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Elements of robustness and optimal control for infrastructure networks

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Elements of robustness and optimal control for infrastructure networks

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Content Elements of Robustness and Optimal Control for Infrastructure Networks by Qin Ba A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Civil Engineering) May 2018 Dissertation Committee Ketan Savla (Chair) Jong-Shi Pang Roger Ghanem Ashutosh Nayyar Copyright 2018 Qin Ba Abstract An infrastructure system is formally modeled as a network, whose topology is described by a directed or undirected graph, and whose nodes and links are associated with physical quantities such as node potential and link ow. Two canonical physical laws occurring in multiple infrastruc- ture networks are considered to relate these physical quantities. The rst one is ow conservation law. It requires zero net ow at every node. The second physical law, referred to as diusion law, requires a strictly monotonic relationship between link ow and the potential dierence between the incident nodes. The linear version of diusion law plus ow conservation law denes the class of DC networks. The ow over a DC network is uniquely determined by node supply-demands and link weights. We are interested in nding a weight that produces a maximal feasible ow, i.e, ow satisfying capacity constraints, for given supply-demands over a DC network. The solution relies on minimizing a congestion function that is neither convex nor dierentiable. Based on the derived directional derivative of the congestion function and properties proved for ow-weight Jacobian matrix, we show that every local minimum of the congestion function is a global minimum. The exibility of choosing weight is also adopted as a control mechanism to maximize the robustness of a DC network toward disturbances to nominal supply-demand. The margin of robustness is dened as the radius of the largest ` 1 ball of disturbances for which there exists a weight within specied bounds such that the resulting ow is feasible. Computation of this margin is posed as a non-convex optimization problem. A multi-level programming approach is developed to solve the problem for reducible networks, i.e., a class of networks with certain sparsity properties. The approach is based on a novel notion of equivalent capacity function, i which is proven to possess a strong quasi-concavity property for link reducible networks. This property further facilitates an easy solution to the multilevel programming formulation for a subclass of reducible networks, that we call type I tree reducible networks. Another major consideration in robust operation of infrastructure networks is to prevent cas- cading failure, i.e., a phenomenon under which the initial failure of a small number of links due to an exogenous event is followed by a sequence of link failures due to the interplay between physics, network topology, and controls. We consider DC networks and model cascading failure as a discrete-time dynamical system featuring a hybrid state consisting of a set variable for the active link and a continuous variable for supply-demand. While link weight is assumed xed, load shedding, i.e., decreasing the magnitude of supply-demand, is adopted as the control mechanism to stop failure propagation. The control objective is to minimize the amount of load to be shed. We propose two approaches for computing an optimal control, and provide time complexity anal- ysis for these approaches. The rst approach, geared towards a certain class of networks that we call type II tree reducible networks, decomposes the global non-convex problem into a system of coupled local non-convex problems, which can be solved to optimality in two iterations. The sec- ond approach transforms the continuous reachable set equivalently into a nite set by leveraging and extending tools for arrangement of hyperplanes and convex polytopes. Finally, we adapt the Alternating Direction Method of Multipliers (ADMM) to develop a dis- tributed algorithm for computing nite horizon optimal control for trac ow over transportation networks and present its convergence properties. ii Acknowledgements I would like to thank my advisor Ketan Savla for his patience in guidance and input in this thesis. From him, I have learnt the importance of being rigorous in thinking and clear in writing. I am also grateful to have a number of respectable professors teach me classes over the last four years of my graduate studies: Prof. Jong-Shi Pang and Suvrajeet Sen, who taught me optimization, Prof. Ashutosh Nayyar and Rahul Jain, who taught me stochastic control and dynamic programing, Prof. Shang-Hua Teng, who taught me algorithm design, and Prof. Sheldon Mark Ross, who taught me stochastic processes. Special thanks are given to Prof. Jong-Shi Pang. His encouragement and enthusiasm in classes has inspired me to pursue a career in academia. I also thank Prof. Roger Ghanem for being in my committee and for expressing his appreciation of my work. Over the last four and half years of living away from families, I feel grateful to have many friends by my side. Friendship with them brings joy to my everyday life. Even though it is not possible to list them all, I would like to name a few as follows: Guanbo Bian, Mohammad Ali Motie, Mahsa Moslehi, Zheng Yang, Meida Chen, Zhongzhe Yang, Pouya Vahmani, Mohammad S. Al-Shaiji, Arash Mohegh, Pouyan Hosseini, Bin Tian, Jian Wang, Xin Huang, Huachao Mao, Zinan Xiang, Xuanpu Zhang, Xize Wang, Yifan Song. Finally, I would like to thank my girlfriend Yun Li. Her company and caring gives me strength and peace for overcoming stress and diculties. I also would like to thank my mom in China. Her continuous support and caring formed a precondition for the completion of this thesis. iii Table of Contents Abstract i Acknowledgements iii List Of Tables vii List Of Figures viii Chapter 1: Introduction 1 1.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Chapter 2: Network Model 9 2.1 Basic Concepts from Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Laplacian Matrix of Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Network Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.1 Physical Quantities for Networks . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.2 Physical Laws for Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.3 Flow Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3.4 Diusion Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.4 DC Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Chapter 3: Maximal Flow Over DC Networks with Controllable Weights 25 3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Relationship between Maximal Flow over Weight Controlled DC Networks and over Flow Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.1 Relationship between feasible ow sets . . . . . . . . . . . . . . . . . . . . . 28 3.2.2 Geometry of DC Flow And Least Congested Flow . . . . . . . . . . . . . . 30 3.3 Oblique Projection Matrix of DC Network . . . . . . . . . . . . . . . . . . . . . . . 31 3.4 Flow-weight Jacobian and Optimality Conditions . . . . . . . . . . . . . . . . . . . 35 3.4.1 The Flow-weight Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.4.2 The Directional Derivative of Congestion Function . . . . . . . . . . . . . . 41 3.4.3 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4.4 A Descent Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.5 Flow Redistribution Under Weight Perturbations . . . . . . . . . . . . . . . . . . . 46 3.5.1 A Multigraph Perspective for Weight Perturbations . . . . . . . . . . . . . 46 3.5.2 Pseudo-inverse of Laplacian Matrix Under Weight Perturbations . . . . . . 48 3.5.3 Flow Redistribution Under Weight Perturbations . . . . . . . . . . . . . . . 52 3.5.4 Interpretation of Link Removal and Disconnected Case . . . . . . . . . . . . 56 3.6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 iv Chapter 4: Robustness of DC Networks With Controllable Weight 60 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3 Relationship Between Margin of Robustness and Min-cut Capacity . . . . . . . . . 67 4.4 The Multiplicative Disturbance Case . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.5 The Nongenerative Disturbance Case: A Multilevel Programming Approach . . . . 73 4.5.1 A Novel Network Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.5.2 An Equivalent Bilevel Formulation . . . . . . . . . . . . . . . . . . . . . . . 82 4.5.3 A Nested Bilevel Approach for Multilevel Formulation . . . . . . . . . . . . 86 4.6 An Ecient Solution Methodology for Tree Reducible Networks . . . . . . . . . . . 88 4.6.1 Tree Reducible Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.6.2 Input-output Properties of the Simplied Version of the Reduction Problem 92 4.6.3 Series Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.6.4 Parallel Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.6.5 Computing Margin of Robustness for Tree Reducible Networks . . . . . . . 105 4.7 Decentralized Control Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.7.1 Optimal Centralized Control for Parallel Networks . . . . . . . . . . . . . . 107 4.7.2 A Memoryless Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.7.3 Controller with Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.8 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.8.1 Margin of Robustness Estimates . . . . . . . . . . . . . . . . . . . . . . . . 120 4.8.2 Equivalent Capacities for Link Reducible Networks . . . . . . . . . . . . . . 121 4.9 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Chapter 5: Computing Optimal Control of Cascading Failure in DC Networks 124 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.2 Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.2.1 Cascading Failure Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.2.3 Solution by Optimal Tree Search . . . . . . . . . . . . . . . . . . . . . . . . 133 5.3 Analytical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.3.1 Parallel Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.3.2 A Decomposition Approach for Type II Tree Reducible Networks . . . . . . 140 5.3.3 Input-output Properties of the Subproblem for Constant Control . . . . . . 145 5.3.4 Optimal Constant Control Action for Type II Tree Reducible Networks . . 151 5.4 An equivalent state aggregation approach . . . . . . . . . . . . . . . . . . . . . . . 152 5.4.1 A State Aggregation Approach . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.4.2 Optimal Control Synthesis: From Aggregated to the Original State Space . 157 5.4.3 Ecient Aggregated Tree Search . . . . . . . . . . . . . . . . . . . . . . . . 159 5.5 Computing Aggregation Through Arrangement of Hyperplanes . . . . . . . . . . . 160 5.5.1 Arrangement of Hyperplanes, Polytope and Incidence Graph . . . . . . . . 161 5.5.2 On Construction of the Incidence Graph of U(E;P ) and U(E;P ) . . . . . . 164 5.5.3 Constructing the Incidence Graph of cubeP from the Incidence Graph of P 165 5.6 Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 5.6.1 Complexity of Optimal Control of Cascading Failure . . . . . . . . . . . . . 174 5.6.2 Time Complexity of the Solution Methods . . . . . . . . . . . . . . . . . . . 176 5.7 Approximation Algorithm and Simulations . . . . . . . . . . . . . . . . . . . . . . 178 5.7.1 Approximation Algorithm via Projection . . . . . . . . . . . . . . . . . . . 178 5.7.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 5.8 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 v Chapter 6: On Distributed Computation of Optimal Control of Trac Flow over Networks 183 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 6.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.2.1 Network Flow Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.2.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 6.3 The Alternating Direction Method of Multipliers (ADMM) . . . . . . . . . . . . . 189 6.3.1 Piecewise Constant Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.3.2 ADMM Iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.4 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.5 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Chapter 7: Conclusions and Future Work 202 Reference List 204 Appendix A A.1 Minimizing A Quasi-concave Function over A Polytope . . . . . . . . . . . . . . . . 210 A.2 Derivative of Pseudoinverse of Laplacian Matrix . . . . . . . . . . . . . . . . . . . 210 A.3 Inverse of Sums of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 A.4 Extension of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 A.5 Derivatives of g(w eq ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 A.6 Proof of Proposition 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 vi List Of Tables 4.1 Explicit characterization of and w opt from (4.52) and (4.53), respectively, for n = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.1 Comparison of time complexity of several algorithms. . . . . . . . . . . . . . . . . 176 5.2 Optimal residual load under (5.7) and under the projection-based approximations in (5.27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6.1 The ADMM iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 vii List Of Figures 3.1 Illustration of the unit ow and congestion function. (a) A graphG containing a single supply node 1 and a single demand node 6; (b) The unit ow function; (c) The congestion function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Illustration of DC ow and least congested ow . . . . . . . . . . . . . . . . . . . . 30 3.3 Illustration of oblique projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.4 Illustration of signs of @f=@w i : red arrows alongside each link denote the ow di- rection on the corresponding link under the supply-demand vector f i A i ; for every link6= i, if the red arrow alongside a link aligns with the link direction, then the corresponding component of @f=@w i is negative, and positive otherwise. Corre- spondingly,@f k1 =@w i > 0,@f k4 =@w i < 0,@f k2 =@w i > 0,@f k3 =@w i > 0. We always have @f i =@w i > 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.5 A multigraph perspective for weight perturbations. (a) the initial networkG has weight w; (b) the network ~ G post-perturbations, where the dashed links are links to be removed, the change of weight is4w: 4w 1 < 0,4w 2 = 0,4w 3 > 0 and 4w 4 =w 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.1 Network used in Examples 3 and 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 Illustration of a reducible network. Please see Remark 32 for clarication regarding the dierence in depiction of v 1 and v 2 infG; ~ G 1 g andG 2 . . . . . . . . . . . . . . . 75 4.3 Illustration of recursive network reduction, where the supply node set isf1; 4g and the demand node set isf2; 3g; the thick edges denote the equivalent links. The original networkG is reduced to the terminal networkG T in three reductions: (i) subnetworkG (0) 2 ! link (4; 3); (ii) subnetworkG (1) 2 ! link (5; 6); (iii) subnetwork G (2) 2 ! link (3; 2). G k := ~ G (k1) 1 is the resulting network after k-th reduction, k = 1; 2; 3. Notice that the rst and the second reductions can be implemented in parallel, and the terminal networkG T =G (3) is not reducible. . . . . . . . . . . . . 87 4.4 A candidate graph topology for tree reducible network . . . . . . . . . . . . . . . . 89 4.5 A sampleS 0 function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.6 A three link series network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 viii 4.7 A two link parallel network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.8 Equivalent capacity function for a parallel network consisting of links with constant capacities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.9 The IEEE 39 bus system (left) and its terminal network (right). . . . . . . . . . . 120 4.10 The equivalent capacity functions in the process of reduction for the network shown in Fig. 4.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.1 The sequence of ow redistribution, control action, and link failures, under the proposed cascading dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.2 The graph topology for the network used in Example 7 to illustrate that the feasible control action setD is not necessarily closed. . . . . . . . . . . . . . . . . . . . . . 132 5.3 The tree composed of states reachable from (E 0 ;p 0 ) in at most N time steps. . . . 133 5.4 A parallel network with two links. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.5 The graph topology for the network used in Example 8 to illustrate that the set of feasible one-shot control actions is neither connected nor closed. . . . . . . . . . . . 140 5.6 (a) a tree reducible networkG; and (b) a reduced treeT ofG. . . . . . . . . . . . . 141 5.7 The four local (star) sub-networks corresponding to Figure 5.6b. . . . . . . . . . . 144 5.8 Illustration of (x) dened in (5.16). . . . . . . . . . . . . . . . . . . . . . . . . . 146 5.9 (a) Input functions to ; (b) Output function from each subproblem. . . . . . . . 150 5.10 (a) A network (V;E 0 ) withV s = f1; 2g, V d = f4g, w = 1, c = [10; 3; 3; 6] T , p 0 = [5;5; 0; 10] T ; (b) Projection of U(E 0 ;p 0 ) and u2B E jf i (E 0 ;u) =c i , i2f2; 3; 4g, onu 1 u 2 plane; (c) Incidence graph ofU(E 0 ;p 0 ); (d) incidence graph of U 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.11 Two possible scenarios for projection: (a) and (b) show the projection onto a space of the same dimension; and (c) and (d) show projection onto a space of lower dimension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.12 Illustration of sweep in scenario I: (a) the geometrical graph; and (b) the incidence graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.13 Dierent facets according to the direction 1 k . . . . . . . . . . . . . . . . . . . . . 