IEEE Trans. Wireless Commun., vol. 9, no. 11, pp. 3501?3511, Nov. 2010.
[78] Y. Zhao, S. Mao, J. H. Reed, and Y. Huang, ?Experimental study of utility function selection
for video over cognitive radio networks,? in Proc. TRIDENTCOM 2009, Washington D.C.,
Apr. 2009, pp. 1?10.
[79] ??, ?Utility function selection for streaming videos with a cognitive engine testbed,?
ACM/Springer Mobile Networks and Applications Journal, vol. 15, no. 3, pp. 446?460,
June 2010.
[80] Y. Huang, S. Mao, and R. M. Nelms, ?Smooth electric power scheduling in power distribu-
tion networks,? in Proc. IEEE GLOBECOM 2012 ? Workshop on Smart Grid Communica-
tions: Design for Performance, Anaheim, CA, Dec. 2012, pp. 1?5.
[81] M. LeMay, R. Nelli, G. Gross, and C. A. Gunter, ?An integrated architecture for demand
response communications and control,? in HICSS ?08, Washington, DC, 2008, pp. 174?183.
[82] Y. Huang, S. Mao, and R. M. Nelms, ?Adaptive electricity scheduling in microgrids,? in
Proc. IEEE INFOCOM 2013, Turin, Italy, Apr. 2013.
181
[83] Y. Huang and S. Mao, ?Adaptive electricity scheduling with quality of usage guarantees in
microgrids,? in Proc. IEEE GLOBECOM 2012, Anaheim, CA, Dec. 2012, pp. 1?6.
[84] ??, ?On qualify of usage provisioning for electricity scheduling in microgrids,? IEEE
Systems Journal, Special Issue on Smart Grind Communications Systems, 2013, to appear.
[85] S. Mao, Y. Huang, and Y. Li, ?Downlink vbr video scheduling in cellular networks with
orthogonal channels,? IEEE E-Letter for MMTC, vol. 6, no. 9, pp. 40?42, Sept. 2011.
[86] Y. Huang, S. Mao, and Y. Li, ?A majorization approach to downlink power control for vbr
videos over cellular networks,? Elsevier Computer Communications, vol. 35, no. 15, pp.
1828?1837, Sept. 2012.
[87] ??, Green video streaming over cellular networks. New York, NY: CRC Press, 2012,
ch. 23, pp. 659?690.
[88] G. T. Heydt, ?The next generation of power distribution systems,? IEEE Trans. Smart Grid,
vol. 1, no. 3, pp. 225?235, Dec. 2010.
[89] S. Shao, M. Pipattanasomporn, and S. Rahman, ?Demand response as a load shaping tool
in an intelligent grid with electric vehicles,? IEEE Trans. Smart Grid, vol. 2, no. 4, pp.
624?631, Dec. 2011.
[90] A. H. Mohsenian-Rad and A. Leon-Garcia, ?Optimal residential load control with price
prediction in real-time electricity pricing environments,? IEEE Trans. Smart Grid, vol. 1,
no. 2, pp. 120?133, Sept. 2010.
[91] C. Ibars, M. Navarro, and L. Giupponi, ?Distributed demand management in Smart Grid
with a congestion game,? in Proc. IEEE SmartGridComm?10, Gaithersburg, MD, Oct. 2010,
pp. 495?500.
[92] N. Li, L. Chen, and S. H. Low, ?Optimal demand response based on utility maximization in
power networks,? in 2011 IEEE PES General Meeting, Detroit, MI, Jul. 2011, pp. 1?8.
[93] M. G. Kallitsis, G. Michailidis, and M. Devetsikiotis, ?Optimal power allocation under com-
munication network externalities,? IEEE Trans. Smart Grid, vol. 3, no. 1, pp. 162?173, Mar.
2012.
[94] J. Medina, N. Muller, and I. Roytelman, ?Demand response and distribution grid operations:
Opportunities and challenges,? IEEE Trans. Smart Grid, vol. 1, no. 2, pp. 193?198, Sept.
2010.
[95] J. Salehi, Z.-L. Zhang, J. Kurose, and D. Towsley, ?Supporting stored video: reducing rate
variability and end-to-end resource requirements through optimal smoothing,? IEEE/ACM
Trans. Networking., vol. 6, no. 4, pp. 397?410, Aug. 1998.
[96] E. Jorswieck and H. Boche, ?Optimal transmission strategies and impact of correlation in
multiantenna systems with different types of channel state information,? IEEE Trans. Signal
Processing, vol. 52, no. 12, pp. 3440?3453, Dec. 2004.
182
[97] A. Papavasiliou and S. S. Oren, ?Supplying renewable energy to deferrable loads: Algo-
rithms and economic analysis,? in Proc. IEEE Power & Energy Society General Meeting?10,
Jul. 2010, pp. 1?8.
[98] B. C. Arnold, Majorization and the Lorenz Order: A Brief Introduction. New York, NY:
Springer-Verlag, 1987.
[99] A. Keyhani, Design of Smart Power Grid Renewable Energy Systems. Hoboken, NJ:
Wiley-IEEE Press, 2011.
[100] D. P. Palomar and M. Chiang, ?A tutorial on decomposition methods for network utility
maximization,? IEEE J. Sel. Areas Commun., vol. 24, no. 8, pp. 1439?1451, Aug. 2006.
[101] M. Fahrioglu and F. L. Alvarado, ?Designing incentive compatible contracts for effective
demand management,? IEEE Trans. Power Systems, vol. 15, no. 4, pp. 1255?1260, Nov.
2000.
[102] P. Samadi, A. Mohsenian-Rad, R. Schober, V. Wong, and J. Jatskevich, ?Optimal real-
time pricing algorithm based on utility maximization for smart grid,? in IEEE SmartGrid-
Comm?10, Oct. 2010, pp. 415?420.
[103] A. J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control. New York:
John Wiley & Sons, 1984.
[104] F. Rahimi and A. Ipakchi, ?Demand response as a market resource under the smart grid
paradigm,? IEEE Trans. Smart Grid, vol. 1, no. 1, pp. 82?88, Jun. 2010.
[105] S. Amin and B. Wollenberg, ?Toward a smart grid: power delivery for the 21st century,?
IEEE Power and Energy Mag., vol. 3, no. 5, pp. 1?37, Sept.-Oct. 2005.
[106] A. Mohsenian-Rad, V. Wong, J. Jatskevich, R. Schober, and A. Leon-Garcia, ?Autonomous
demand-side management based on game-theoretic energy consumption scheduling for the
future smart grid,? IEEE Trans. Smart Grid, vol. 1, no. 3, pp. 320?331, Dec. 2010.
[107] H. K. Nguyen, J. B. Song, and Z. Han, ?Demand side management to reduce peak-to-
average ratio using game theory in smart grid,? in Proc. IEEE INFOCOM Workshops?12,
Orlando, FL, Mar. 2012, pp. 91?96.
[108] J. Huang, C. Jiang, and R. Xu, ?A review on distributed energy resources and microgrid,?
ELSEVIER Renewable and Sustainable Energy Reviews, vol. 12, no. 9, pp. 2472?2483, Dec.
2008.
[109] E. Fumagalli, J. W. Black, I. Vogelsang, and M. Ilic, ?Quality of service provision in elec-
tric power distribution systems through reliability insurance,? IEEE Trans. Power Systems,
vol. 19, no. 3, pp. 1286?1293, Aug. 2004.
[110] J. Slotine and W. Li, Applied Nonlinear Control. Prentice Hall, 1991.
[111] T. T. Kim and H. H. V. Poor, ?Scheduling power consumption with price uncertainty,? IEEE
Trans. Smart Grid, vol. 2, no. 3, pp. 519?527, Sept. 2011.
183
[112] M. J. Neely, E. Modiano, and C. E. Rohrs, ?Dynamic power allocation and routing for time-
varying wireless networks,? IEEE J. Sel. Areas Commun., vol. 23, no. 1, pp. 89?103, Jan.
2005.
[113] M. J. Neely, ?Stock market trading via stochastic network optimization,? in Proc. IEEE
CDC?10, Atlanta, GA, Dec. 2010, pp. 1?8.
[114] M. J. Neely, A. S. Tehrani, and A. G. Dimakis, ?Efficient algorithms for renewable energy
allocation to delay tolerant consumers,? in IEEE SmartGridComm?10, Oct. 2010, pp. 549?
554.