170 5.14 Illustration of sweep: (a) sweep of U 1 ; (b) the incidence graph of sweep 2 (U 1 ); and (c) cubeU 1 as sweep of sweep 2 (U 1 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.15 The IEEE 39 bus network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 6.1 Illustration of variable separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 ix Chapter 1 Introduction Societal functions highly rely on networked infrastructure systems such as transportation, power, water, and data networks. Due to increasing load, scale and interconnectedness, these systems are frequently overloaded and tend to exhibit complex behaviors, which poses considerable new challenges in their design and operation. In part due to the potential catastrophic consequences caused by ensuing cascading failures which can also aect other dependent systems, robust oper- ation and control of these infrastructure systems are becoming a major concern. The goal of this dissertation is to promote a formal and unied framework for optimal control synthesis for infras- tructure networks. Towards this goal, we consider several problems for dierent infrastructure networks and develop mathematical tools to solve these problems. The rst step in analytical study of physical systems is modeling. Due to intrinsic dierences in underlying physics, it is challenging to study dierent infrastructure networks within a com- mon modeling framework. One possible strategy is to resort to complex network theory [10, 56]. There does exist a large body of literature on the use of this theory to analyze the properties of infrastructure networks and their resiliency [42]. One such approach is to use percolation model to analyze cascading failure [33,71,88]. However, due to lack of proper physical interpretations, these approaches present severe limitations, as demonstrated by both real data [9, 65] and simulation studies [13, 14]. Even worse, these generic black box approaches impose an impenetrable mask 1 over system physics that hampers true understanding. This dissertation focuses on developing network models, particularly, static deterministic models, based on rst principles. Network models usually have intuitive characters and are promising tools for describing real world problems. The best example to illustrate this is the ow network model. Network ow problems such as shortest path, assignment, max- ow, transportation, transhipment, spanning tree, matching, traveling salesman, generalized assignment, vehicle routing, and multicommodity ow constitute the most common class of practical optimization problems [15]. With the existence of extensive studies based on ow network model, its application spectrum is ever expanding. One important aspect is distributed optimization and control over ow networks. In Chapter 6, we propose a distributed algorithm, based on the well known Alternating Direction Method of Multiplier (ADMM) [21], and show its convergence properties for computing nite horizon optimal assignment for trac ow. The rigorous convergence poof dierentiates this work from that in [74], though ADMM is used as well in the latter. The majority of the dissertation focuses on infrastructure networks that can not be suciently modeled using the ow network paradigm. Prime examples are power grids and gas networks. In order to formally dene a network model for such infrastructure networks, we dierentiate the topological structure from the governing physics. While it is natural to describe the topology of an infrastructure network using a directed or undirected graph, the physics is often complex and also related to practical operational concerns. In order to describe the physics, we endow nodes and links with variables, such as node potential and link ow, that represent physical quantities. We focus on two canonical physical laws that appear in multiple infrastructure networks, namely ow conservation law and diusion law. The ow conservation law requires the net ow at every node to be equal to zero. The diusion law requires a strictly monotonic relationship, described by a kernel function, between ow on a link and the potential dierence between its two incident nodes. We call a network a diusion network if it respects both physical laws. The exibility of choosing kernel functions reinforces the expressive power of diusion network model. Common examples of 2 diusion networks are DC power network [84], gas network [8] and the steady coupled-oscillator model [23]. Among the dierent types of diusion networks, this dissertation is most devoted to DC networks, a subclass of diusion networks whose kernel functions are linear. There are at least two reasons for special focus on DC networks in this dissertation. Firstly, DC networks are a good modeling framework for power grids, and hence of great practical importance. Secondly, DC networks can be interpreted as the simplest linear diusion networks, and hence their study is a natural rst step towards understanding nonlinear diusion networks. In Chapter 2, we show that every diusion network admits a unique link ow distribution that depends on both the supply-demand vector and link weights. The supply-demand vector uses positive and negative numbers to represent supplies and demands at nodes, respectively, and link weights quantify key link properties. In case of DC power networks, the weight of a link is the negative of susceptance of a power line. Due to uniqueness of ow distribution, all variables of a diusion network are coupled. Any local change in the network, such as changing the weight on a single link or changing the supply-demand at a single node, lead to ow redistribution over the entire network. This coupling makes distributed optimization or control over a diusion network challenging, which is to be contrasted with ow networks. Nevertheless, in Section 4.5, Section 4.6 and Section 5.3, we provide distributed computational approaches to solve relevant non-convex problems, for DC networks having certain graph topologies. The technical approach for all these problems involves topological reduction of networks. It works by taking advantage of the locality nature of diusion laws and constitutes a promising tool for diusion networks in general. Furthermore, Section 4.7 provides decentralized control policies in the context of robust weight control, which are proven to be optimal for special classes of networks. Apart from the physics, we also take into account the operational constraints of an infras- tructure network. Specically, we consider that a feasible ow has to satisfy capacity constraints, i.e., to stay within the capacity bounds. As ow depends on the supply-demand vector and link 3 weight vector, a fundamental problem is to nd a feasible weight such that certain cost function of ow is minimized for a xed supply-demand vector. The problem is non-convex because ow is a non-convex function of link weights. This dissertation contributes towards novel understanding of the dependence of ow on link weights, and its implication in optimal control design. The motivation is to enable solutions to a variety of problems for DC networks. The maximal DC ow problem discussed in Chapter 3 is to nd a feasible weight that maxi- mizes the amount of supply and demand a DC network can support without violating the capacity constraints for a given direction of supply-demand vector. It has strong connection to its coun- terpart for an undirected ow network, i.e., the maximal ow problem. It is clear that the later problem provides an upper bound for the former, as additional physical constraints are imposed on the DC network. In this dissertation, we identify conditions under which these two problems have the same optimal value. Furthermore, we transform the maximal DC ow problem into a problem that minimizes a certain congestion function over box constraints. The congestion func- tion relates to the least congested ow over the DC network. In spite of the non-convexity and non-dierentiability, we show that, for the unconstrained minimization of the congestion function, every local minimum is a global minimum. Moreover, useful results are derived for several im- portant objects such as the ow-weight Jacobian matrix, pseudo-inverse of Laplacian matrix and an oblique projection matrix. The latter is a novel concept that we introduce in this dissertation, and is shown to contain rich information about the underlying DC network. While these results may have interests on their own, the expression derived from them for ow redistribution under weight perturbation is already an extension of recent work [82]. The investigations on congestion function and the three relevant matrices mentioned in last paragraph are directed towards an iterative weight update algorithm to solve the maximal DC ow problem, which can be interpreted in dierent ways. From an algorithm design perspective, it resonates with the seminal work [26] on solving the maximal ow problem using fast Laplacian solvers [62,83] and multiplicative weight update method [2]. Note that, in these two last methods, 4 the term "weights" is not meant to have the same physical meaning as we imply in this dissertation. From an optimal control perspective, the algorithm, if convergent, can be used to construct an optimal oblivious control policy for weight control problem, as we would discuss next. The weight control problem in Chapter 4 studies robustness of DC networks towards balanced disturbances to a nominal supply-demand vector. The margin of robustness is dened as the radius of the largest ` 1 ball of disturbances for which there exists a weight to produce a feasible ow. Computation of this margin is a non-convex optimization problem. We propose an equivalent multi-level programming approach for a class of networks in which the nodes with non-zero demand or supply are relatively sparse. This approach is based on recursive application of equivalent bilevel formulation for a relevant class of, possibly non-convex, optimization problems. The lower level problem in each recursion corresponds to replacing a sub-network by a (virtual) link with equivalent weight and capacity functions. The equivalent capacity function for a link reducible network possesses a strong quasi-concavity property. This property facilitates easy solution to the multilevel programming formulation for a certain class of networks, that we call type I tree reducible networks. Chapter 5 studies the optimal control of cascading failure in DC networks. This problem was formulated in [18,19], where a restricted set of control policies were considered. In this disserta- tion, we further formalize it as a nite horizon optimal control problem and provide a thorough characterization of the problem complexity for general control polices and arbitrary DC networks. Specically, we consider discrete-time dynamics, for cascading failure in DC networks, whose map is composition of failure rule with control actions. Supply-demand at the nodes is monotonically non-increasing under admissible control. Under the failure rule, a link is removed permanently if its ow exceeds capacity constraints. We consider nite horizon optimal control to steer the network from an arbitrary initial state, dened in terms of active link set and supply-demand at the nodes, to a feasible state, i.e., a state which is invariant under the failure rule. There is no running cost and the reward associated with a feasible terminal state is the associated cumulative 5 supply-demand. We propose two approaches for computing optimal control, and provide time complexity analysis for these approaches. The rst approach, geared towards a certain class of networks that we call type II tree reducible networks, decomposes the global problem into a system of coupled local problems, which can be solved to optimality in two iterations. When restricted to the class of constant control actions, the optimal solutions to the local problems possess a piecewise linear property, which facilitates analytical solution. The second approach computes optimal control by searching over the reachable set, which is shown to admit an equivalent nite representation by aggregation of control actions leading to the same reachable active link set. An algorithmic procedure to construct this representation is provided by leveraging and extending tools for arrangement of hyperplanes [38,85] and convex polytopes [49,92]. To best of our knowl- edge, this dissertation provides the rst instance of the application of the elegant computational geometric tools related to arrangement of hyperplanes to an engineering application, beyond path planning and related topics in robotics, e.g., see [50]. In summary, this dissertation makes several important contributions in the domain of robust and optimal operation of infrastructure networks. First, it provides a formal description of diu- sion network model that can be used for various infrastructure networks. Second, it formulates the maximal DC ow problem, identies a congestion function useful for its solution, and prove that, despite non-convexity and non-dierentiability, every local minimum is a global minimum of the congestion function. Third, it derives useful insights into the relationship between ow and weight of a DC network, which is related to both solving the maximal ow problem through itera- tive weight update and improving robustness of the DC network through feedback weight control. Fourth, it formulates a novel robust weight control problem for DC networks, and proposes a multi-level programming approach that has the potential to solve other problems related to DC networks. Fifth, it provides a much needed formalism to the optimal control problem of cascad- ing failure in DC power networks and provides two novel solutions. One exploits the structures of networks with special topology and the other relates and extends tools from computational 6 geometry. Sixth, it provides a distributed algorithm with convergence guarantee for computing nite-horizon optimal control of trac ow over freeway networks. 1.1 Notations We next dene a few key notations used in the dissertation. R, R 0 , R >0 , R 0 and R <0 will stand for real, non-negative real, strictly positive real, non-positive real, and strictly negative real, respectively. I, 0 and 1 will denote the identity matrix and vector of all zeros and all ones, respectively, where the sizes of the matrix and vectors will be clear from the context. For an integer n, [n] :=f1; 2;:::;ng. For two vectors x and y of the same size, xy means x i y i for all i. The same convention is adopted for, < and >. For a vector x2 R n , diag (x)2 R nn denotes the diagonal matrix whose diagonal entries are those of x. The same notation extends to a block diagonal matrix. jSj denotes the cardinality of set S. We use the shorthand notation R S1S2 := R jS1jjS2j for index sets S 1 and S 2 . For a matrix M 2 R S1S2 and two subsets S 0 1 S 1 and S 0 2 S 2 , M S 0 1 S 0 2 2 R S 0 1 S 0 2 is the submatrix ofM that contains entries in theith row and thejth column for alli2S 0 1 andj2S 0 2 . In addition, M S 0 2 := M S1S 0 2 . If S 0 1 =fig and S 0 2 =fjg, then M S 0 1 S 0 2 M ij and M S 0 2 M j for simplicity. Therefore, M j is the jth column of matrix M. Similarly for a vector x2 R S , x S 0 denotes the appropriate sub-vector for S 0 S. Therefore, for some set S [n], 1 S 2 R n denotes a vector that has ones on components corresponding to indices in S and zeros elsewhere. In particular, 1 i denotes theith unit coordinate vector, which has a single one on theith component and zeros elsewhere. Given a mapf :X!Y ,R(f) will denote the range off. We will also useR(M) to denote the range of matrix M. The support of a vector x2 R n is the set of indices associated with nonzero components, i.e.,fi2 [n]jx i 6= 0g. The sign function, denoted by sign(x), is dened to be equal to +1 if x > 0,1 if x < 0, and 0 if x = 0. Given a point set S, convS and aS denotes 7 the convex hull and ane hull of S, respectively. For a point setfx 1 ;:::;x n g R d , we use the shorthand notation conv(x 1 ;:::;x n ) := conv(fx 1 ;:::;x n g). Given point sets S 1 R n and S 2 R n ,S 1 +S 2 denotes the Minkowski sum of S 1 andS 2 , that is,fx 1 +x 2 jx 1 2S 1 ;x 2 2S 2 g. The notations related to graphs and networks are introduced in Section 2. While the notations within a chapter are consistent, we shall occasionally use the same letter to denote dierent quantities in dierent chapters. 8 Chapter 2 Network Model A network is considered to be a mathematical abstraction of a physical system, whose topology is described by a graph and whose nodes and links are associated with physical quantities related by physical laws and operational constraints. 2.1 Basic Concepts from Graph Theory We start by reviewing several key concepts and notations related to graphs. While reader new to this led can nd a thorough overview from standard textbooks on graph theory, e.g., [37], experienced readers may nd it useful to skim the section for possible nonconventional denitions and notations. A graphG = (V;E) is a collection of two sets, whereV denotes the set of vertices (or nodes) andEVV denotes the set of edges (or links) 1 . We denote the edge inE that connects two verticesv i andv j bye ij = (v i ;v j ). At the same time, we also label edges and denote thekth edge by e k . These two notations for edges are used interchangeably. Therefore, by e k = (v i ;v j ) =e ij , we mean that thekth edge connectsv i tov j . In this case, we say that vertices v i ,v j and edgee k are incident, and we callv i andv j the ends (or end nodes) of edgee k . In this dissertation, we do not allow self loops, i.e., edges whose ends are identical. 1 Despite the fact that vertex (resp. edge) and node (resp. link) are usually used interchangeably, we use vertex and edge for graphs and node and link for networks in this dissertation. 9 A graph can be either undirected or directed. An undirected graph has unorientated edges which correspond to unordered sets, i.e., e ij = (v i ;v j ) = (v j ;v i ) = e ji 2 . A directed graph has orientated edges whose directions are described by the order of the incident vertices. For example, edge e ij = (v i ;v j ) denotes an edge from v i to v j and is dierent from e ji = (v j ;v i ). In general, an undirected graph can be represented as a directed graph by replacing each undirected edge with two directed edges in opposite directions. As would be shown later, an edge is usually associated with a physical quantity called ow. In order to describe the direction of ow over an undirected graph, it is convenient to assign directions to its edges (cf. Remark 4). In a directed graph or an undirected graph with direction convention, the head and tail vertex of edge e are denoted by (e) and (e), respectively, and the set of incoming and outgoing edges are denoted byE v :=fe2Ej(e) = vg andE + v =fe2Ej(e) = vg, respectively. LetE v :=E + v [E v . The number of edges incident with a vertex v, i.e.,jE v j, is called the degree of a vertex v. A path in a graphG = (V;E) is a subgraphP = (V 0 ;E 0 ) (V 0 V,E 0 E) of the following form: V 0 =fv 1 ;v 2 ;:::;v k g; E 0 =f(v 1 ;v 2 ); (v 2 ;v 3 );:::; (v k1 ;v k )g: The edges inP are connected in series from vertex v 1 to vertex v k , where v 1 and v k are referred to as the ends of pathP. A path in a directed graph is directed and all its edges have the same direction. However, depending on direction convention, the direction of an edge in a path of an undirected graph can agree or disagree with the direction of the path. The length of a pathP, denoted byjPj =jE 0 j, is the number of edges it contains. A path is simple if it does not visit any vertex more than once. A path is called a cycle if the ends of the path are identical. A undirected graphG is connected if there exists a path from v i to v j for all v i ;v j 2G. A directed graph is strongly connected if it is connected and weakly connected if its undirected version is connected. 2 Precisely speaking, an edge connectingv i andv j in an undirected graph should be denoted asfv i ;v j g. (v i ;v j ) is used here for convenience. 10 A cut ofG = (V;E) is a partition of the vertex setV into two nonempty subsets: V 0 and its complementVnV 0 [15]. Such a cut, referred to asV 0 VnV 0 cut, is uniquely determined by setV 0 . For a directed graph, the partition is ordered in the sense that cutV 0 VnV 0 is distinct from cutVnV 0 V 0 . However, for an undirected graph, we consider thatV 0 VnV 0 and VnV 0 V 0 denote the same cut. We also refer to cutV 0 VnV 0 byC V 0, whereC V 0 denotes the set of edges connecting from vertices inV 0 to vertices inVnV 0 . For a directed graphG = (V;E), C V 0 :=fe2Ej(e)2V 0 ;(e)2EnV 0 g. For an undirected graphG = (V;E), regardless of the direction convention,C V 0 :=fe2Ej(e)2V 0 ;(e)2EnV 0 g[fe2Ej(e)2V 0 ;(e)2EnV 0 g. Several types of graphs used in the dissertation are summarized here. Simple Graph and Multigraph A simple graph is a graph that does not contain any multiple edges, i.e., edges with identical ends. A multigraph is a graph permitted to have multiple edges. Multigraphs include simple graphs as special cases. Unless stated otherwise, we consider a graph to be a multigraph in the dissertation. Series Graph A series graph is a simple path. Parallel Graph A parallel graph consists of two vertices which are connected by multiple edges. Tree A tree is a connected undirected graph that does not contain a cycle. Any two vertices in a tree are connected exactly by one path, and the number of vertices is one more than the number of edges in a tree. 11 Directed Acyclic Graph (DAG) A Directed Acyclic Graph (DAG) is a directed graph that does not contain a directed cycle. While an undirected graph with no cycle has a simple tree structure, it is possible for a DAG to have a complex structure. A DAG has a topological ordering, which is an ordering of its vertices as v 1 ;v 2 ;:::;v n so that every edge (v i ;v j ) has to satisfy i<j. 2.2 Laplacian Matrix of Graph The node-link incidence matrix A2f0; 1;1g VE is a concise algebraic representation of a di- rected graph or an undirected graph with certain direction convention. It is dened such that A vi is equal to1 if v = (i), equal to +1 if v = (i), and equal to zero otherwise. Therefore, the kth column of A, i.e., A k corresponds to edge e k . Remark 1. 