[115] M. J. Neely and R. Urgaonkar, ?Opportunism, backpressure, and stochastic optimization
with the wireless broadcast advantage,? in Asilomar Conference on Signals, Systems, and
Computers?08, Pacific Grove, CA, Oct. 2008, pp. 1?7.
[116] The National Renewable Energy Laboratory, ?Western wind resources dataset,? [online]
Available: http://wind.nrel.gov/Web nrel/.
[117] S. B. Peterson, J. F. Whitacre, and J. Apt, ?The economics of using plug-in hybrid electric
vehicle battery packs for grid storage,? J. Power Sources, vol. 195, no. 8, pp. 2377?2384,
2010.
[118] ?The Electric Reliability Council of Texas,? [online] Available: http://www.ercot.com/.
[119] M. He, S. Murugesan, and J. Zhang, ?Multiple timescale dispatch and scheduling for
stochastic reliability in smart grids with wind generation integration,? in Proc. IEEE IN-
FOCOM?11, Shanghai, China, Apr. 2011, pp. 461?465.
[120] B. Karimi and V. Namboodiri, ?Capacity analysis of a wireless backhaul for metering in the
smart grid,? in Proc. IEEE INFOCOM?12, Orlando, FL, Mar. 2012, pp. 61?66.
[121] Z. Lu, W. Wang, and C. Wang, ?Hiding traffic with camouflage: Minimizing message delay
in the smart grid under jamming,? in Proc. IEEE INFOCOM?12, Orlando, FL, Mar. 2012,
pp. 3066?3070.
[122] M. A. Rahman, P. Bera, and E. Al-Shaer, ?Smartanalyzer: A noninvasive security threat
analyzer for AMI smart grid,? in Proc. IEEE INFOCOM?12, Orlando, FL, Mar. 2012, pp.
2255?2263.
[123] H. Ma, H. Li, and Z. Han, ?A framework of frequency oscillation in power grid: Epidemic
propagation over social networks,? in Proc. IEEE INFOCOM?12, Orlando, FL, Mar. 2012,
pp. 67?72.
[124] H. Liang, B. J. Choi, W. Zhuang, and X. Shen, ?Towards optimal energy store-carry-and-
deliver for PHEVs via V2G system,? in Proc. IEEE INFOCOM?12, Orlando, FL, Mar. 2012,
pp. 1674?1682.
[125] X. Liu, ?Economic load dispatch constrained by wind power availability: A wait-and-see
approach,? IEEE Trans. Smart Grid, vol. 1, no. 3, pp. 347?355, Dec. 2010.
184
[126] X. Fang, D. Yang, and G. Xue, ?Online strategizing distributed renewable energy resource
access in islanded microgrids,? in IEEE GLOBECOM?11, Huston, TX, Dec. 2011, pp.
1931?1937.
[127] I. Koutsopoulos, V. Hatzi, and L. Tassiulas, ?Optimal energy storage control policies for the
smart power grid,? in IEEE SmartGridComm?11, Oct. 2011, pp. 475?480.
[128] S. X. Chen, H. B. Gooi, and M. Q. Wang, ?Sizing of energy storage for microgrids,? IEEE
Trans. Smart Grid, vol. 3, no. 1, pp. 142?151, Mar. 2012.
[129] R. Urgaonkar, B. Urgaonkar, M. J. Neely, and A. Sivasubramaniam, ?Optimal power cost
management using stored energy in data centers,? in Proc. ACM SIGMETRICS?11, San Jose,
CA, Jun. 2011, pp. 221?232.
[130] H. Sistek, ?Green-tech base stations cut diesel usage by 80 percent,? Apr. 2008, [online]
Available: http://news.cnet.com/8301-11128 3-9912124-54.html.
[131] M. W. Garrett and W. Willinger, ?Analysis, modeling and generation of self-similar VBR
video traffic,? ACM SIGCOMM Comput. Commun. Rev., vol. 24, no. 4, pp. 269?280, 1994.
[132] J. Beran, R. Sherman, M. Taqqu, and W. Willinger, ?Long-range dependence in variable-bit-
rate video traffic,? IEEE Trans. Commun., vol. 43, no. 2/3/4, pp. 1566?1579, Feb./Mar./Apr.
1995.
[133] D. P. Heyman and T. V. Lakshmanr, ?What are the implications of long-range dependence
for VBR-video traffic engineering?? IEEE/ACM Trans. Networking, vol. 4, no. 3, pp. 301?
317, June 1996.
[134] T. Lakshman, A. Ortega, and A. Reibman, ?VBR video: Trade-offs and potentials,? Proc.
IEEE, vol. 86, no. 1, pp. 952?973, May 1998.
[135] S. Liew and H. Chan, ?Lossless aggregation: a scheme for transmitting multiple stored VBR
video streams over a shared communications channel without loss of image quality,? IEEE
J. Sel. Areas Commun., vol. 15, no. 6, pp. 1181?1189, Aug. 1997.
[136] S. Sen, D. Towsley, Z. Zhang, and J. K. Dey, ?Optimal multicast smoothing of streaming
video over the internet,? IEEE J. Sel. Areas Commun., vol. 20, no. 7, pp. 1345?1359, Sep.
2002.
[137] J. Lee and J. Kwon, ?Utility-based power allocation for multiclass wireless systems,? IEEE
Trans. Vehic Tech., vol. 58, no. 7, pp. 3813?3819, Sep. 2009.
[138] A. Gjendemsj, D. Gesbert, G. Oien, and S. Kiani, ?Binary power control for sum rate max-
imization over multiple interfering links,? IEEE Trans. Wireless Commun., vol. 7, no. 8, pp.
3164?3173, Aug. 2008.
[139] M. Reisslein, ?Video trace library,? Arizona State University, [online] Available:
http://trace.eas.asu.edu/.
185
[140] UMTS World, UMTS Power Control, [online] Available: http://www.umtsworld.com/
technology/power.htm.
[141] J. M. Mcmanus and K. W. Ross, ?A dynamic programming methodology for managing
prerecorded vbr sources in packet-switched networks,? in Proc. SPIE, Performance and
Control of Network Systems, 1997, pp. 140?154.
[142] S. Deng, T. Webera, and A. Ahrens, ?Capacity optimizing power allocation in interference
channels,? AEU - International Journal of Electronics and Communications, vol. 63, no. 2,
pp. 139?147, Jan. 2009.
[143] T. Stockhammer, H. Jenkac, and G. Kuhn, ?Streaming video over variable bit-rate wireless
channels,? IEEE Trans. Multimedia, vol. 6, no. 2, pp. 268?277, Apr. 2004.
[144] M. Chen and A. Zakhor, ?Multiple TFRC connections based rate control for wireless net-
works,? IEEE Trans. Multimedia, vol. 8, no. 5, pp. 1045?1062, Oct. 2006.
[145] H. D. Sherali and W. P. Adams, A Reformulation-Linearization Technique for Solving Dis-
crete and Continuous Nonconvex Problems. Dordrecht, Boston, London: Kluwer Aca-
demic Publishers, 1999.
[146] S. Kompella, S. Mao, Y. Hou, and H. Sherali, ?On path selection and rate allocation for
video in wireless mesh networks,? IEEE/ACM Trans. Networking., vol. 17, no. 1, pp. 212?
224, Feb. 2009.
[147] M. Andersin, Z. Rosberg, and J. Zander, ?Gradual removals in cellular PCS with constrained
power control and noise,? in Proc. IEEE PIMRC?95, Toronto, Canada, Sept. 1995, pp. 56?
60.
[148] A. Naman and D. Taubman, ?JPEG2000-based scalable interactive video (JSIV),? IEEE
Trans. Image Process., vol. 20, no. 5, pp. 1435?1449, May 2011.
[149] W.-K. Liao and Y.-H. Lai, ?Type-aware error control for robust interactive video services
over time-varying wireless channels,? IEEE Trans. Mobile Comput., vol. 10, no. 1, pp. 136?
145, Jan. 2011.
[150] G. Cheung, A. Ortega, and N.-M. Cheung, ?Interactive streaming of stored multiview video
using redundant frame structures,? IEEE Trans. Image Process., vol. 20, no. 3, pp. 744?761,
Mar. 2011.
[151] A. Dan, P. Shahabuddin, D. Sitaram, and D. Towsley, ?Channel allocation under batching
and vcr control in movie-on-demand servers,? J. Parallel Distrib. Comput., vol. 30, pp. 168?
179, 1995.
[152] buto.tv, ?Feature: Clickable videointeractive, clickable video that anyone can build in min-
utes,? [online] Available: http://get.buto.tv/features/clickable-video/.
186