1. The node-link incidence matrix of an undirected graph is not unique and depends on the direction convention adopted for its edges. 2. The node-link incidence matrix of a tree has independent columns [17, Chap 7.1]. A graphG = (V;E) can be weighted, in which case each edge e i 2E is associated with a weight w i > 0. For convenience, let W := diag(w)2 R EE >0 . The Laplacian matrix of a weighted undirected graph is dened as follows. Denition 1 (Laplacian Matrix). Given an undirected graphG = (V;E) with node-link incidence matrix A2f1; 0; +1g VE and weight w2 R E >0 , its weighted Laplacian matrix is dened as L G (w) :=AWA T : The characteristics of a Laplacian matrix reveal rich properties of the associated graphs. For example, a graph is connected if and only if the second smallest eigenvalue of the Laplacian matrix 12 is positive [86]. The study of the relationship between the structural properties of a graph and the algebraic properties of the Laplacian matrix is called spectral graph theory [27]. For brevity in notation, we shall drop explicit dependence of L on w orG when clear from the context. Remark 2. The Laplacian of a network does not depend on the specic choice of directionality for edges. This is to be contrasted with node-link incidence A as per Remark 1. While the Laplacian matrix is usually dened for simple graphs, Denition 1 considers the general multigraph setting. 2.3 Network Models Graph is a simple yet expressive mathematical abstraction of objects and their binary relation- ships. Its vertices and edges are usually related to physical objects naturally. As a modeling tool, it is of signicant value to produce theoretical and algorithmic results in numerous applications. Infrastructure systems such as transportation networks, power networks, gas or water networks belong to the domains where graphic model can play an important role. However, studies on graphs most focus on their topological properties [46]. Interpretation of these results may not be straightforward for physical systems. Furthermore, infrastructure systems are usually equipped with complicated physical laws, dynamics and operational constraints. In order to include these in the model, we formally dene a network to be the combination of a graph topology and the associated physical quantities that are related by physical laws. While the type of physical quanti- ties and physical laws depends on specic physical systems, we aim at identifying and formulating basic features shared by common infrastructure networks. While the underlying physics of infras- tructure networks can be dynamic processes in general, we only focus on their equilibrium states in this chapter. The resulting network model is hence static. 13 2.3.1 Physical Quantities for Networks We start by inspecting several major infrastructure networks. In a transportation network, a node can be an intersection and a link can be a street or a highway. A link conveys certain amount of trac ow, which can be quantied by the number of vehicles passing per hour. In a power network, a node is a bus and a link is a power line. A node operates at certain voltage, frequency and phase angle and a link transfers certain amount of power ow. In a gas network, a node can be a gas station or a user terminal and a link can be a pipeline between two stations. In that case, gas ows through links under the pressure dierence between nodes. Considering the similarity and dierence among these infrastructure networks, we assign the following physical attributes to nodes and links in a graphG = (V;E). node v i 7! i (potential);p i (supply-demand) link e i 7! f i ( ow);w i (weight) where the weightw is assumed to be nonnegative and the owf has the same unit as the supply- demand p. The weight denotes relevant physical properties of links in a physical network, such as the length of a highway or the negative of susceptance of a power line. A node v i is called source or supply node if p i > 0, and sink or demand node if p i < 0. If a node is neither a supply nor demand node, then it is called a transmission node. Supply and demand nodes are called uniformly non-transmission node. The magnitudejp i j equals to the amount of supply (or in ow) and demand (or out ow) on node v i . We note the above physical quantities do not mean to be necessary nor sucient for a physical network. Furthermore, as each high way has a trac ow limit and each compressor can provide no more than certain maximal value of pressure, physical quantities nearly always come together 14 with physical constrains. While this can be true for all the above physical quantities, in this dissertation, we shall only focus the following capacity constraint: c l fc u or jfjc (2.1) where c l and c u are the lower and upper capacity, respectively. For symmetric capacities, i.e., c l =c u , we use single notation c and constraintjfjc for simplicity. Remark 3. In general, capacities are not necessarily constants. They can be functions depending on some varying variables. For example, capacities for trac ow depend on link density in the dynamic network loading models for transportation networks (see. Chapter 6.2.1). Moreover, depending on the operational constraints, a network can have directed or undirected underlying graph. For example, a transportation network is usually modeled by a directed graph because vehicles are allowed to go in one direction only at a certain high way. Whereas a power network is usually modeled by an undirected graph because power ow on a power line can go both directions. Remark 4. Direction convention is assumed to assign directions to links in an undirected network. The sign of components off are to be interpreted as being consistent with the direction convention chosen for the links. For example, f i < 0 denotes a ow of magnitude ofjf i j on linke i from node (e i ) to node (e i ). 2.3.2 Physical Laws for Networks In this section, we introduce two physical laws that are common in multiple physical networks. The rst one is the ow conservation law as shown in Denition 2. 15 Denition 2 (Flow conservation law). For a networkG = (V;E) with ow vector f2 R E and supply-demand vectorp2 R V , it is said to satisfy ow conservation law if the following equations hold. X i2E + v f i X i2E v f i =p i 8v2V It is clear that the ow conservation law in Denition 2 is mass conservation law for trans- portation networks and Kichho's Current Law (KCL) for electrical networks. Using the node-link incidence matrix A of graphG, the ow conservation law can be succinctly written as: Af =p (2.2) The second physical law for networks is diusion law as shown in Denition 3. Denition 3 (Diusion law). For a networkG = (V;E) with potential vector 2 R V and ow vector f2 R E , and for continuous and strictly increasing functions i : R! R for all i2E, the network is said to satisfy diusion law with kernel function if the following equations hold. i (f i ) = (i) (i) 8i2E (2.3) Remark 5. We emphasize that in order for a function : R n ! R n to be a valid kernel function for a networkG = (V;E),n must be equal tojEj and every component function i ,i2 [n], has to be a continuous and strictly increasing scalar function. In this dissertation, we are most interested in the diusion law with kernel function being linear functions of positive slopes, i.e., i (f i ) = f i =w i for w i > 0;i2E. We refer this special diusion law as Ohm's law. The term is adopted from circuit theory with f, and w being interpreted as the current, voltage and conductance, respectively. Ohm's law can be compactly written as: f =WA T (2.4) 16 where we recall W := diag(w)2 R EE . Based on whether these two physical laws are satised or not, we dene ow networks and diusion networks. 2.3.3 Flow Network Denition 4 (Flow network). A network is called a ow network if the links and nodes are endowed with ow and supply-demand variables, respectively, and ow conservation law (as per Denition 2) is satised. Remark 6. A ow network can be either directed or undirected. While ows can be positive or negative in an undirected ow network, they must be nonnegative in a directed ow network. In a ow networkG = (V;E), the imposed supply-demand vector p2 R V has to be balanced over all connected components ifG is undirected, or over all weakly connected components ifG is directed, that is, p2B E := 8 < : x2 R V j X v2V (i) x v = 08i 9 = ; (2.5) whereV (i) denotes the node set ofith (weakly) connected components ofG. The balance condition (2.5) can be derived for a connected network as follows: 1 T p = 1 T Af = 0, where the second equality is due to the denition of A. The same argument can be made for every connected component of a disconnected network. Note as the connectivity ofG depends on the link setE, setB E depends onE as well. For an undirected ow networkG = (V;E) with balanced supply-demandp2 R V , (2.2) shows that the set of admissible ow is an ane space. IfG is connected, the rank ofA is equal tojVj1 and the dimension of the ane space equals tojEjjVj + 1, which is also equal to the number of independent cycles ofG. 17 2.3.4 Diusion Network Denition 5 (-diusion network). For a valid kernel function (cf. Remark 5), an undirected ow network is called a -diusion network 3 if it has additional potential variables on nodes and the diusion law (2.3) is satised with respect to function . Diusion network model includes DC network model and gas network model as special cases. While gas network model is provided in the next example, DC network model will be discussed separately in Section 2.4. Example 1 (Gas network [8]). A gas network is an undirected ow network with each link repre- senting a pipe through which gas ows in one direction or the other. We consider a gas network G = (V;E) with ow f2 R E and supply-demand vector p2 R V . Then f i and p v represent the gas ow on pipe i2E and the gas supply or demand on node v2V. Gas ows through pipes under pressure dierence between nodes. s v is used to denote the pressure of node v2V. Apart from the ow conservation equation (2.2), the following equation has to be satised: l i d 5 i f i jf i j =s 2 (i) s 2 (i) 8i2E (2.6) wherel i andd i are the length and inner diameter of pipe i2E, respectively, and is a technical coecient, which is the same for all pipes. l i , d i and are considered to be constants. Since the function (l i =d 5 i )f i jf i j is continuous and strictly increasing in f i for all i2E , gas network is a diusion network with s 2 v playing the role of the potential variable v in (2.3). The next result shows that a diusion network has unique ow distribution for a given supply- demand vector. Theorem 1. Consider a -diusion networkG = (V;E) with an arbitrary valid kernel function . For every balanced supply-demand vector p2 R V , the ow distribution overG is unique. 3 The network science community uses the term \diusion" to refer to information propagation, whereas in the diusion network dened in this dissertation, the term \diusion" is used to refer to distribution of energy or resources. 18 Proof. Since is a valid kernel function for networkG, i is continuous and strictly increasing for all i2E. Therefore, there exist strictly convex functions h i (f i ) = R i (x)dx for all i2E. Consider the following convex problem. min f2R E X i2E h i (f i ) s.t. Af =p (2.7) whereA2 R VE is the node-link incidence matrix ofG. The Lagrangian of the above problem is: P i2E h i (f i ) + T (pAf), where 2 R V is the Lagrangian multiplier. The optimality condition is given by the following KKT conditions for convex problem. Af =p i (f i ) ( (i) (i) ) = 0 8i2E where we use the fact that @hi(fi) @fi = i (f i ). It is straightforward to see that the second equation represents the diusion law in Denition 3, if is treated as the potential variable. The assertion then follows from the fact that the solution to a convex problem with strictly convex objective function is unique. Remark 7. 1) If the function h i (f i ) in the proof of Theorem 1 is to be interpreted as the energy on link e i , then the diusion law can be seen as a result of the minimum total potential energy principle. 19 2) For ohm's law in (2.4), i.e., linear kernel functions with positive slope, the potential energy is h i (f i ) = 1 2 f 2 i =w i , which represents heat loss from a resistor in a circuit network. In this case, (2.7) can be equivalently written as the following minimum weighted norm problem. min f2R E kW 1=2 fk 2 s.t. Af =p (2.8) As oppose to a ow network for which the set of admissible ow under a given supply-demand vector is an ane space, the set of admissible ow over a diusion network is a singleton. This distinction has considerable in uence on control synthesis and design of these two type of systems. While one is allowed to choose ow distribution in a ow network directly as a means to achieve optimal operation, the control of ow over a diusion network is usually indirect and in a form of adjusting the supply-demand vector or network parameters such as link weights. Furthermore, in a diusion network, the ow distribution, due to its uniqueness, relates all physical quantities to each other. Disturbances on supply-demand of a single node or removal of a single link lead to changes of ow over the entire network. This poses a big challenge for distributed operation and control of diusion networks. It is therefore desirable to gain a good understanding on the factors that aect the ow distribution, and its implication on control design, for diusion networks. This dissertation studies these by focusing on the class of diusion networks with linear kernel functions. 2.4 DC Network Denition 6 (DC network). A DC network is a diusion network with a linear kernel function. The slope of every component kernel function is given by the reciprocal of the corresponding link 20 weight. Specically, for a DC networkG = (V;E) with weight w2 R E >0 , supply-demand p2 R V , ow f2 R E and potential 2 R V , the following equations have to be satised: Af =p f =WA T (2.9) It is straightforward that a DC circuits network containing only resistors and ideal current sources belongs to the class of DC networks. In that case, link weight refers to conductance. Another example of DC networks is the transmission power network under DC approximation. It is explained in the following remark. Remark 8. The DC approximation to power ow is a linear approximation of the AC Branch-Flow model [66]. It only considers the real power ow equations and assumes that the transmission lines are lossless and the voltage magnitudes at nodes are constant at 1:0 unit [45,84]. Under the DC approximation, the physical quantities of a power system, including the real power ow f, bus voltage angle , the negative of susceptances w and real power supply-demand p, are related by (2.9). Such a power system is referred to as a DC power network. DC power network model is widely and even increasingly used in practice [19,45,84]. We make the following remark on zero weight on a link. Remark 9. (2.9) implies that setting the weight of a link to zero is equivalent to removing the link. Moreover, if a link is not contained in any path from a supply node to a demand node, then it carries zero ow regardless of the values of the supply-demand vector and link weight vector. We hence make the following standing assumption on the links and nodes of a network. Assumption 1. For a DC networkG = (V;E), every link i2E and nodev2V are contained in at least one path from a supply node to a demand node ofG. 21 Theorem 1 implies that a DC network has unique ow distribution satisfying (2.9), which we refer to as DC ow. Lemma 1 provides an explicit expression for DC ow. Lemma 1. For a connected DC networkG = (V;E) with weight w2 R E and balanced supply- demand vector p2 R V , there exists a unique f2 R E satisfying (2.9), and is given by: f =WA T L y (w)p =:f G (w;p) (2.10) where L y (w) is the Moore-Penrose pseudo-inverse of the Laplacian L(w), A2f1; 0; 1g VE is the node-link incidence matrix ofG, and W = diag(w)2 R EE . Proof. Substituting the second equation into the rst in (2.9), we get L =p. Note that the null space of L and L y is spanf1g [86]. Since p is balanced (cf. (2.5)), it is in the range space of L, and hence the solution to L =p is given by =L y p + 1, where is an arbitrary scalar. Since A T 1 = 0, the ow solution is unique: f =WA T =WA T L y p. We shall drop the explicit dependence of f onG, w or p when clear from the context. Remark 10. 1) Function f(w;p) is linear with respect to p for all w2 R E >0 . 2) As w appears both outside and inside L y (w), the dependence of f(w) on w is complex. However, note that f(w;p) =f(w;p) for all p2 R E and > 0, that is, f(w) is positively homogeneous of degree 0. 3) In [20], a result similar to (2.10) is provided as f =W ~ A T ( ~ AW ~ A T ) 1 ~ p, where ~ A and ~ p are one row reduced versions ofA andp, respectively. The equivalence can be shown by setting the corresponding component of the solution of to 0. We note that Lemma 1 applies to unconnected networks, as long as the supply and demand are balanced for each of these components. This is true due to the following observation. For 22 a networkG containing multiple components G (1) ;:::;G (r) , L G = diagfL G (1);:::;L G (r)g and L y G = diagfL y G (1) ;:::;L y G (r) g if nodes are labeled accordingly. The following results show that the ow on every link, under a DC power ow model, is no greater than the total supply/demand. The latter is equal tokpk 1 =2. Lemma 2. For a network with graph topologyG = (V;E), link weights w2 R E >0 and balanced supply-demand vector p2 R V , the following are true. 1) The unique solution f 2 R E to (2.9) satisesjf i (w;p)jkpk 1 =2 for all i2E and the inequality is strict if and only if link i alone does not form a cut separating the supply nodes from the demand nodes. 2) IfG contains a single supply demand pair, then all links connecting the supply node carry nonzero ows that ow out from the supply node, and all links connecting the demand node carry nonzero ows that ow into the demand node. Proof. Let ~ E be the union of links inE with positive ows and the reverse of links inE with negative ows. Note that ~ E does not contain links with zero ow, and that the ows on links in ~ E are positive, i.e., ~ f > 0. Therefore, in order to show the assertion, we need to show that ~ f i kpk 1 =2 for all i2 ~ E. It is easy to see that the directed graph ~ G := (V; ~ E) does not contain cycles. This is because, for every cycleC2 ~ E, one can construct a dierent ow ~ f 0 := ~ f 1 C min j2C ~ f j for ~ E, and hence the corresponding owf 0 for the original graphG. This construction of ~ f 0 implies thatjf 0 jjfj, with the inequality being strict on at least one component, and hencef 0 T W 1 f 0 <f T W 1 f, and that f 0 also satises ow conservation, i.e., it is a feasible point for (2.8). Theorem 1 then leads to a contradiction that f is the solution to (2.9). Since ~ G does not contain cycles, every path in ~ G containing i2 ~ E is a supply-demand path. Therefore, for all i2 ~ E, ~ f i is no greater than the sum of supply/demand associated with paths 23 containing i, which in turn is no greater than the sum of total supply/demand in the network, i.e.,kpk 1 =2. Moreover, let = L y (w)p2 R E be a phase angle associated with f(w;p). Without loss of generality, assume f i (w;p) =kpk 1 =2, then (i) > (i) . LetV 1 := v2Vj v (i) . We claim that theV 1 contains all the supply nodes and cutV 1 EnV 1 contains the single linki, that is,C V1 =fig. This is because, by denition ofV 1 , all nodes inV 1 have a bigger phase angle than nodes inEnV 1 , hence all links inC V1 must contain ow in the direction fromV 1 toEnV 1 . The claim then follows straightforwardly from the fact that f i (w;p) =kpk 1 =2. As for 2), we note that a node has a phase angle no smaller than all its neighboring nodes and strictly greater than at least one neighboring node only if it is a supply node. Since the network contains a single supply node, then among all nodes the supply node has the largest value of phase angle. Therefore, links connecting to the supply node carry nonzero ows sourcing from the supply node. The proof for links connecting to the demand node is similar. 24 Chapter 3 Maximal Flow Over DC Networks with Controllable Weights The goal of this chapter is to study the dependence of DC owf(w;p) on weightw. Towards this goal, we consider a DC network with xed balanced supply-demand vector p and exible weight w that can be adjusted within a given range. For convenience, we often drop the dependence of f(w;p) on p and write it as f(w). Furthermore, we consider symmetric capacity denoted by c, and we do not assume connectivity of networks unless stated otherwise. 3.1 Problem Formulation For a DC networkG = (V;E) with balanced supply-demand vector p2 R E and link capacity c2 R E >0 , the maximal DC ow problem is formulated as follows. max 2R;w2R E s:t: jf(w)jc w l ww u (3.1) 25 where w l 2 R E 0 and w u 2 R E >0 are given lower and upper bounds of w. For a given weight w2 R E >0 , (3.1) is a linear program and yields an analytical solution. Dene congestion of a link i2E to be i (w) :=jf i (w)j=c i and congestion function of networkG to be %(w) :=k(w)k 1 =kC 1 f(w)k 1 (3.2) Then it is straightforward to see that 1=%(w) is the optimal value of (3.1) for the given w. Therefore, (3.1) is equivalent to the following problem: min w l ww u %(w) (3.3) Solving (3.1)-(3.3) requires understanding of the properties of function f(w) and %(w). As illus- trated in Example 2, f(w) is non-convex and %(w) is neither convex nor dierentiable. Example 2. Consider the network shown in Fig. 3.1a. Nodes 1 and 6 are the only supply and demand nodes, respectively. The capacity is c = 1. Let w 1 = [70; 15; 10; 80; 80; 50; 10; 80] T , w 2 = [40; 40; 70; 1; 1; 70; 60; 50] T . Fig. 3.1b and Fig. 3.1c shows, respectively, the unit ow function f(w()) = f(w 1 + (1)w 2 ) for link (1; 2) and (1; 3), and the congestion function %(w()) = %(w 1 + (1)w 2 ), for all 2 [0; 1]. 1 2 3 4 5 6 e1 e2 e7 e3 e4 e8 e5 e6 (a) 0 0.5 1 θ 0.3 0.4 0.5 0.6 0.7 f(w(θ)) f1 f2 (b) 0 0.5 1 θ 0.5 0.6 0.7 0.8 0.9 1 ϱ(w(θ)) (c) Figure 3.1: Illustration of the unit ow and congestion function. (a) A graph G containing a single supply node 1 and a single demand node 6; (b) The unit ow function; (c) The congestion function 26 In general, it is dicult to solve a non-convex and non-dierentiable problem. However, adjusting ow distribution by changing weight to minimize the congestion of a DC network is similar to the following maximal ow problem for an undirected ow network: max 2R;f2R E s:t: Af =p; jfjc (3.4) Note that the standard maximal ow problem considers a single source sink pair and can be seen as a special case of (3.4). Moreover, it is straightforward to see that (3.4) is equivalent to the following least congested ow problem. min f2R E kC 1 fk 1 s:t: Af =p (3.5) It is straightforward to see that (3.5) provides a lower bound for (3.3). The formal investigation on the relationship between (3.3) and (3.5) is shown in next section. 3.2 Relationship between Maximal Flow over Weight Controlled DC Networks and over Flow Networks In this section, we show that (3.3) and (3.5) are equivalent under appropriate conditions. We begin by exploring the relationship between the feasible sets of (3.3) and that of (3.5). 27 3.2.1 Relationship between feasible ow sets Let us dene the feasible sets as follows: F 1 :=ff2 R E jAf =pg (3.6a) F 2 :=ff2 R E j9w2 [w l ;w u ]; 2 R V ; s.t. Af =p;f =wA T g (3.6b) whereF 1 is for (3.5), andF 2 is for (3.3). The following remark is straightforward. Remark 11. Since F 2 has additional constraints, it is straightforward to see that F 2 F 1 . Moreover, using the notations ofF 1 andF 2 , (3.5) and (3.3) can be written as min f2F1 kC 1 fk 1 and min f2F2 kC 1 fk 1 . For a ow networkG = (V;E), a cycleO is a subset ofE that forms a loop. O consists of forward link setO F and backward link setO B , where the forward links and backward links are the links along clockwise and counter-clockwise direction ofO, respectively [17]. A circulation is a ow f2 R E that satises Af = 0. It is straightforward to see that a ow f 0 2 R E contains a circulation if there exists a cycleO such thatf 0 i > 0 for alli2O F andf 0 i < 0 for alli2O B . Let F 0 :=ff2 R E jf does not contain a circulationg We then have the following relationship betweenF 0 ,F 1 andF 2 . Proposition 1. For a DC networkG = (V;E) with balanced supply-demand vector p2 R E and link weight bounds w l 2 R E 0 and w u 2 R E >0 , F 2 F 1 \F 0 (3.7) Moreover, 1. ifG is a tree, thenF 1 =F 2 28 2. if w l = 0, thenF 2 =F 1 \F 0 Proof. SinceF 2 F 1 , in order to prove (3.7), it is sucient to prove thatf2F 0 for allf2F 2 , i.e., , a feasible ow for a DC network does not contain a circulation. This is proven by contradiction as follows. For a owf2F 2 , suppose there exists a circulation on a cycleO. Applying Ohm's law on all the links inO, we get that f i =w i = (i) (i) for all i2O F , andf i =w i = (i) (i) for all i2O B . Taking summation over all links inC, we get that 0< X i2O F f i =w i X i2O B f i =w i = X i2O F (i) + X i2O B (i) X i2O F (i) X i2O B (i) = 0: where the inequality is due to the denition of circulation, and the last equality to zero is due to the denition of a cycle. This leads to a contradiction. In order to prove 1), it is sucient to prove thatF 1 F 2 , i.e., , f2F 2 for any f2F 1 for a tree network. Pick arbitrary f2F 1 and w2 [w l ;w u ]. It is sucient to show that the constraint f = wA T is satised for some 2 R V . Let A be the subvector and submatrix of and A respectively with the rst row removed. SinceG is a tree, A has independent columns, and A is full rank. Let := ( A T ) 1 w 1 f2 R jVj1 . It is then easy to see that f =wA T is satised for := [0 T ] T . In order to prove 2), it is sucient to prove thatF 1 \F 0 F 2 . Pick arbitraryf2F 1 \F 0 . To provef2F 2 is to show that there existw2 [w l ;w u ] and such that the constraintf =wA T is satised. We now construct suchw and as follows. Maintain the directions of links with positive ow and reverse the directions of links with negative ow. Sincef2F 0 , there is no directed cycle in the network with the new direction assigned. Hence, there exists a topological ordering of the nodes inV. Pick a strictly decreasing sequence ( 1 ;:::; jVj ), and assign it the nodes as per the topological ordering. Let ~ w i :=f i =( (i) (i) )> 0 for alli2E. Finally, choose the link weights as: w = ~ w, where = min i2E w u i =w i > 0. The following result follows immediately from Remark 11 and Proposition 1. 29 Corollary 1. Consider a networkG = (V;E) with balanced supply-demand vector p2 R E , link weight bounds w l 2 R E 0 and w u 2 R E >0 , and link capacities c2 R E >0 . Then, the optimal value of (3.1) is upper bounded by the optimal value of (3.4). Moreover, ifG is a tree or w l = 0, then (3.1) and (3.4) have equal optimal values. 3.2.2 Geometry of DC Flow And Least Congested Flow f 1 f 2 Af =p f( ^ w) f(w) f Figure 3.2: Illustration of DC ow and least congested ow Geometry can be useful for understanding the relationship between a least congested ow f 2F 1 for (3.5) and a feasible DC ow f(w)2F 2 for (3.3). F 1 is a special ane space in R E , whose associated hyperplanes have coecients A ij 2 f0; 1;1g. A least congested ow f , that (3.5) computes, is a point in F 1 that has the smallest innite norm weighted by C 1 . This is illustrated in Fig. 3.2, where the box with dashed boundary lines is the point set n f2 R E jkC 1 fk 1 =kC 1 f k 1 o . On the other hand, Remark 7 shows that, for a given weight w, the DC ow f(w) is the unique point inF 1 that has minimum l 2 norm weighted by W 1=2 . Let f w := W 1=2 f be the weighted ow and A :=AW 1=2 be the weighted node-link incidence matrix. For a given balanced supply-demand vector, the ow conservation equation (2.2) becomes Af w =p, and (2.8) becomes min Afw =p kf w k 2 . Therefore, in thef w -space, the weighted DC owW 1=2 f(w;p) is the orthogonal projection of the origin 0 onto the ane space n f w 2 R E j Af w =p o . Back in the f-space, the 30 DC ow can be interpreted in two equivalent ways, as illustrated in Fig. 3.2 for two dierent weights w and ^ w, where the ellipsoids are point sets n f2 R E jkW 1=2 fk 2 =kW 1=2 f(w)k 2 o and n f2 R E jk ^ W 1=2 fk 2 =k ^ W 1=2 f( ^ w)k 2 o . On one hand, the DC ows f(w) and f( ^ w) can be seen as the two points of tangency between the ane spaceF 1 and the two ellipsoids. On the other hand, f(w) and f( ^ w) can be seen as the oblique projections of 0 ontoF 1 in dierent directions. The directions depend onw and ^ w. With the latter perspective in mind, we introduce the oblique projection matrix in the next section. Therefore, (3.3) is to nd a weight in [w l ;w u ], or an associated projection direction, such that the projection of 0 onF 1 has the smallest innite norm weighted by C 1 . In particular, Corollary 1 implies that, for w l = 0, there exists a weight w and f 2F 1 such that f has both the smallest innite norm weighted by C 1 and the smallest l 2 norm weighted by (W ) 1=2 . 3.3 Oblique Projection Matrix of DC Network For a networkG = (V;E) with weight w2 R E , we dene the oblique projection matrix to be: K(w) :=WA T L y (w)A (3.8) We shall drop explicit dependence of K(w) on w when clear from the context. Consider an arbitrary vector f 0 2 R E . It is straightforward to see from the denition in (3.8) that K(w) converts f 0 into the unique DC ow f(w;Af 0 ). This is summarized as follows. Lemma 3. Consider a DC networkG = (V;E) with link weights w2 R E >0 . Let K(w) be as dened in (3.8). For an arbitrary vector f 0 2 R E , K(w)f 0 =f(w;Af 0 ). Remark 12. Lemma 3 implies that for any f2F 1 and w2 [w l ;w u ], K(w)f2F 2 . Remark 7 implies that f = K(w)f 0 is the projection of the origin 0 onto the hyperplane n f2 R E jAf =Af 0 o and among all points in the hyperplane n f2 R E jAf =Af 0 o , f has 31 the minimum weighted l 2 norm. This is illustrated in Fig 3.3, where the dashed lines denote the projection direction. f 1 f 2 Af =Af 0 f =Kf 0 f 0 Ax = 0 (IK)f 0 Figure 3.3: Illustration of oblique projection Furthermore, (IK(w))f 0 is the optimal solution to the following problem, which is obtained from (2.8) by changing variable x :=f 0 f. min x2R E kW 1=2 (xf 0 )k 2 s.t. Ax = 0 This further implies that, as shown in Fig 3.3, (IK(w))f 0 is the projection of f 0 onto the hyperplane n x2 R E jAx = 0 o . Among all points in the hyperplane n x2 R E jAx = 0 o , (I K(w))f 0 is the closest one to f 0 in terms of weighted l 2 norm. Remark 13. Both K(w) and IK(w) are oblique projection matrices and play important roles in results provided in the following sections. A similar notion of oblique projection, called cutset projection, is dened asA T L y (w)AW in [57]. It is straightforward to see that the cutset projection matrix is similar to K(w). The components of the oblique projection matrix K2 R EE of a networkG = (V;E) have the following physical interpretation. K ij is the DC ow on link i when a unit supply and a unit demand are imposed on the tail and head nodes, respectively, of link j, i.e., when the supply-demand vector isA j . We further dene the normalized oblique projection matrixK sym := 32 W 1=2 A T L y A T W 1=2 = W 1=2 KW 1=2 . Note that K is similar to K sym . Let A = AW 1=2 be the weighted node-link incidence matrix. Then K sym = A T L y A = A T ( A A T ) y A is the orthogonal projection matrix onto the row space of A [53]. Hence, it only has eigenvalues of 1 or 0. The next set of results on the oblique projection matrix follow straightforwardly either from its denition or from Lemma 2. Lemma 4. For a DC networkG = (V;E) with link weights w2 R E >0 , the following is true for the oblique projection matrix K as dened in (3.8): 1. The only eigenvalues of K are 0 and 1; 2. Kf = 0 if and only if f = 0 or f 2 R E is a circulation; K is singular if and only ifG contains at least one cycle; 3. Kf = f if and only if f2 R E is a DC ow overG, that is, if f satises (2.9), for some supply-demand vector p2 R V ; 4. For all i;j2E, i6= j, iffig is a cut ofG, then K ii = 1 and K ij = 0; otherwise, then 0<K ii < 1 andjK ij j 1K ii . Remark 14. 1. Lemma 4 implies that the range ofK is the set of DC ows under all possible supply-demand vectors, and the null space of K is set of circulation ows. 2. LetCE be a cut ofG and let 1 C 2f1; 0g E have ones on components associated with links inC and zeros elsewhere. Since W1 C is a valid DC ow that satises (2.9) with v = 0 for all nodes v on one side of cutC and v 0 = 1 for all nodes v 0 on the other side of cutC, the third item of Lemma 4 implies that KW1 C = 1 C . Lemma 3 and 4 imply that K(w) is capable of removing the circulation from any arbitrary ow f2F 1 and renders the unique DC ow f(w). The next result is on the principal submatrices of K. 33 Lemma 5. Consider a networkG = (V;E) with link weights w2 R E >0 . Then, for every link set IE, 2 (K II ) = max x2R I kx 0 I k 2 2 kx 0 I k 2 2 + 2kx 0 I k 2 2 +kxx 0 I k 2 2 1 (3.9) where (K II ) is the spectral radius of K II , x 0 := K sym I x and I :=EnI. Moreover, (3.9) holds true with equality if and only ifI contains a cut ofG. Proof. Since K sym II = W 1=2 II K II W 1=2 II is similar to K II , it is sucient to focus on proving the results for K sym II . It is straightforward that K sym II = A T I L y A I is positive semi-denite, where A I is theEI submatrix of A. For arbitrary x2 R I , let ^ x = [x T 0 T ] T 2 R E . Then,kxk 2 =k^ xk 2 , and x 0 =K sym I x =K sym ^ x. Since K sym = A T ( A A T ) y A is an orthogonal projection matrix, k^ xk 2 2 =kK sym ^ xk 2 2 +k^ xK sym ^ xk 2 2 =kx 0 k 2 2 +k^ xx 0 k 2 2 =kx 0 I k 2 2 +kx 0 I k 2 2 +kxx 0 I k 2 2 +kx 0 I k 2 2 =kx 0 I k 2 2 + 2kx 0 I k 2 2 +kxx 0 I k 2 2 (3.10) Since K sym II is symmetric, (K sym II ) = max x2R I kK sym II xk 2 kxk 2 = max x2R I kx 0 I k 2 k^ xk 2 Substitutingk^ xk 2 2 from (3.10) gives (3.9). (3.9) holds true with equality if and only ifkx 0 I k 2 =k^ xk 2 . According to (3.10), it implies that x 0 I = 0 andx =x 0 I . Furthermore, it is equivalent to the existence of some nonzero vector y2 R E with y I = 0 such that Ky = y. In other words, if one imposes y i unit supply and demand on the tail and head nodes of link i for all i2I, the resulting ow on link i must be y i for all i2I and 0 for all i2 I. The suciency of the conditionI containing a cut ofG follows immediately from Remark 14. The necessity of the condition is proved by contradiction. Suppose there exists a satisfyingy andI does not contain a cut ofG, it is possible to pick an arbitrary link i2I with 34 y i 6= 0. SinceI does not contain a cut, there exist a path inEnI connecting the end nodes of i. By assumption, the path conveys zero ow. Applying \Ohm's law" on all its links, the end nodes of i must have the same potential. However, this contradicts with the assumption y i 6= 0. Remark 15. IfG is connected, Lemma 5 implies that (K II ) = 1 if and only if the network (V;EnI) is disconnected. Lemma 3, Lemma 4 and Lemma 5 demonstrate that the oblique projection matrixK contains rich information on the corresponding DC network. In next section, the above properties ofK are used to study ow variation due to innitesimal weight changes and for derivation of optimality conditions for (3.3). 3.4 Flow-weight Jacobian and Optimality Conditions The ow variation due to innitesimal changes in weight is studied through the ow-weight Jaco- bian matrix, which we discuss rst. Then, the directional derivative of %(w) is derived based on the ow-weight Jacobian, and helps to formulate the optimality conditions for (3.3). 3.4.1 The Flow-weight Jacobian Let J(w) := h @f(w) @w i 2 R EE be the ow-weight Jacobian for function f(w). We provide an explicit expression for J(w) in the next result, whose proof depends on [47, Theorem 4.3]. For the sake of completeness, we reproduce this result from [47] and also provide a concise proof in Appendix A.2. Proposition 2. Consider a networkG = (V;E) with balanced supply-demand vector p2 R V . Let f(w) :=f(w;p), where f(w;p) is dened as in (2.10). Function f(w) is Fr echet dierentiable at all w2 R E >0 and the ow-weight Jacobian is given by: J(w) = (IK(w))W 1 diag(f(w)) (3.11) 35 where K(w) is dened as in (3.8). Proof. The Laplacian L(w) =AWA T is Fr echet dierentiable [12] and has constant rank for all w2 R E >0 , Theorem 7 then implies that L y (w) is Fr echet dierentiable. It is then straightforward to see thatf(w) is Fr echet dierentiable. We then focus on proving (3.11). The partial derivative of L(w) with respect to w i is given by @L @w i = @(AWA T ) @w i =A i A T i (3.12) where A i is the i-th column of matrix A. Since L(w) is a Laplacian, it has a constant rank =jVj 1 for all w2 R E >0 . Therefore, Theorem 7 in the Appendix implies that the derivative of L y is given by: @L y @w i =L y @L @w i L y +L y L y T@L T @w i (ILL y ) + (IL y L) @L T @w i L y T L y (3.13) In order to simplify (3.13), using singular value decomposition, one can writeLL y =L y L =UU T , whereU is an(n1) orthogonal matrix, whose columns are all orthogonal to 1, wheren =jVj. Therefore,ILL y andIL y L are both projection matrices onto 1. That is,ILL y = 1 nn =n = IL y L, where 1 nn is a matrix all of whose entries are one. Therefore, using (3.12), and noting that A T i 1 = 0, @L T @w i (ILL y ) =A i A T i 1 nn n = 0 = (IL y L) @L T @w i (3.14) Substituting (3.12) and (3.14) in (3.13), we get that @L y @w i =L y @L @w i L y =L y A i A T i L y (3.15) 36 Therefore, the i-th column of the Jacobian is: J i (w) = @f(w) @w i = @W @w i A T L y p +WA T @L y @w i p =A T i L y p 1 i WA T L y A i A T i L y p (3.16) where 1 i is the vector whosei-th component is equal to one, and all other entries are zero. When written in matrix form and substituting A T i L y p =f i =w i , (3.16) gives (3.11). Remark 16. The expression for thei-th column of Jacobian, as given in (3.16), has the following useful interpretation. J i (w) = f i (w) w i (1 i K i (w)) = f i (w) w i (1 i f(w;A i )) (3.17) Recall that the entries of the column J i (w) give the sensitivities of ows on various links with respect to change in weight on link i. The rst term on the right hand side of (3.17) is non-zero only when computing sensitivity of ow on linki with respect to changes inw i , and hence is local in nature. The non-locality in the sensitivity comes from the second term, which is equal to the ow distribution in the network under supply-demand vector A i . Computing sensitivity of ows with respect to weights, via (3.11), requires considerable com- putation, especially for large networks. However, the ow changes in predictable patterns for weight changes in several directions related to the cuts and paths of the network, as stated in the next result. Proposition 3. For a connected DC networkG = (V;E) with link weights w2 R E >0 and balanced supply-demand vector p2 R E , the following are true about the ow-weight Jacobian J(w) given in (3.11): 1. J(w)w = 0. 37 2. LetCE be a cut ofG. If f i (w)6= 0 for all i2C, then J(w)d = 0, where d i =w 2 i =f i (w) for all i2C and d i = 0 elsewhere. 3. ConsiderG to be simple and contain single supply-demand pair. LetP be a simple path from the supply node to the demand node. Without loss of generality, assume that all the links inP have the same direction asP. Then, there exists a direction d2 R E such that sgn [J(w)d] i = sgnf i for all i2EnP and [J(w)d] i 0 for all i2P. Moreover, if f i (w)6= 0 for all i2P, one such d is d i =w i =f i (w) for all i2P and d i = 0 elsewhere. Proof. Firstly, 1) and 2) follow immediately from Remark 14 as follows. Substituting (3.11), J(w)w = (IK(w))f(w) = 0 J(w)d = (IK(w))W 1 diag(f(w))d = (IK(w))W1 C = 0 where 1 C is equal to 1 for all components associated with links inC and 0 elsewhere. As for 3), we rst consider the case thatf i (w)6= 0 for alli2P. In this case, letd i :=w i =f i (w) for all i2P. Substituting (3.11), we get J(w)d = (IK(w))1 P = 1 P f(w; 2p=kpk 1 ) = 1 P 2f(w)=kpk 1 , where the second equality follows from the fact thatP is a simple path from the supply to the demand and A1 P = 2p=kpk 1 and the third equality follows from the linearity of f(w;p) with respect to p. It is then straightforward that [J(w)d] i =2f i (w)=kpk 2 and hence sgn [J(w)d] i = sgnf i (w) for all i2EnP. The assertion that [J(w)d] i 0 for all i2P follows from Lemma 2. In order to prove that the nonzero ow condition can be removed for some direction, we need to nd a direction d2 R E with d i = 0 wherever f i (w) = 0 such that J(w)d has the properties in 3). This is achieved with the help of 2). Let d 0 2 R E be such that d 0 i := w i =f i (w) for i2 fj2Pjf j (w)6= 0g andd 0 i := 0 elsewhere. Let :=L y (w)p be an associated phase angle with ow f(w) and letE 0 :=fi2Pjf i (w) = 0g. For all k2E 0 , letV k := v2Vnf(k)gj v (k) ; since (k) = (k) , (k)2V k Vnf(k)g and k2C V k , where we recall thatC V k denotes the 38 cutV k EnV k . Denition ofE 0 implies that a single link k2E 0 can not form a cut. Denition ofV k , fork2E 0 , implies that (j) 6= (j) andf j (w)6= 0 for allj2C V k nfkg. It is then valid to dened k 2 R E to be such thatd k j :=w 2 j = (f j (w)w k ) for allj2C V k nfkg andd k j := 0 elsewhere. Let d :=d 0 + P k2E0 d k . By denition, it is clear that d i = 0 wherever f i = 0. It remains to be shown that J(w)d satises the properties in 3). Let k := W 1 diag(f(w))d k for all k2E 0 [f0g and := P k2E0[f0g k . As d k = 0 for all k2E 0 , k = 0 can be written as k = 1 1. By combining the rst term into 0 and the second term into k , we obtain the following. = 0 + X k2E0 1 k ! + X k2E0 k w k w k = 1 P X k2E0 1 w k W1 C V k The assertion then follows from the fact that (IK(w))W1 C V k = 0 for all k2E 0 . Remark 17. 1. The rst item of Proposition 3 agrees with Remark 10 on the homogeneity of f(w;p) with respect to w. 2. If the gradient component associated with link i2E has opposite sign of f i 6= 0, then the magnitude of ow on link i, i.e.,jf i j would not increase along the gradient direction. Therefore, Proposition 3 implies that there exists a direction of weight change such that the magnitude of ow would not increase for all links except for the links in a simple path from the supply node to the demand node. 3. Extension of the third item of Proposition 3 to multi-graphs is shown in Proposition 25 in Appendix A.4. Furthermore, the following result provides a weight direction that brings the DC ow closer to a given ow f2F 1 . 39 Proposition 4. For a DC network with simple connected graphG = (V;E), link weightsw2 R E >0 and balanced supply-demand vector p2 R E , letF 1 be as dened in (3.6a). For all f 0 2F 1 , there exists a direction d2 R E such that J(w)d = f 0 f(w). Moreover, if f i (w)6= 0 for all i2E 0 := j2Ejf 0 j 6= 0 , one such d is d i = f 0 i w i =f i (w) for all i2E 0 and d i = 0 for all i2EnE 0 . Proof. We shall only consider the case when f i (w)6= 0 for all i2E 0 . If this condition is not satised, then a satisfying d can be found in the same way as in the proof of Proposition 3. The following is straightforward. J(w)d = (IK(w))f 0 =f 0 f(w) where the second equality follows from Lemma 3 and the fact that f 0 2F 1 . Remark 18. 1. Proposition 4 implies that the dierence betweenf 0 andf(w), i.e.,jf(w)f 0 j would decrease on every link i2 j2Ejf j (w)6=f 0 j in some direction d2 R E of weight change. 2. The results in Proposition 4 can be extended to networks with multigraphs, in the same way as Proposition 3 is extended to Proposition 25. Moreover, some entries of the Jacobian in (3.11) exhibit sign-deniteness, as illustrated in Fig. 3.4, and stated in Proposition 5. Proposition 5. For a network with directed multigraph G = (V;E), weights w 2 R E >0 , and balanced supply-demand vector p2 R V , the ow-weight Jacobian in (3.11) satises the following for all i2E: sign(J ki (w))2 sign(f i )[f0g for all k2fig[E (i) [E + (i) and sign(J ki (w))2 sign(f i )[f0g for all k2fE + (i) [E (i) gnfig. 40 1 2 3 4 k1 ! k2 i " k3 k4 ! fi fi Figure 3.4: Illustration of signs of@f=@w i : red arrows alongside each link de- note the ow direction on the corresponding link under the supply-demand vec- torf i A i ; for every link6=i, if the red arrow alongside a link aligns with the link direction, then the corresponding component of @f=@w i is negative, and posi- tive otherwise. Correspondingly,@f k1 =@w i > 0,@f k4 =@w i < 0,@f k2 =@w i > 0, @f k3 =@w i > 0. We always have @f i =@w i > 0. Proof. We provide proof for the case when f i > 0; the case when f i 0 follows along similar lines. (3.17) implies that J ii w i =f i f i w i A T i L y A i ; J ki w i =f i w k A T k L y A i (3.18) for all k6= i characterized in the lemma. The proof then follows directly from Lemma 2 and Lemma 4. Remark 19. Proposition 5 can be interpreted as generalization of existing results, e.g., see [63], that study the eect of removal of a link on ows in neighboring links. 3.4.2 The Directional Derivative of Congestion Function We rst recall the denition of directional derivative, as well as several relevant results. Let : D R n ! R m be a vector-valued function dened on an open subsetD of R n . The directional derivative of at a point x2D along the direction d2 R n , if it exists, is the limit 0 (x;d) := lim #0 (x +d) (x) 41 If 0 (x;d) exists for alld2 R d , then is said to be directionally dierentiable atx. For a nite set of functions i (x) : R n ! R directionally dierentiable at x2 R n for all i2 [q], the directional derivative of the function (x) := max i2[q] i (x) is given as follows: 0 (x;d) = max i2S 0 i (x;d); 8d2 R n (3.19) whereS =fi2 [q]j i (x) = (x)g. Furthermore, : D R n ! R m is said to be B(ouligand)-dierentiable at x2D if is Lipschitz continuous near x 1 and directionally dierentiable at x. If : R n ! R m and : R m ! R q are B-dierentiable at x2 R n and (x)2 R m , respectively, then the composition map := : R n ! R q is B-dierentiable at x and the following rule holds [40]. 0 (x;d) = 0 ((x); 0 (x;d)); 8d2 R n (3.20) Proposition 6. For a connected DC networkG = (V;E) with link capacitiesc2 R E >0 and supply- demand vector p 2 R V , the directional derivative of the congestion function at w 2 R E >0 in direction d2 R E is given by: % 0 (w;d) =%(w) max i2S(w) [(IK(w))] i f i (w) (3.21) whereS(w) :=fi2Ej i (w) =%(w)g, i =f i (w)d i =w i . 1 is Lipschitz continuous near x if there exists a constant > 0 and a neighborhoodN of x such that k(y) (z)kkyzk for all y;z2N . 42 Proof. Let (f) :=C 1 jfj, then is B-dierentiable and 0 i (f i ;4f i ) =4f i sgn(f i )=c i for f i 6= 0 and i2E. = f. Since f(w) is Frechet-dierentiable and f 0 (w;d) =J(w)d, using the chain rule in (3.20), we obtain for every i2E and w such that f i (w)6= 0, 0 i (w;d) = 0 i (f i (w);f 0 i (w;d)) = 0 i (f(w);J(w)d) = sgn(f i (w)) c i [J(w)d] i By denition, %(w) = max i2E i (w). % 0 (w;d) = max i2S(w) 0 i (w;d) = max i2S(w) sgnf i (w) c i [J(w)d] i = max i2S(w) %(w) f i (w) [(IK)W 1 diag(f(w))d] i =%(w) max i2S(w) [(IK(w))] i f i (w) where the rst equality is due to (3.19); the second equality is straightforward substitution for f i (w)6= 0 for all i2S(w); the third is due to that %(w) = i (w) = f i sgnf i =c i for all i2S(w); and the fourth equality is obtained by substituting =W 1 diag(f)d. Remark 20. For a weight w 2 R E >0 and direction d 2 R E , if % 0 (w;d) < 0, then the vector (IK(w)) has dierent signs from f(w) on all the components in the index setS(w). Note that f i (w)6= 0 for all i2S(w) and w2 R E >0 . 3.4.3 Optimality Conditions We study optimality conditions for (3.3) under the special case where w l = 0. The homogeneity of functionf(w) (cf. Remark 10) implies homogeneity of%(w), that is,%(w) =%(w) for all> 0. As a result, for anyw> 0 andw u > 0, there exists ^ w2 [0;w u ] such that%(w) =%( ^ w). Therefore, for w l = 0 and arbitrary w u > 0, (3.3) is equivalent to the problem min w0 %(w). 43 Theorem 2. For a connected DC networkG = (V;E) with balanced supply-demand vectorp2 R E and capacity c2 R E >0 , w 2 R E >0 is a global minimizer of %(w) if and only if it satises the following: % 0 (w ;d) 0 8d2 R E (3.22) where %(w) is as dened in (3.2). Proof. We prove the contrapositive. Letw not be a global minimizer. Then, there exists ^ w6=w such that %( ^ w)<%(w ). This implies the following. jf j ( ^ w)j c j max i2E jf i ( ^ w)j c i < max i2E jf i (w )j c i = jf j (w )j c j 8j2S(w ); which is to say,jf j ( ^ w)j <jf j (w )j for all j2S(w ). As f( ^ w)2F 1 , Proposition 4 shows that there exists a directiond2 R E such thatJ(w )d =f( ^ w)f(w ). For suchd and allj2S(w ), if f j (w )> 0, thenf j ( ^ w)<f j (w ) and [J(w)d] i =f i (w ) = (f j ( ^ w)f j (w ))=f j (w )< 0; iff j (w )< 0, then f j ( ^ w) > f j (w ) and [J(w)d] i =f i (w ) = (f j ( ^ w)f j (w ))=f j (w ) < 0. Combining these two gives that % 0 (w ;d)< 0 for some d2 R E . We make the following remarks on the implication of Theorem 2. Remark 21. 1. Theorem 2 implies that every local minimum of function %(w) is a global minimum. 2. IfG contains a single supply demand pair, Theorem 2 can also be proved using Proposition 3 as follows: if w is not a global minimizer, there exists an under capacitated path from the supply node to the demand node such that changing weights appropriately on links in the path could lead to decreases in magnitude of ows on all over capacitated links. Changing weights of an under capacitated path is equivalent to \push" ow to an augmented path in the well known Ford-Fulkerson algorithm [41,59]. 44 3.4.4 A Descent Algorithm Inspired by Theorem 2, we have the following descent algorithm for (3.3), despite its non- convexity and non-dierentiability. Let w(t) denote the weight vector at iteration step t. If min kdk21 % 0 (w(t);d) 0, then w(t) is an optimal solution; otherwise, update weight as follows: w(t + 1) =w(t) + t d(t) (3.23) where t > 0 is the step-size and d(t)2 argmin kdk21 % 0 (w(t);d). Additionally, if w l 6= 0, at each step of weight update, w(t + 1) needs to be projected, if necessary, onto the feasible set [w l w u ]. For simplicity, we substitute (3.21) and let i (t) :=d i (t)=w i (t). Along the lines of [16, Section 2.1.2], we obtain the following projected descent algorithm equivalent to (3.23): w(t + 1) =w(t) +w(t)(t) (3.24) where (t) is the solution to the following strictly convex quadratic program for given w(t) at iteration step t: min 2R E %(w(t)) max i2S(w(t)) 0 @ i X j2E K ij (w(t)) f j (w(t)) f i (w(t)) j 1 A + 1 2 t X j2E w 2 j (t) 2 j s:t: w l j (1 + j )w j (t)w u j 8j2E Note t > 0 is the same step-size used in (3.23) and w(t), %(w(t)), K(w(t)) and f(w(t)) are considered as constants for the given w(t) at iteration step t. (3.24) gives an unweighted version of projected gradient iteration { it can be generalized by incorporating an appropriate positive denite weighting matrix into the regularization term, e.g., see [16, Section 2.1.2]. Convergence questions of the above descent algorithm used in non-dierentiable functions are delicate and should not be treated lightly, even in the case of convex problems [16], let alone for 45 the case of non-convex problems. A rigorous convergence analysis of the above algorithm, as well as other potential algorithms, is still an ongoing work. Towards this goal, the results in the next section are useful. 3.5 Flow Redistribution Under Weight Perturbations The ow-weight Jacobian matrix in (3.11) describes howf(w) changes due to innitesimal changes in weight. This section provides exact formula that relatesf(w+4w) tof(w) for non-innitesimal weight perturbation4w. In order to discuss this, we rst introduce the multigraph perspective for weight perturbations. 3.5.1 A Multigraph Perspective for Weight Perturbations We already know from Remark 9 that removing a link is equivalent to setting its weight to zero. We use this equivalence to establish the connection between arbitrary weight perturbations and topology changes of the network. For this purpose, we need to introduce the notion of reduced simple graph of a multigraph. Recall that a undirected multi-graph is a graph that allows multiple edges connecting the same incident nodes. Denition 7 (Reduced Simple Graph). Given a multigraphG = (V;E), the corresponding re- duced simple graph is denoted asG s = (V s ;E s ), whereV s =V, andE s E is constructed as follows. For every node pairfv 1 ;v 2 g2VV, for all the links fromv 1 tov 2 inE, there exists only one link from v 1 to v 2 inE s ; if there is no link from v 1 to v 2 inE, then there is no link from v 1 to v 2 inE s . For every i2E s , letM i be the corresponding links inE. The weight matrix forG s , denoted as W s 2 R E s E s >0 , is dened as w s i := P j2Mi w j for all i2E s . Let A G s denote the node-link incidence matrix ofG s . The next result states thatG andG s have the same weighted Laplacian matrices and DC ows. 46 Lemma 6. Consider a DC network with multigraphG = (V;E), weight w2 R E >0 and its simple graphG s = (V;E s ) with weight w s 2 R E s . Let L G and L G s be the weighted Laplacian matrices, respectively, and f G 2 R E and f G s 2 R E s be the DC ows, respectively, under the same supply- demand vector, then 1. L G =L G s; 2. f G k =f G s i w k = P j2Mi w j for all i2E s and k2M i . Proof. 1) follows from the Denition of Laplacian matrix in Denition 1. L G s =A Gs W s A T Gs = X i2E s A s i w s i A s i T = X i2E s A s i ( X j2Mi w j )A s i T = X j2E A j w j A T j =L G where A s i is the i-th column of A G s, A j is the j-th column of A, w s i is the i-th diagonal element of w s , and the fourth equality is due to the fact that A j =A s i for all j2M i , i2E s . In order to prove 2), let p2 R V be the common supply-demand vector onG andG s , and G := L y G p and G s := L y G sp be the corresponding solutions for potentials satisfying (2.9). 1) implies that G = G s . 2) then follows from the second equation of (2.9) and the denition of w s i in Denition 7. Lemma 6 indicates that a Laplacian matrix is associated with a class of graphs which have identical reduced simple graph. It also implies that a single link in a network can be split into multiple links, each with fractions of weight, without aecting the ow distribution over the network. Based on this, weight perturbations can be seen as topology changes. Consider the networkG = (V;E) in Fig. 3.5 and consider the weight perturbation from w to w +4w, where ~ G denotes the network after the weight perturbation. For a link whose weight decreases, e.g., 4w 1 < 0, we split it into two links inG, of weight w 1 +4w 1 and4w 1 , respectively. Weight decrease on link 1 by4w i amounts to removing the link of weight4w i in ~ G, as indicated using dashed link in Fig. 3.5b. If the weight decrease of a link is equal to the weight of the link, e.g., 47 4w 4 =w 4 , then the link is considered to be completely removed in ~ G. Similarly, weight increase of a link i, e.g.,4w 3 > 0, is equivalent to adding a parallel link of weight4w i onto the same incident nodes. For links whose weights remain the same, no changes are to be made on them in ~ G. G w1 +4w1 4w1 w4 w2 w3 (a) ~ G w1 +4w1 w2 w3 4w3 (b) Figure 3.5: A multigraph perspective for weight perturbations. (a) the initial networkG has weight w; (b) the network ~ G post-perturbations, where the dashed links are links to be removed, the change of weight is4w:4w 1 < 0, 4w 2 = 0,4w 3 > 0 and4w 4 =w 4 . With the equivalence of weight decrease and link removal established, the following is a straightforward result from Lemma 5 and Remark 15. Lemma 7. Consider a connected network G = (V;E) with node-link incidence matrix A 2 f0; 1;1g VE and link weights w2 R E >0 . For an arbitrary weight decrease4w2 [0;w], ifG remains to be connected with weight w4w, then the matrix I4W II A T I L y (w)A I is nonsin- gular, whereI :=fi2Ej4w i 6= 0g. 3.5.2 Pseudo-inverse of Laplacian Matrix Under Weight Perturbations The next result establishes the relationship between L y (w +4w) and L y (w) for perturbation 4ww. 48 Theorem 3. Consider a connected networkG = (V;E) with node-link incidence matrix A2 f0; 1;1g VE and link weights w2 R E >0 . For an arbitrary weight perturbation4ww, ifG remains to be connected with weight w +4w, then L y (w +4w) =L y (w) I +L(4w)L y (w) 1 = I +L y (w)L(4w) 1 L y (w) (3.25a) =L y (w)L y (w)A I I +4W II A T I L y (w)A I 1 4W II A T I L y (w) (3.25b) where L(4w) :=A4WA T andI :=fi2Ej4w i 6= 0g. Proof. Let4w + i := maxf4w i ; 0g and4w i := minf4w i ; 0g for all i2E, then4w =4w + 4w . LetI 0 := i2Ej4w i 6= 0 . We show (3.25) by applying Theorem 9 twice. The rst time gives L y (w +4w + ) and the second time gives L y (w +4w). Firstly, since matrix (4W + ) 1=2 AL y (w)A T (4W + ) 1=2 is positive semi-denite, matrix I + (4W + ) 1=2 AL y (w)A T (4W + ) 1=2 is nonsingular. Theorem 8 then implies that I +L(4w + )L y (w) is nonsingular. Moreover, L(w)L y (w)A =A. Theorem 9 then implies the following. L y (w +4w + ) = (L(w) +L(4w)) y =L y (w) I +L(4w + )L y (w) 1 (3.26) We then consider that networkG decreases its weight from w +4w + to w +4w + 4w = w+4w. SinceG is connected with both weightw+4w + and weightw+4w, Lemma 7 implies that matrixI4W I 0 I 0A T I 0L y (w +4w + )A I 0 is nonsingular, which further leads toIL(4w )L y (w + 49 4w + ) being nonsingular, with the help of Theorem 8. We also haveL(w +4w)L y (w +4)A =A. Therefore, applying Theorem 9 for the second time, we obtain: L y (w +4w) = (L(w +4w + )L(4w )) y =L y (w +4w + ) IL(4w )L y (w +4w + ) 1 =L y (w) I +L(4w + )L y (w) 1 IL(4w )L y (w +4w + ) 1 =L y (w) (IL(4w )L y (w +4w + ))(I +L(4w + )L y (w)) 1 =L y (w) I +L(4w + )L y (w)L(4w )L y (w) 1 =L y (w) I +L(4w)L y (w) 1 where the third and forth equalities follow by substituting L y (w +4w + ) from (3.26) and the rst and last equalities follow from the fact that L(w +4w) =L(w +4w + )L(4w ). The second equality in (3.25a), as well as (3.25b), can be established by applying Theorem 8 and noticing that L(4w) =A I 4W II A T I . Remark 22. Though being equivalent, (3.25b) is computationally advantageous over (3.25a) to the extent thatI is a proper subset ofE, i.e.,jIj<E, in which case, I +4W II A T I L y (w)A I 1 in (3.25b) has a smaller size and will be more readily computable than I +L(4w)L y (w) 1 and I +L y (w)L(4w) 1 in (3.25a). When restricted to the special case of decreasing weights to zero in Theorem 3, we obtain the following result on change in pseudo-inverse of Laplacian matrix under link removal. This result has also appeared in [7] and [82]. Corollary 2. Consider a connected networkG = (V;E) with link weights w2 R E >0 . Let ^ G = (V;EnI) be the residual network obtained by removing links inIE. Let L and ^ L denote the weighted Laplacian matrices forG and ^ G, respectively. If ^ G is connected, then: ^ L y =L y +L y A I (IK II ) 1 W II A T I L y (3.27) 50 where recall that K II =W II A T I L y A I 2 R II . A special case of Corollary 2 is forjIj = 1. As shown in Corollary 3, in this case, computing the pseudo-inverse of Laplacian matrix of the residual network does not involve computing matrix inversion. Corollary 3 appeared previously in [3], where it was proven using a standard rank one perturbation method [70]. Corollary 3. Consider a connected networkG = (V;E) with link weights w2 R E >0 . Let ^ G := (V;Enfjg) be the residual network ofG obtained by removing link j2E. Let L and ^ L denote the weighted Laplacian matrices forG and ^ G, respectively. If ^ G is connected, then: ^ L y =L y + 1 1K jj L y A j w j A T j L y (3.28) Remark 23. It is easy to see from (3.28) that the condition on connectivity of ^ G is necessary. As shown in Lemma 4, if the removal of a link j2E disconnectsG, thenK jj = 1 and the expression on the right hand side of (3.28) is not valid. The next result addresses the case when the removal of a single linkj disconnectsG. Repeated application of this result and Theorem 3 allows us to consider arbitrary weight perturbations. Proposition 7. Consider a connected networkG = (V;E) with link weights w 2 R E >0 . Let ^ G := (V;Enfjg) be a residual network ofG obtained by removing link j2E. Let L and ^ L denote the weighted Laplacian matrices forG and ^ G, respectively. If ^ G is disconnected, then ^ L y =L y hh T L y L y hh T + (h T L y h)hh T (3.29) where h :=L y A j =kL y A j k 2 . Proof. The proposition follows directly from [70, Theorem 6]. One only needs to show that A j is in the range space of L, which is straightforward. 51 Remark 24. In Proposition 7, if the nodes of the two connected components are labeled in a way such that [n 1 ] andfn 1 + 1;:::;n 1 +n 2 g denote the two node sets, then h =L y A j =kL y A j k 2 = 1 p n 1 n 2 (n 1 +n 2 ) 2 6 6 4 n 2 1 n1 n 1 1 n2 3 7 7 5 and furthermore, ^ L y is a block diagonal matrix in which the diagonal elements are two matrices of size n 1 n 1 and n 2 n 2 . 3.5.3 Flow Redistribution Under Weight Perturbations We use Theorem 3 to study the ow redistribution under weight perturbations. Specically, we obtain a formula relatingf(w +4w) tof(w). In order to computef(w +4w), Lemma 3 implies that it is sucient to focus on K(w +4w). For this purpose, we dene K(w;w 0 ) for a network G = (V;E) and two weights w;w 0 2 R E >0 as follows: K(w;w 0 ) :=W 0 A T L y (w)A (3.30) By this denition, K(w)K(w;w). Lemma 8. Consider a connected DC networkG = (V;E) with link weights w2 R E >0 and weight perturbation4ww. IfG is connected with weight w +4w, then the following are true. K(w +4w) =K(w;w +4w) (I +K(w;4w)) 1 (3.31a) =K(w;w +4w)K EI (w;w +4w)(I +K II (w;4w)) 1 K IE (w;4w) (3.31b) = I (IK(w))4W (W +4W ) 1 1 K(w) (3.31c) K I I (w +4w) =K I I (w) +K II (w) (IK II (w)) 1 K I I (w) (3.31d) 52 whereI :=fi2Ej4w i 6= 0g, I :=EnI, K(w;w +4w) and K II (w;4w) are as dened in (3.30), (3.31c) holds only for4w>w and (3.31d) holds only for4w I =w I . Proof. (3.31a) is proved using (3.25a). K(w +4w) = (W +4W )A T L y (w +4w)A = (W +4W )A T L y (w) I +A4WA T L y (w) 1 A = (W +4W )A T L y (w)A I +4WA T L y (w)A 1 =K(w;w +4w) (I +K(w;4w)) 1 where the third quality uses (A.6). (3.31b) is proved using (3.25b). K(w +4w) = (W +4W )A T L y (w)A (W +4W )A T L y (w)A I I +4W II A T I L y (w)A I 1 4W II A T I L y (w)A =K(w;w +4w)K EI (w;w +4w)(I +K II (w;4w)) 1 K IE (w;4w) If4w>w, then w +4w> 0, (3.31c) is proved as follows. K(w +4w) = (W +4W )A T I +L y (w)A4WA T 1 L y (w)A = (W +4W ) I +A T L y (w)A4W 1 A T L y (w)A = W I +A T L y (w)A4W (W +4W ) 1 1 WA T L y (w)A = I (IK(w))4W (W +4W ) 1 1 K(w) 53 If4w i =w i for all i2I, then w i +4w i = 0 for all i2I, K II (w;4w) =K II (w), K IE (w;4w) =K IE (w), and K EI (w;w +4w) = [0; K II (w)] T . Substituting into (3.31b), K(w +4w) =K(w;w +4w)K EI (w;w +4w)(I +K II (w;4w)) 1 K IE (w;4w) = 2 6 6 4 0 0 K II (w) K I I (w) 3 7 7 5 2 6 6 4 0 K II (w) 3 7 7 5 [IK II (w)] 1 [K II (w) K I I (w)] = 2 6 6 4 0 0 K II (IK II ) 1 K I I +K II (IK II ) 1 K I I 3 7 7 5 Therefore, K I I (w +4w) =K I I (w) +K II (w) (IK II (w)) 1 K I I (w). Remark 25. 1. The expression on the right hand side of (3.31b) is the Schur complement of the upper left block of the following matrix. 2 6 6 4 I +K II (w;4w) K IE (w;4w) K EI (w;w +4w) K(w;w +4w) 3 7 7 5 Equivalently,IK(w4w) is the Schur complement of the upper left block of the following matrix. I 2 6 6 4 4w II 0 0 w4w 3 7 7 5 [A I A] T L y (w)[A I A] where the second term can be seen as the oblique projection matrix of the network obtained from the original network by splitting every link i2I into two links of weight4w i and w4w i , respectively. 54 2. Equation (3.31d) provides the formula for updating K matrix under the condition that removing all links inI does not disconnect the network. Note the expression on the right hand side of (3.31d) is the Schur complement of the upper left block of the following matrix 2 6 6 4 IK II (w) K I I (w) K II (w) K I I (w) 3 7 7 5 Equivalently,IK I I (w+4w) is the Schur complement of the blockIK I I (w) ofIK(w), where matrix I has proper dimension in each expression. Lemma 8 leads to the following result on ow redistribution under weight perturbations. Proposition 8. Consider a connected DC networkG = (V;E) with balanced supply-demand vector p2 R V , link weights w2 R E >0 and weight perturbation4ww. IfG is connected with weight w +4w, then the following are true. f(w +4w) = (I +4WW 1 ) I +K(w)4WW 1 1 f(w) (3.32a) = I (IK(w))4W (W +4W ) 1 1 f(w) (3.32b) f I (w +4w) =f I (w) +K II (w)(IK II (w)) 1 f I (w) (3.32c) whereI :=fi2Ej4w i 6= 0g, I :=EnI, (3.32b) holds only for4w >w and (3.32c) holds only for4w I =w I . 55 Proof. (3.32b) and (3.32c) follow from (3.31c) and (3.31d), respectively, in a straightforward manner. We only show (3.32a). f(w +4w) = (W +4W )A T L y (w +4w)p = (W +4W )A T I +L y (w)A4WA T 1 L y (w)p = (W +4W ) I +A T L y (w)A4W 1 A T L y (w)p = (W +4W )W 1 W I +A T L y (w)A4W 1 W 1 WA T L y (w)p = (I +4WW 1 ) I +K(w)4WW 1 1 f(w) Remark 26. 1. It is simple algebra to obtain equations relating (w +4w) with(w) and%(w +4w) with %(w), using (3.32). For example: for4w>w, (3.32b) implies: (w +4w) = I IC 1 K(w)C 4W (W +4W ) 1 1 (w) where recall that C := diagc2 R EE . 2. It is possible to use (3.32) to prove Proposition 2, by letting4w go to 0. 3.5.4 Interpretation of Link Removal and Disconnected Case Equation (3.32c) has an interesting interpretation when the network remains connected under link removal. Letting := (IK II (w)) 1 f I (w); (3.33) 56 and substituting into (3.32c), we get the following: 4f I =f I (w +4w)f I (w) =K II (3.34) (3.34) implies that the ow change due to removal of links inI is equal to that induced by the additional supply-demand vector A I added to the original network. Since A I = P i2I A i i , adding A I is equivalent to adding i unit of supply (demand) on the tail (head) node of link i, for every i2I. In other words, acts the source that leads to the ow change due to link removal, for which we make the following remark. Remark 27. Lemma 5 implies that the spectral radius of K II (w) is less than 1. Therefore, = (IK II ) 1 f I =f I +K II f I +K 2 II f I +::: Recall the physical interpretation of matrix K in Section 3.3. K II f I refers to the ow over link setI. Therefore, can be interpreted as a result from an echo eect of f I that intensies itself and K n II f I can be seen as the gain at the n-th re ection. Additionally, (3.33) can be re-written as =f I (E;p) +K II =f I (E;p +A I ) (3.35) (3.35) means that for the network under the original supply-demand vector p and the additional supply-demand vector A I , the ow on every link i2I is equal to the additional supply and demand added on the incident nodes of i. This motivates the following alternate approach for nding ow change due to link removal. Proposition 9. Consider a connected DC networkG = (V;E) with link weights w2 R E >0 and balanced supply demand vector p2 R V . If there exists 0 2 R I ,IE, satisfying (3.35), then 57 the ow changes on the residual network due to the removal of links inI is4f I =K II 0 , where I :=EnI . Proof. It is sucient to prove thatf I =f I (E;p+A I 0 ) for 0 satisfying (2.9). Consider a network (V;E) with supply-demand vector p +A I 0 and let f2 R E and 2 R V be the ow and phase angle satisfying (2.9). Combination of the rst equality in (2.9) and (3.35) implies thatA I f I =p, which can be seen as the ow conservation constraint for network (V; I) with supply-demand vector p. It straightforward to see that f I and satisfy the \ohm's law" (the second equality in (2.9)) as well. This is to say, f I =f( I;p). Remark 28. The proof of Proposition 9 does not require the residual network (V; I) to be con- nected. However, if the network got disconnected, then a 0 satisfying (3.35) would exist if and only if f I (E;p) = 0. If (V; I) was connected, then a 0 satisfying (3.35) would be unique and given in (3.33). Flow redistribution in the scenario where link removal disconnects the network is not nec- essarily well-posed, and requires additional specications. In order to illustrate this point, for simplicity, consider the case when only one link j2E is removed, due to which ^ G = (V;Enfjg) has two connected components. If j does not carry ow before link removal, then trivially there is no redistribution of ow upon removal of link j. However, if j carries ow, then the demand- supply vector p is not balanced with respect to the two connected components in ^ G. One needs to specify a protocol for inducing such a balance, which will then determine new ows in the two components. One such protocol is to proportionally reduce supply or demand at nodes, depending on whether there is excessive supply or demand respectively. If ^ p is the balanced supply-demand rendered by such a balancing protocol, then the new ow can be computed using (2.10) and (3.29). 58 3.6 Conclusions and Future Work We formulate the maximal DC ow problem and show its strong connection to the maximal ow problem for undirected ow networks. While the latter provides an upper bound for the former, we identify conditions under which these two problems have the same optimal value. We also identify a congestion function that plays a key role in solving the maximal DC ow problem. In spite of the non-convexity and non-dierentiability, we show that every local minimum is a global minimum of the congestion function. Moreover, useful results are derived for the ow-weight Jacobian matrix, pseudo-inverse of Laplacian matrix and the oblique projection matrix. Based on them, we obtain an explicit expression for ow redistribution under weight perturbations. Such an expression is expected to be useful for our future work on designing an iterative algorithm and proving its convergence properties for the maximal DC ow problem. We also plan to investigate the relationship between such an algorithm with the one in [26]. The latter solves the maximal ow problem eciently using algebraic methods. 59 Chapter 4 Robustness of DC Networks With Controllable Weight In this and the next chapter, we project our investigations of DC networks onto the application domain of DC power networks (cf. Remark 8). 4.1 Introduction Robustness to man-made and natural disturbances is becoming an important consideration in the design and operation of critical infrastructure networks, such as the power grid. Disturbances to power networks are usually in the form of line failures and uctuations in the supply-demand prole, e.g., , due to renewables. From a control design perspective, the objective is to ensure that variations in power ow quantities caused by such external disturbances do not violate physical constraints, such as exceeding line thermal limits or voltage collapse. Our primary motivation in this chapter is DC power networks, whose line susceptances or weights are controllable. We study robustness of such DC networks towards balanced distur- bances to the supply-demand vector, i.e., disturbance vectors whose entries add up to zero. Such disturbances can result, e.g., , from the tripping of an active line [89, Chap 11]. Alternately, balanced disturbances are also relevant when weight control is used to increase the feasible region (with respect to link capacity constraints) for some other primary control mechanism, such as economic dispatch, e.g., see [64]. Such primary mechanisms give balanced supply-demand vector 60 in response to any (i.e., not necessarily balanced) disturbance. One way to quantify the feasible region is using the notion of margin of robustness. Consider the set of balanced disturbances for which there exist feasible link weights so that the resulting link ows are within specied capacity bounds. The margin of robustness is dened as the radius of the largest ` 1 ball around the origin which belongs to this set. This notion is related to system loadability, e.g., , see [60]. System loadability quanties deviations in supply- demand vector in terms of percentage of the nominal, i.e., pre-disturbance, value under which the system remains feasible. The objective of this chapter is to compute margin of robustness and the associated weight control policy. The computation of margin of robustness can be posed as an optimization problem, which is non-convex in general. We propose a multilevel programming approach to reduce the complexity for a certain class of networks, in which the nodes with non-zero demand or supply are relatively sparse. Specically, we identify a class of, possibly non-convex, optimization problems, each of which can be equivalently converted into a bilevel formulation over two sub-networks. The upper level problem corresponds to one sub-network, and the lower level problem to the other. The sub-network associated with the lower level problem does not contain any supply-demand nodes, with the possible exception of the nodes common to the two sub-networks. Interestingly, the lower level problem can be interpreted as dening a novel notion of capacity for the corresponding (sub- )network, analogous to capacity for a single link. This notion of network capacity is expectedly dependent on the weights of the constituent links. We show that, for the weight control problem, it is sucient to express network capacity in terms of equivalent weight of the network. The notion of equivalent weight is reminiscent of equivalent resistance from circuit theory. The proposed bilevel formulation can be applied recursively in a nested fashion to yield a mul- tilevel framework, where each iteration of bilevel formulation results in additional computational savings. The computational savings are with respect to the brute force search method. In the general case, the computational gains possible by the proposed multilevel formulation depend on 61 the relative sparsity of the nodes with non-zero supply or demand. In the particular case, when the network is tree reducible, i.e., when it can be reduced to a tree by sequentially replacing series and parallel sub-networks with equivalent capacities and weights, then we prove that an analyt- ical solution can be obtained. This is because, for series and parallel networks, the dependence of network capacity on equivalent weight, i.e., the equivalent capacity function, is shown to be strongly quasi-concave. Such a strong quasi-concavity property also holds true for meta-networks, where a series or parallel sub-network is replaced with a single link endowed with the equivalent capacity function. Recursive application of this procedure then proves that the equivalent capac- ity function of a link reducible network is strongly quasi-concave. Such a property then leads to an analytical solution of the margin of robustness for tree reducible networks. The control strategy in this chapter is partially motivated by FACTS devices, which allow online control of line properties in power networks. Prior work on FACTS devices has been performed in the context of optimal placement, e.g., see [43], coordinated control, e.g., see [55], and optimal power ow problem, e.g., see [80]. The weight control problem in this chapter is related to the so called impedance interdiction problem which has been studied for DC power ow models in [20], and can also be considered to be relaxation of the transmission switching and network topology optimization problem for power networks, e.g., , see [51]. In the topology control problem, the control actions associated with every link only take binary values corresponding to on/o status of the link, or equivalently corresponding to the weight of that link being equal to either zero or its nominal value. On the other hand, in our problem, we allow a continuum of control actions that includes these two values. In summary, the chapter makes several new contributions. First, a novel robust control prob- lem is formulated where the control strategy consists of changing link weights in response to disturbances. We then establish connection between the margin of robustness and cut capacities of associated ow network based on the results in Section 3.2. Second, we identify a class of, possibly non-convex, optimization problems, each of which can be equivalently formulated as a 62 bilevel problem. Such a reformulation gives computational gains with respect to the brute-force search method. We identify conditions under which the robust weight control problem belongs to this class. Third, a notion of network capacity, analogous to link capacity, is introduced. The lower level problems in our proposed multilevel hierarchy imply that the dependency of this no- tion of capacity on the link weights can be expressed in terms of the equivalent weight of the (sub-)network. This relationship between network capacity and equivalent weight is shown to be strongly quasi-concave for series and parallel networks, which facilitates analytical computation of the margin of robustness for tree reducible networks. 4.2 Problem Formulation In this section, we formulate the problem of robust weight control, and provide preliminary results. Throughout this chapter, we shall consider a connected DC network (cf. Denition 6) with balanced supply-demand vector (cf. (2.5)). Recall that the ow solution of power grid is unique and given by function f G (w;p) = WA T L y (w)p (cf. Lemma 1 ). We consider asymmetrical capacities and are interested in feasible ows, i.e., ows that satisfy the following lower and upper line capacity constraints c l fc u (4.1) We call a network feasible if the ows on all its links are feasible. Throughout this chapter, we make the following rather natural assumption on line capacities: Assumption 2. c l < 0<c u We note the capacitiesc l andc u are typically symmetrical about 0. We implicitly assume that the nominal (i.e., pre-disturbance) ow satises Assumption 3. c l f(w 0 ;p 0 )c u . where w 0 and p 0 are the nominal values of weight and supply-demand vector, respectively. 63 We are interested in quantifying disturbances on the nominal supply-demand vector under which (4.1) continues to be satised, using w as control. Disturbances will be modeled by change in the supply-demand vector. We assume that the disturbance induces a one-shot change to the system. Formally, under a disturbance, the supply-demand vector changes irreversibly from nom- inal value p 0 top 4 =p 0 +4, with 1 T 4 = 0, and hence 1 T p 4 = 0. A natural source of balanced disturbance is the removal of a link [89, Chap 11]. Alternately, the balanced disturbance feature is also relevant when weight control is used to increase the feasible region (with respect to line capacity constraints) for some other primary control mechanism, such as economic dispatch, e.g., see [64]. Such primary control mechanisms give a balanced supply-demand vector in response to any (i.e., not necessarily balanced) disturbance. Specically, line capacity constraints in primary control design can be replaced with feasible region under controllable link weights. Since the primary control design optimizes over balanced supply-demand proles (even for unbalanced dis- turbance), it is desired to nd a characterization of feasible region under controllable link weights in the space of balanced disturbances. One way to characterize such a feasible region is through the notion of robustness, and more specically the margin of robustness. We now formalize these notions. The network responds to a disturbance by changing the weights, which in turn also changes link ows due to (2.10). Under a robust control policy, the new link weights w () in response to disturbance are chosen to be any w2 R E 0 satisfying: c l f(w;p )c u ; 0w l ww u (4.2) if (4.2) is feasible, and (arbitrarily) equal to w u otherwise. Here, w l and w u are the lower and upper limits, respectively, for the operation range of the weight controller, and f(w;p ) is as given in (2.10). 64 Given a DC networkG = (V;E) with link weight bounds w l 2 R E 0 and w u 2 R E >0 , link capacity bounds c l 2 R E <0 and c u 2 R E >0 , and nominal supply-demand vector p 0 2 R V , the margin of robustness is dened as (G;w l ;w u ;c l ;c u ;p 0 ) := supf 0 :8 balanced4 satisfyingk4k 1 ;9w2 R E satisfying (4.2)g (4.3) The choice of the ` 1 norm in (4.3) is justied because we consider only balanced disturbances, and thereforek4k 1 is equal to twice the cumulative deviation in supply (or demand) caused by disturbances. The following example provides a simple illustration of the increase in margin of robustness when the line weights are controllable. 1 2 3 4 e 1 e 2 e 5 e 3 e 4 Figure 4.1: Network used in Examples 3 and 4. Example 3. Consider the network shown in Figure 4.1 with w u = [1 3 1 1 1] T , w l the same as w u , except forw l 2 = 0;c u =c l = [1 1 1 0:5 1] T andp 0 = [1 0 01] T . The ow corresponding to weight w u and load p 0 is f(w u ;p 0 ) = [0:33 0:67 0:44 0:56 0:11] T which is infeasible due to the excessive ow on link e 4 . However, the ow under the same p 0 but with weight w l is f(w l ;p 0 ) = [1:00 0 0:67 0:33 0:33] T which is feasible. Choosing w i = 0 for some link i, e.g., , for link 2 in Example 3, corresponds to disconnecting that link. Such line tripping strategies have been considered in the context of cascading failures [1,90]. Our objective in this chapter is to provide a framework for tractable computation of (G;w l ;w u ;c l ;c u ;p 0 ) 65 (4.3) can be easily seen to be equal to the following: := min :kk1=1; 1 T =0 () (4.4) where () = max w2R E 0 ;0 subject to c l f(w;p 0 +)c u w l ww u (4.5) For brevity, the explicit dependence of and onG, w l , w u , c l , c u or p 0 is dropped when clear from the context. (4.3) diers from (4.4)-(4.5) only in parameterization of the set of disturbances in terms of disturbances on a unit ` 1 -ball and magnitude . (4.5) only considers disturbances along the direction and () gives the maximal magnitude of such disturbances under which link ows are feasible. (4.4) then considers all possible directions of balanced disturbances. Lemma 9. For a DC networkG = (V;E) with link weight bounds w l 2 R E 0 and w u 2 R E >0 , link capacity bounds c l 2 R E <0 and c u 2 R E >0 , and nominal supply-demand vector p 0 2 R V , there exists a42 R V withk4k 1 arbitrarily greater than such that (4.2) is infeasible. The next example shows thatk4k 1 > is not sucient for infeasibility. Example 4. Consider the network shown in Figure 4.1, with c u = c l = 5:5 1, and w l = w u = [1 3 3 1 1] T = w (say). The ow corresponding to p 0 = [8 0 0 8] T is f(w;p 0 ) = [3:2 4:8 4:8 3:2 1:6] T which is feasible. Consider two perturbations4 1 = [1:5 0:5 0:5 1:5] T and4 2 = [2 2 2 2] T . Note thatk4 1 k 1 = 4 < 8 = k4 2 k 1 , and that, element- wise,4 1 and4 2 have the same signs. The ows under these perturbations are f(w;p 41 ) = [3:95 5:55 5:55 3:95 2:1] T and f(w;p 42 ) = [4:6 5:4 5:4 4:6 2:8] T . Since f(w;p 41 ) is infeasible, k4 1 k 1 = 4. However, the ow under4 2 , whose norm is greater than is feasible. That is, even if4 2 dominates4 1 element-wise, the system is feasible under4 2 , but not under4 1 . Such 66 non-monotonicity is directly attributable to non-monotonicities of ow distribution with respect to the supply-demand vector in power networks. 4.3 Relationship Between Margin of Robustness and Min- cut Capacity The robust weight control problem (4.4)-(4.5) is to maximize the margin of robustness, considering all possible disturbances. Based on the discussion in Section 3.2 on the relationship between admissible ow over ow network and DC network with exible weights, we formally investigate the relationship between the weight control problem (4.4)-(4.5) and a standard network ow problem in this section. We show that, under appropriate conditions, the weight control problem (4.5) is equivalent to a network ow problem with the margin of robustness for a given disturbance being the objective function. We use this relationship to make connections between the margin of robustness of a given DC power network and the min cut capacity of a certain associated ow network. These results are reminiscent of previous work in [31, 32] on robustness of transport networks. Proposition 1 implies that, for a network whose underlying graph is a tree, (4.4)-(4.5) is equivalent to: 0 := min :kk1=1; 1 T =0 0 () (4.6) where 0 () := max 0;f subject to Af =p 0 + c l fc u (4.7) If the underlying graph is not a tree, a feasible ow f 2F 1 can contain circulations, i.e., f = 2F 0 , and hence f = 2F 2 by Proposition 1. In this case, it is possible to eliminate circulations 67 from f to obtain a ~ f2F 1 \F 0 as follows. Set ~ f = f. While ~ f contains a circulation for some cycleO, update ~ f = ~ f min i2O ~ f i 1 O , where 1 O is a binary vector containing one for entries corresponding toO, and zero otherwise. Moreover, it is easy to see that, if (;f) is feasible for (4.7), then (; ~ f) is also feasible. Proposition 1 implies that the ow obtained by removing circulation satises ~ f 2F 2 if w l = 0. Therefore, (4.4)-(4.5) is equivalent to (4.6)-(4.7) when w l = 0. In summary, if the underlying graph of a network is a tree or w l = 0, then the nonconvex problem (4.4)-(4.5) is equivalent to (4.6)-(4.7), whose inner problem (4.7) is convex. Indeed, (4.7) is a classical network ow problem and can be solved eciently for a given disturbance . However, computational tractability of the minimax problem (4.6)-(4.7) is not readily apparent. The next result establishes a useful property of 0 (), which in turn will lead to an ecient solution methodology for (4.6)-(4.7). Lemma 10. For a DC networkG = (V;E) with link capacity bounds c l 2 R E <0 and c u 2 R E >0 , and initial supply-demand vector p 0 2 R V , 0 () dened in (4.7) is quasiconcave. Proof. Given arbitrary 1 and 2 , we show that 0 ( 1 + (1) 2 ) minf 0 ( 1 ); 0 ( 2 )g for all 2 [0; 1]. Let u 1 = 0 ( 1 ) and u 2 = 0 ( 2 ). Without loss of generality, assume u 1 u 2 and we need to prove 0 ( 1 + (1) 2 ) u 1 . It is sucient to show that u = u 1 is feasible to (4.7) when = 1 + (1) 2 . When = 1 + (1) 2 , u =u 1 , the equality constraint becomes Af =p 0 +u 1 ( 1 + (1) 2 ) =(p 0 +u 1 1 ) + (1 2 )(p 0 +u 1 2 ) =Af 1 + (1)Af 0 2 where f 1 and f 0 2 are some ow on the network under disturbed supply-demand vector p 0 +u 1 1 and p 0 +u 1 2 , respectively. By setting f = f 1 + (1)f 0 2 , the ow conservation constraint is satised. For feasibility of ( 1 + (1) 2 ;u 1 ), what remains to be shown is that such f satises 68 the capacity constraint. It is sucient to show that there existf 1 andf 0 2 that are feasible. f 1 can be selected as the optimal solution to (4.7) corresponding to u 1 and hence feasible. In order to see that there exists feasible f 0 2 , note that the feasible set of (4.7) is a polyhedron, = 0, f =f 0 and = 2 and f = f 2 are feasible, where f 2 is the optimal ow solution corresponding to u 2 , andu 1 u 2 is convex combination of 0 and u 2 . Therefore, 0 ( 1 + (1) 2 ;G t )u 1 and 0 () is quasiconcave. Lemma 11. Consider a DC networkG = (V;E) with link capacity boundsc l 2 R E <0 andc u 2 R E >0 , and initial supply-demand vector p 0 2 R V . Then, 0 dened in (4.6) is equal to min 20 0 (), where 0 :=f2 R V j9s;t2V; s = 1=2; t =1=2; v = 08v2Vnfs;tgg, and 0 () is as dened in (4.7). Proof. The feasible setf2 R V jkk 1 = 1; 1 T = 0g for (4.6) is a polytope. We now show that f2 R V jkk 1 = 1; 1 T = 0g is the convex hull of set 0 . The result then follows by using Lemma 10, and Lemma 29 (in the Appendix). Pick an arbitrary 2 R V withkk 1 = 1 and 1 T = 0. We now show that there existf k g andf 0 k g satisfying k 0 and 0 k 2 0 for all k, and P k k =kk 1 = 1. Let ~ = , and k = 1. While ~ 6= 0, do the following. LetV + :=fvj ~ v > 0g,V :=fvj ~ v < 0g, and pick v 2 argmin v2V + [V , and let k := 2j ~ v j. If v 2V + , then let ~ v = ~ v k =2, pick arbitrary v 0 2V , and let ~ v 0 = ~ v 0 + k =2. 0 k is then chosen such that 0 k;v = 1=2, 0 k;v 0 =1=2, and 0 k;v = 0 for allv2Vnfv ;v 0 g. One can similarly choose 0 k whenv 2V . We then setk =k +1, and repeat the process for selecting 0 k and k while ~ 6= 0. Lemma 11 implies that, in order to solve (4.6)-(4.7), it is sucient to consider a nite number of disturbance directions 2 0 , each with only one positive and one negative component. Then a naive solution strategy to compute for a network with tree topology or w l = 0 is to solve (4.7) for all the disturbance directions in 0 and then take the minimum. However, by using the Max-Flow-Min-Cut theorem, chapter [61, Theorem 8.6], one can execute this step in a simpler 69 way as we describe next. In order to do this, we rst construct a ow network associated with the given network, where we recall the standing Assumption 3. Denition 8 (Associated Flow Network). Consider a DC networkG = (V;E) with link weight bounds w l 2 R E 0 and w u 2 R E >0 , link capacity bounds c l 2 R E <0 and c u 2 R E >0 , initial supply- demand vectorp 0 2 R V , and initial weightsw 0 2 [w l ;w u ]. Letf 0 be the corresponding initial ow, as given by (2.10). The associated ow network (G ;c ) consists of a directed graphG = (V;E ), whereE is the union ofE and as well as reversed versions of links inE, and link capacities c dened as c i := c u i f 0;i if i2E, and c i :=c l i +f 0;i if i2E nE. Assumption 3 imply that c 0. Cut capacity is a function C : 2 V nf;[Vg R E 0 ! R 0 over the cuts and ow capacities and dened as: C(V c ;c ) = X i:(i)2Vc;(i)= 2Vc c i where with a little abuse of notation, we use the same letter for both cut capacity function and cut matrix, when the meaning is clear from context. The min-cut capacity C min (G ;c ) ofG is the minimum cut capacity among all cuts inG , chapter C min (G ;c ) = min ;(Vc(V C(V c ;c ). The next proposition relates the margin of robustness to the min-cut capacity of the associated ow network. Proposition 10. Consider a DC networkG = (V;E) with link weight bounds w l 2 R E 0 and w u 2 R E >0 , link capacity bounds c l 2 R E <0 and c u 2 R E >0 , initial supply-demand vector p 0 2 R V , and initial link weights w 0 2 [w l ;w u ]. Then, its margin of robustness (G) is upper bounded as (G) 2C min (G ;c ), where (G ;c ) is the associated ow network (cf. Denition 8). Moreover, ifG is a tree or w l = 0, then (G) = 2C min (G ;c ). In particular, ifG is a tree, then (G) = 2 min i2E ff 0;i c l i ;c u i f 0;i g, where f 0 is the initial ow, as given by (2.10). Proof. We rst prove the equality for the case whenG is a tree or w l = 0. In this case, (G) = 0 , and hence it is equivalent to proving 0 = 2(G ;c ). Following Lemma 11, for a given 70 2 0 with s = 1=2, t =1=2 and v = 0 for all v 2Vnfs;tg, the Max-Flow-Min-Cut theorem, e.g., [61, Theorem 8.6], implies that 0 () = 2 min Vc:s2Vc;t= 2Vc C(V c ;c ). Therefore, 0 = min 20 0 () = 2 min ;(Vc(V C(V c ;c ) = 2C min (G ;c ) . It is easy to see that (G) is upper bounded by the margin of robustness for a network with the same attributes for (G;w u ;c l ;c u ;p 0 ;w 0 ) andw l = 0 (since it expands the feasible set in (4.5)). We have already shown in the previous paragraph that the latter is equal to 2C min (G ;c ). The exact expression of (G) whenG is a tree follows from the fact that, in this case, each link separates the network, and hence C min (G ;c ) = min i2E flc i . There exists an extensive literature on ecient computation of min-cut capacity, which can be used to provide upper bound or exact characterization of the margin of robustness under special cases, as per Proposition 10. However, computing the exact value of margin of robustness in the general case requires solution to the non-convex problem (4.4). In Section 4.5, we propose a multilevel programming approach for more general networks under nongenerative disturbances. Before that, we shall consider the case of multiplicative disturbances in the next section. In this case, (4.4) is related with the maximal DC ow problem (3.1) and (3.3) in Section 3.1 and the descent algorithm provided in Section 3.4.4 can be used accordingly to obtain a solution. 4.4 The Multiplicative Disturbance Case In this section, we restrict our attention to the class of disturbances that are multiplicative. Formally, we let the set of over which the minimum is taken in (4.4) befp 0 =kp 0 k 1 ;p 0 =kp 0 k 1 g. Let M denote the corresponding solution to (4.4) for such a restriction of . For = p 0 =kp 0 k 1 and =p 0 =kp 0 k 1 , the set of disturbed supply-demand vectors can be parameterized as (1 + 71 =kp 0 k 1 )p 0 and (1=kp 0 k 1 )p 0 , respectively. Therefore, letting = 1 +=kp 0 k 1 and = =kp 0 k 1 1, respectively, for these two cases, solution to (4.4) can be obtained from: max w2R E >0 ;0 subject to c l f(w;p 0 )c u w l ww u (4.8) and a counterpart of (4.8) where p 0 is replaced withp 0 as follows. Let + denote the optimal solution to (4.8), and let denote the optimal solution to the counterpart of (4.8) where p 0 is replaced withp 0 . Then, M can be written as: M =kp 0 k 1 minf + 1; + 1g: (4.9) The assumed feasibility of the pre-disturbance state of the network (cf. Assumption 3) implies that + 1, and hence (4.9) is well-dened. Remark 29. When the ow capacities are symmetrical, chapterjc l j =jc u j, we have + = , and (4.9) is reduced to =kp 0 k 1 ( + 1). For the general case of asymmetrical ow capacities, + 6= . Small disturbances in thep 0 direction decrease the supply and demand and hence the link ows, and are therefore favorable. However, if < + 2, then (4.9) implies that the margin of robustness under disturbances in thep 0 direction is less than that under disturbances in the +p 0 direction. In order to re-write (4.8) and its counter part forp 0 succinctly, we consider the following notion of eective line capacity. Given ow f, for all i2E: ~ c i (f i ) := 8 > > < > > : c u i if f i 0 c l i if f i < 0 for = p 0 kp 0 k 1 ; ~ c i (f i ) := 8 > > < > > : c l i if f i 0 c u i if f i < 0 for = p 0 kp 0 k 1 : (4.10) 72 (4.8) can then be equivalently written as: minimize w2R E >0 max i2E f i (w) ~ c i (f i ) subject to w l ww u (4.11) It is straightforward to see that (4.8) and (4.11) become (3.1) and (3.3), respectively, if c u = c l . The complication brought by asymmetric capacities will not cause any serious problems as long as Assumption 2 is in place. A similar descent algorithm as shown in Section 3.4.4 can be obtained in the same way. In order to make clear the changes required in the algorithm to suit the asymmetric capacity setting, we provide the following simplied version: w(t + 1) = argmin w l ww u max i2S(w(t)) J iE (w(t)) (ww(t)) ~ c i (f i ) + 1 2 t (ww(t)) T (ww(t)) (4.12) where t > 0 is the step-size andS(w) := argmax i2E f i (w)=~ c i (f i ) has similar meaning as the same quantity dened in Proposition 6 and J iE (w) is the ith row of the ow-weight Jacobian given in (3.11). 4.5 The Nongenerative Disturbance Case: A Multilevel Programming Approach The optimization problem in (4.4)-(4.5) is non-convex in general. In this section, we propose a network reduction methodology to reduce the computational complexity, for nongenerative disturbances. Denition 9. For a DC networkG = (V;E) with supply-demand vector p2 R V , a balanced disturbance42 R V is called nongenerative with respect to p if p v = 0 implies4 v = 0 for all v2V. 73 Let the set of all nongenerative balanced disturbances with respect to p2 R V be denoted as NG (p) R V . Remark 30. Note that Denition 9 implies that the dependence of NG (p) on p is only through the support of p, i.e., only the nodes in p which correspond to supply and demand. We consider disturbances arising due to uctuations in demand and supply, and hence it is natural to attribute such disturbances to generation and load equipments. Insofar as every node with such an equip- ment has a non-zero entry in the vectorp, the set of nongenerative disturbances NG (p) includes all possible (balanced) disturbances. Indeed, this is the implicit assumption in Denition 9. How- ever, one can modify Denition 9 to call a disturbance4 non-generative even if it has non-zero entries corresponding to a specied set of transmission nodes in p, i.e., nodes v with p v = 0. The tradeo resulting from such a generalization is discussed in Remark 36. 4.5.1 A Novel Network Reduction We relate savings in computational complexity to relative sparsity of non-transmission nodes in the network. The computational savings are due to the reformulation of (4.4)-(4.5), as described in Sections 4.5.2 and 4.5.3. The key to this reformulation is the notion of reducible networks, which we dene next. Denition 10. A DC networkG = (V;E) is called reducible about v 1 2V and v 2 2V under supply-demand vector p2 R V if there existG 1 = (V 1 ;E 1 ) andG 2 = (V 2 ;E 2 ) such that all of the following conditions are satised: 1. V 1 [V 2 =V,V 1 \V 2 =fv 1 ;v 2 g ; 2. E 1 [E 2 =E,E 1 \E 2 =;,jE 2 j 2 ; 3. G 1 andG 2 are both weakly connected ; 4. the supply-demand vector p is supported only onV 1 . 74 HereG 2 is referred to as the `reducible component' and ~ G 1 = (V 1 ; ~ E 1 ) is referred to as the `reduced network' ofG, where ~ E 1 =E 1 [ (v 1 ;v 2 ), and (v 1 ;v 2 ) is an additional (virtual) link, not originally present inG. The direction (v 1 ;v 2 ) is chosen arbitrarily. Remark 31. In Denition 10, (a) reducibility of a network depends both on its graph topology and the location of the sup- ply and demand nodes; in particular, the dependence on the supply-demand vector p is only through its support { we however choose to state this dependence on the entire p in Denition 10 for simplicity in presentation; (b) if (v 1 ;v 2 ) is a link (or corresponds to several links), then it can be assigned arbitrarily to eitherE 1 orE 2 ; in this case the reduction process will result in an additional link (v 1 ;v 2 ) in ~ E 1 ; (c) v 1 and v 2 can be supply, demand or transmission nodes inG. " = ,ℇ ' = ( " = " ,ℇ ) " " ' + -. ' = ' ,ℇ ' " ' " ' Supply node Demand node Transmission node Figure 4.2: Illustration of a reducible network. Please see Remark 32 for clarication regarding the dierence in depiction of v 1 and v 2 infG; ~ G 1 g and G 2 . We now describe a reduction procedure for a reducible network, e.g., , as illustrated in Fig. 4.2. Specically, this network is decomposed into two smaller sub-networks ~ G 1 andG 2 ; (v 1 ;v 2 ) is a virtual link with equivalent weight w eq =H(w E2 ;G 2 ;v 1 ;v 2 ) as dened in Denition 11, andG 2 has a virtual supply-demand vector supported on nodes v 1 and v 2 . The reduction is equivalent (cf. Lemma 12) in the sense that, the ows on links inE obtained from Lemma 1, is the same as 75 the ows on corresponding links inE 1 andE 2 by applying Lemma 1 to sub-networks ~ G 1 andG 2 , respectively. Remark 32. The status of v 1 and v 2 as supply, demand or transmission nodes in fG; ~ G 1 g is determined by p. However, their status inG 2 is determined by the choice of the direction of the virtual link between v 1 and v 2 in ~ G 1 . For the choice (v 1 ;v 2 ), as it is in Figure 4.2, v 1 and v 2 are, respectively, supply and demand nodes inG 2 . This is independent of their status infG; ~ G 1 g. In order to emphasize this dierence, we do not associate v 1 and v 2 with supply, demand or transmission, infG; ~ G 1 g, but explicitly show their status inG 2 in Figure 4.2. Let us denote the net ow throughG 2 as f eq = X i2E + v 1 \E2 f i X i2E v 1 \E2 f i = X i2E v 2 \E2 f i X i2E + v 2 \E2 f i (4.13) The next lemma shows that f eq is indeed the ow on the virtual link in ~ G 1 . Lemma 12. Consider a DC networkG = (V;E) with link weightsw2 R E >0 and a supply-demand vector p2 R V . IfG is reducible about v 1 ;v 2 2V under p (cf. Denition 10), the link ows f G in G, are equal to the corresponding link ows f ~ G1 and f G2 in sub-networks ~ G 1 andG 2 , respectively: f eq =f ~ G1 v1v2 (w ~ E1 ;p V1 ) f G i (w;p) =f ~ G1 i (w ~ E1 ;p V1 ) 8i2E 1 f G i (w;p) =f G2 i (w E2 ;a v1v2 )f eq 8i2E 2 (4.14) where w ~ E1 and w E2 are the link weights associated with sub-networks ~ G 1 andG 2 , respectively, p V1 is the sub-vector of p corresponding to nodes inV 1 , and a v1v2 2f1; 0; +1g V2 is such that its v 1 -th component is +1, v 2 -th component is1, and all the other components are zero. 76 Proof. Noting that the components of the supply-demand vector at nodes inV 2 nfv 1 ;v 2 g are zero, Ohm's and Kirchho's laws for linksE 2 and nodesV 2 inG 2 can be written as: w i ( (i) (i) ) =f G i 8i2E 2 ; X i2E + v f G i X i2E v f G i = 0 8v2V 2 nfv 1 ;v 2 g (4.15) where (i) and (i) are, respectively, the tail and head nodes of link i. (4.15), along with (4.13), are the same equations as one would get by writing Kirchho's and Ohm's law forG 2 under supply-demand vector f eq a v1v2 . Taking this latter interpretation of (4.13) and (4.15), Lemma 1 and its proof then gives the ow solution on links inG 2 in (4.14), i.e., f G i =f G2 i (w E2 ;f eq a v1v2 ) = f eq f G2 i (w E2 ;a v1v2 ). Moreover, if V2 denotes the sub-vector of corresponding to nodes inV 2 , then we have V2 =f eq L y G2 a v1v2 , and hence v1 v2 =a T v1v2 V2 =f eq a T v1v2 L y G2 a v1v2 (4.16) f eq and (4.16) can be seen as the ow on, and Ohm's law for the virtual link (v 1 , v 2 ), respec- tively. Now writing Ohm's and Kirchho's laws forG 1 , we get w i ( (i) (i) ) =f G i 8i2E 1 X i2E + v f G i X i2E v f G i =p v 8v2V 1 nfv 1 ;v 2 g X j2E + v 1 \E1 f G i X i2E v 1 \E1 f G i +f eq =p v1 X i2E + v 2 \E1 f G i X i2E v 2 \E1 f G i f eq =p v2 (4.17) where we use the denition of f eq in (4.13). As f eq is interpreted to be the ow on a virtual link (v 1 ;v 2 ), (4.16) and (4.17) become Ohm's and Kirchho's laws for ~ G 1 . The expression forf G i , i2E 1 and f eq , in (4.14) now follows from Lemma 1 and its proof. 77 (4.16) motivates the following denition of equivalent weight, which is reminiscent of resistance distance in graph theory [58], and equivalent resistance in circuit theory. Denition 11 (Equivalent Weight). Given a DC networkG = (V;E) with link weightsw2 R E >0 , the equivalent weight between v 1 ;v 2 2V is dened as: H(w;G;v 1 ;v 2 ) := 1 a T v1v2 L y G a v1v2 (4.18) where a v1v2 is as dened in Lemma 12. Remark 33. Denition 11 is well-posed, i.e., a T v1v2 L y G a v1v2 > 0. This is because, for connected networkG, L y G is positive denite in space R E n spanf1g. At times, when the graphG and nodesv 1 andv 2 are clear from the context, we shall denote the equivalent weight simply byH(w) for brevity.H(w) is generalization of rather standard formulae for equivalent resistances for series and parallel connections from circuit theory, which we brie y state next for completeness. Example 5. For a network consisting of m links with weights w i ;i = 1; 2;:::;m, in parallel or series from node v 1 to node v 2 , the equivalent weight between v 1 and v 2 , as given by (4.18), can be shown to be P m i=1 w i and ( P m i=1 1=w i ) 1 , respectively. These are reminiscent of standard expressions for equivalent resistance from circuit theory. Remark 34. (a) Monotonicity ofH(w) with respect to components ofw follows from Rayleigh's monotonicity law, e.g., , see [81, Section 1.4]. Example 5 implies that the equivalent weight function is strictly monotone for series and parallel networks. (b) Monotonicity ofH along with its continuity implies thatH(w) is a, not necessarily one-to- one, map from [w l ;w u ] R E >0 to [H(w l );H(w u )] R >0 . Lemma 12 illustrates that link ow computations for a reducible network can be decomposed into computations for two sub-networks, thereby giving computational savings. As an intermediate 78 step in extending this decomposition to (4.4)-(4.5), we decompose the ow capacity constraints in (4.5). First note that, if a network is reducible with respect to a nominal supply-demand vector p 0 , then the same reduction is valid also for p 4 = p 0 +4 under any nongenerative disturbance 4 with respect to p 0 . Consequently, the ow capacity constraints in (4.5) can be written as: c l i f ~ G1 i (w ~ E1 ;p 4;V1 )c u i 8i2E 1 (4.19a) c l i f eq f G2 i (w E2 ;a v1v2 )c u i 8i2E 2 (4.19b) f eq =f ~ G1 v1v2 (w ~ E1 ;p 4;V1 ); w eq =H(w E2 ) (4.19c) where p 4;V1 is the subvector corresponding to node setV 1 of p 4 . (4.19) is simply a re-statement of capacity constraints in (4.5) in terms of the notations used in (4.14). We intend to replace (4.19b)-(4.19c) with capacity constraint for the virtual link (v 1 ;v 2 ), or equivalently of the sub-networkG 2 between v 1 andv 2 . In order to do this, we next introduce the notion of equivalent capacities. Denition 12 (Equivalent Capacities). Consider a DC networkG = (V;E) with lower and upper bounds on link weights w l 2 R E >0 and w u 2 R E >0 , respectively, and lower and upper capacity functions c l i : [w l i ;w u i ]! R <0 and c u i : [w l i ;w u i ]! R >0 , i2E, respectively. Given two nodes v 1 ;v 2 2V, and an equivalent weightw eq 2 [H(w l E2 ;G;v 1 ;v 2 );H(w u E2 ;G;v 1 ;v 2 )], the corresponding equivalent lower capacityC l (w eq ) and upper capacityC u (w eq ) from v 1 to v 2 are dened as: C l (w eq ) := minG(w eq ); C u (w eq ) := maxG(w eq ) (4.20) where G(w eq ) :=fz2 Rjc l zf(w;a v1v2 ) c u for some w2 [w l ;w u ] satisfyingH(w) = w eq g, and a v1v2 is as dened in Lemma 12. Remark 35. 79 (a) Note that the link capacity functions in Denition 12 are assumed to be weight-dependent. This general setup allows denition of equivalent capacity to be applicable to networks whose links themselves could be equivalent links for some underlying sub-network. This feature is specically used in extending the bilevel formulation to a multilevel framework in Section 4.5.3. (b) Remarkably, the equivalent capacity can be expressed concisely in terms of the equivalent weight, as opposed to the entire weight vectorw E2 . This considerably reduces the complexity of the weight control problem (4.4)-(4.5). (c) Computing the equivalent capacities for a given w eq from v 1 to v 2 is equivalent to solv- ing (4.4)-(4.5) with a single supply/demand node pair (v 1 ;v 2 ), and under nongenerative disturbances { however, with the additional equality constraintH(w) =w eq . With Denition 12, consider the following version of capacity constraints in lieu of (4.19): c l i f ~ G1 i (w ~ E1 ;p 4;V1 )c u i 8i2E 1 (4.21a) C l (w eq )f ~ G1 v1v2 (w ~ E1 ;p 4;V1 )C u (w eq ) (4.21b) wherew eq 2 [H(w l E2 );H(w u E2 )]. The equivalence between (4.19) and (4.21) is formally established in the next subsection within a more general class of feasibility problems. Furthermore, this equivalence leads to the following result. Proposition 11. Consider a DCG = (V;E) with lower and upper link weights w l 2 R E >0 and w u 2 R E >0 respectively, and a supply-demand vector p 0 2 R V . IfG is reducible about v 1 ;v 2 2V 80 under p 0 2 R V (cf. Denition 10) and the disturbances are nongenerative with respective to p 0 (cf. Denition 9), then (4.4)-(4.5) is equal to the following min 2 NG (p0) kk1=1 max 0 w E 1 2D E 1 weq2D2 subject to c l i f ~ G1 i (w ~ E1 ;p 0 +)c u i 8i2E 1 C l (w eq )f ~ G1 v1v2 (w ~ E1 ;p 0 +)C u (w eq ) (4.22) where D E1 := [w l E1 ;w u E1 ], D 2 := [H(w l E2 );H(w u E2 )],C l () andC u () are the equivalent lower and upper capacities (cf. Denition 12) ofG 2 from v 1 to v 2 , with c l i (w i )c l i and c u i (w i )c u i for all i2E 2 . Remark 36. Proposition 11 gives the equivalent bilevel formulation for (4.4)-(4.5). The key as- sumptions are on network reducibility and non-generativity of disturbances, both with respect to the nominal supply-demand vector p 0 . It is straightforward to see that Proposition 11 also holds true if the two notions are dened with respect to any ~ p whose support is a superset of the support ofp 0 . Such an extension to network reducibility is consistent with \terminal reducibility" for planar graphs [44]. However, there is a trade-o. Extending non-generativity to ~ p increases the set of disturbances we can incorporate in our analysis. This is useful if one needs to incorporate disturbances which arise due to removal of a link whose end nodes do not lie in the support of p 0 , or disturbances which aect nodes other than the ones in the support of p 0 . On the other hand, extending network reducibility to ~ p shrinks the class of networks which satisfy Denition 10, and hence which can be subject to the bilevel treatment in Proposition 11. We choose to dene these two notions with respect to p 0 , because p 0 has the smallest permissible support to dene these notions. 81 4.5.2 An Equivalent Bilevel Formulation Inspired by (4.19), we consider the following feasibility problem: nd (x I1 ;x I2 ;y 1 ;y 2 )2 D I1 D I2 R R satisfying: q i (x I1 ;y 2 ) 0; 8i2J 1 q i (x I2 ;y 1 ) 0; 8i2J 2 y 1 =h 1 (x I1 ;y 2 ) y 2 =h 2 (x I2 ) (4.23) whereD I1 R I1 andD I2 R I2 are the domain ofx I1 andx I2 , respectively; andq i :D I1 R! R, i2J 1 , q i : D I2 R! R, i2J 2 , h 1 : D I1 R! R and h 2 : D I2 ! R are continuous functions. Remark 37. (4.19) is a special case of (4.23) when x I1 w E1 [f4g, x I2 w E2 , y 1 f eq , y 2 w eq ;h 1 (x I1 ;y 2 )f ~ G1 v1v2 (w ~ E1 ;p 4;V1 ) andh 2 (x I2 )H(w E2 ).J 1 andJ 2 correspond toE 1 and E 2 , respectively, as follows. For every i2E 1 , there are two constraints in (4.23) corresponding to q i (x I1 ;y 2 ) f ~ G1 i (w ~ E1 ;p 4;V1 )c u i and q i (x I1 ;y 2 ) c l i f ~ G1 i (w ~ E1 ;p 4;V1 ). Similarly, for every i2E 2 , there are two constraints in (4.23). We now give an equivalent bilevel formulation for (4.23). Proposition 12. Let q i , i2J 1 [J 2 , h 1 and h 2 be continuous functions. Then, the following are true: (a) there exists a (x I1 ;x I2 ;y 1 ;y 2 )2 D I1 D I2 R R satisfying (4.23) if and only if there exists a (x I1 ;y 2 )2D I1 D 2 satisfying the following: q i (x I1 ;y 2 ) 0 8i2J 1 h 1 (x I1 ;y 2 )2G(y 2 ) (4.24) 82 where G(y 2 ) :=fz2 Rjq i (x I2 ;z) 0 8i2J 2 for some x I2 2 D I2 satisfying h 2 (x I2 ) = y 2 g and D 2 =R(h 2 ) is the domain of y 2 . Moreover, for every y 2 2 D 2 , the set G(y 2 ) is closed. (b) the set G(y 2 ) is connected for all y 2 2D 2 if for all x I2 2D I2 and i2J 2 , q i (x I2 ;z) is quasiconvex with respect to z; and there exists a z 0 2 R such that q i (x I2 ;z 0 ) 0 for all x I2 2D I2 and i2J 2 . Proof. (a) Consider a (~ x I1 ; ~ x I2 ; ~ y 1 ; ~ y 2 ) which satises (4.23). This implies that the rst equation in (4.24) is satised by (~ x I1 ; ~ y 2 ), and that z = ~ y 1 =h 1 (~ x I1 ; ~ y 2 )2G(~ y 2 ) with x I2 = ~ x I2 . Now consider a (^ x I1 ; ^ y 2 ) which satises (4.24). Therefore, (^ x I1 ; ^ y 2 ) readily satises the rst inequality in (4.23). Let ^ y 1 := h 1 (^ x I1 ; ^ y 2 ), then ^ y 1 2 G(^ y 2 ). Therefore, G(^ y 2 ) is not empty and there exists at least one ^ x I2 such that h 2 (^ x I2 ) = ^ y 2 and q i (^ x I2 ; ^ y 1 ) 0 for all i2J 2 . That is to say, (^ x I1 ; ^ x I2 ; ^ y 1 ; ^ y 2 ) satises (4.23). Now we show thatG(y 2 ) is a closed set for every y 2 2D 2 . Pick an arbitrary convergent sequencefz r g in the set G(y 2 ). It is sucient to prove that z = lim r!+1 z r 2 G(y 2 ). Suppose z 62 G(y 2 ), then9k2J 2 s.t. q k (x;z ) > 0;8x2 D I2 satisfying h 2 (x) = y 2 . Continuity of q k then implies that q k (x;z r ) > 0, and hence implies z r = 2 G(y 2 ), for all suciently large r. This leads to a contradiction. (b) The second condition implies that z 0 2 G(y 2 ) R for all y 2 2 D 2 . Since G(y 2 ) is closed for all y 2 2D 2 , let g l (y 2 ) := minG(y 2 ) and g u (y 2 ) := maxG(y 2 ), then g l (y 2 )z 0 g u (y 2 ) for all y 2 2 D 2 . Proving connectedness of the set G(y 2 ) is equivalent to proving that [g l (y 2 );z 0 ]G(y 2 ) and [z 0 ;g u (y 2 )]G(y 2 ). We provide details for the rst set; the proof for the second set follows similarly. For the proof, we assume that1 < g l (y) g u (y) < +1 for simplicity. The proof when either g l (y) =1 or g u (y) = +1 is similar. Consider a x I2 2 D I2 satisfying h 2 (x I2 ) = y 2 and q i (x I2 ;g l (y 2 )) 0 for all i2J 2 ; closedness of the set G(y 2 ) implies 83 well-posedness of x I2 . We also have q i (x I2 ;z 0 ) 0 for all i2J 2 by assumption. Since q i (x I2 ;z) is quasiconvex with respect toz for allx I2 2D I2 andi2J 2 , for all2 [0; 1] and i2J 2 , we have q i (x I2 ;g l (y 2 ) + (1)z 0 ) maxfq i (x I2 ;g l (y 2 ));q i (x I2 ;z 0 )g 0. Since y 2 is arbitrary, [g l (y 2 );z 0 ]G(y 2 ) for all y 2 2D 2 . Proposition 12 can be straightforwardly extended to optimization problems as follows. Proposition 13. Let q i (x I2 ;z) be quasiconvex with respect to z for all x I2 2 R I2 , i2J 2 and let there exist z 0 2 R such that q i (x I2 ;z 0 ) 0 for all x I2 2 D I2 , i2J 2 . Then, for every q 0 :D I1 R! R, max x I1 2D I1 ;x I2 2D I2 y 1 2 R;y 2 2 R q 0 (x I1 ;y 2 ) subject to q i (x I1 ;y 2 ) 0; 8i2J 1 q i (x I2 ;y 1 ) 0; 8i2J 2 y 1 =h 1 (x I1 ;y 2 ) y 2 =h 2 (x I2 ) (4.25) is equal to max x I 1 2D I 1;y22D2 q 0 (x I1 ;y 2 ) subject to q i (x I1 ;y 2 ) 0; 8i2J 1 g l (y 2 )h 1 (x I1 ;y 2 )g u (y 2 ) (4.26) whereg l (y 2 ) := minG(y 2 ),g u (y 2 ) := maxG(y 2 ), and,D 2 ,G(y 2 ) are as dened in Proposition 12. Remark 38. (a) The conditions in Proposition 12(a) do not guarantee that the set G(y 2 ) is non-empty for every y 2 2 D 2 . However, under the additional condition of the existence of z 0 , as in Propositions 12(b) and 13, the set G(y 2 ) is guaranteed to be non-empty for all 84 y 2 2D 2 . In particular, this implies thatg l (y 2 ) andg u (y 2 ) in Proposition 13 are well-dened. Furthermore, connectedness of G(y 2 ) allows us to write the constraint h 1 (x I1 ;y 2 )2 G(y 2 ) as g l (y 2 )h 1 (x I1 ;y 2 )g u (y 2 ). (b) Proposition 13 provides an equivalent bilevel formulation in (4.26) for a class of optimization problems described in (4.25). (x I1 ;y 2 ) and (x I2 ;z) are the upper and lower level variables, respectively. (c) The number of non-redundant variables in (4.25) isjI 1 j+jI 2 j. Therefore, the computational complexity in solving (4.25) by brute-force search is exponential injI 1 j +jI 2 j. On the other hand, the computational complexity associated with computing the functions g l (y 2 ) and g u (y 2 ) by brute-force search in the lower level problem in (4.26) is exponential in jI 2 j. Thereafter, the computational complexity of the upper level problem by brute-force search is exponential injI 1 j + 1. Therefore, the complexity of solving the bilevel problem by brute- force search is exponential in maxfjI 1 j + 1;jI 2 jg, which is much less than that of (4.25). Remark 38(c) implies that the implementation of brute-force search on the bilevel formulation reduces the running time in comparison to brute-force search on the original formulation. This approach to characterizing computational eciency is inspired by [59, Chap 2.1]: \An algorithm is ecient if it achieves qualitatively better worst-case performance, at an analytical level, than brute-force search". The brute-force search here refers to the implementation on the original problem formulation. While ecient solutions can be devised for convex optimization problems, for the nonconvex weight control problem that is studied in this chapter, in general there is no solution method other than brute-force search that provides a global solution. For example, the weight control problem is none of the known nonconvex problems with globally optimal solution that are described in [54]. Therefore, with respect to the notion described in [59, Chap 2.1], and similar in spirit to \divide and conquer" [59, Chap 5], we call our bilevel formulation ecient. 85 Indeed, it is easy to see that the extension of bilevel to multilevel formulation, as we describe in the next subsection, leads to further computational eciencies. Finally, Proposition 11 follows from Proposition 13 because (4.19) is a special case of (4.23) as explained in Remark 37, and because the conditions imposed on functionq i (x I2 ;z) in Proposition 13 are also satised. Recall that, for everyi2E 2 , there are two constraints in (4.23) corresponding to q i (:;y 1 ) y 1 f G2 i (w E2 ;a v1v2 )c u i and q i (:;y 1 ) c l i y 1 f G2 i (w E2 ;a v1v2 ). Therefore, q i (;y 1 ) is linear, and hence quasiconvex, in y 1 for all w E2 2 D E2 (recall D E2 = [w l E2 ;w u E2 ]) and i2J 2 . Additionally, it is straightforward that q i (; 0) 0 8i2J 2 (4.27) 4.5.3 A Nested Bilevel Approach for Multilevel Formulation For a reducible network as per Denition 10, Proposition 11 shows that the weight control problem (4.4)-(4.5) can be transformed into a bilevel optimization problem (4.22), in which the lower level problem involves nding the equivalent lower and upper capacity functions of the corresponding subnetwork. We now extend this to a multilevel framework. A comparison with (4.4)-(4.5) reveals that the upper level problem (4.22) is the same as (4.4)-(4.5) written for the reduced network ~ G 1 , where the equivalent link (v 1 ;v 2 ) has weight w eq =H(w E2 )2 [H(w l E2 );H(w u E2 )] and weight dependent lower and upper capacitiesC l (w eq ) and C u (w eq ), respectively. If the reduced network ~ G 1 =:G (1) is also reducible as per Denition 10, with its sub-networksG (1) 1 andG (1) 2 , one can apply Proposition 11 toG (1) if: (a) q i for i2J (1) 2 are quasiconvex, and (b) the equivalent lower and upper capacity functions for links inE (1) 2 are strictly negative and positive respectively. (b) is required to ensure (4.27). (a) is satised trivially as before because of linearity of q i , and (b) follows from the next result. 86 2 5 6 1 4 3 7 G (0) 2 G (0) 1 G =:G (0) =G (0) 1 [G (0) 2 2 5 6 1 4 3 G (1) 2 G (1) 1 ~ G (0) 1 =:G (1) =G (1) 1 [G (1) 2 2 1 4 3 ~ G (2) 1 =:G (3) =G T 2 5 6 1 4 3 G (2) 2 G (2) 1 ~ G (1) =:G (2) =G (2) 1 [G (2) 2 Figure 4.3: Illustration of recursive network reduction, where the supply node set isf1; 4g and the demand node set isf2; 3g; the thick edges denote the equivalent links. The original networkG is reduced to the terminal networkG T in three reductions: (i) subnetworkG (0) 2 ! link (4; 3); (ii) subnetworkG (1) 2 ! link (5; 6); (iii) subnetworkG (2) 2 ! link (3; 2). G k := ~ G (k1) 1 is the resulting network after k-th reduction, k = 1; 2; 3. Notice that the rst and the second reductions can be implemented in parallel, and the terminal networkG T =G (3) is not reducible. Lemma 13. Consider a DC networkG = (V;E) with lower and upper bounds on link weights w l 2 R E >0 andw u 2 R E >0 , respectively, and link lower and upper capacity functions c l i : [w l i ;w u i ]! R <0 and c u i : [w l i ;w u i ]! R >0 , i2E, respectively. The equivalent lower capacityC l (w eq ) and upper capacityC u (w eq ) between two given nodes v 1 ;v 2 2V satisfy the following: C l (w eq )< 0<C u (w eq ) 8w eq 2 [H(w l );H(w u )] 87 Proof. Since c l i < 0<c u i for all i2E, we have 02G(w eq ) in Denition 12. It is easy to see that, for z 0 = min i2E f max w l i wiw u i c l i (w i ); min w l i wiw u i c u i (w i )g > 0, we haveC l (w eq )z 0 < 0 < z 0 <C u (w eq ) for all w eq 2 [H(w l );H(w u )]. This follows from the straightforward fact that jf i (w;a v1v2 )j 1 for all i2E. A recursive nested application of the bilevel formulation leads to an equivalent multilevel formulation for (4.4)-(4.5); the recursion stops when the reduced network (cf. Denition 10), referred to as the terminal network, is not reducible, as per Denition 10. The resulting multilevel hierarchy consists of a collection of lower level problems, and an upper level problem corresponding to the last recursion. We appropriately then refer to the former as reduction problems and the latter as the terminal problem. In particular, a reduction problem corresponds to computing the equivalent capacities of a reducible component (cf. Denition 12), and the terminal problem is a generalization of (4.4)-(4.5), where capacities are weight-dependent. Figure 4.3 illustrates the reduction procedure for a simple network, where the process of replacingG (0) 2 with an equivalent link inG (1) := ~ G 0 1 corresponds to solving the reduction problem forG (0) 2 , and the terminal problem corresponds to generalized version of (4.4)-(4.5) forG T =G (3) , which then gives the margin of robustness of the original network. 4.6 An Ecient Solution Methodology for Tree Reducible Networks In this section, we show that the two types of problems in the multilevel formulation for the weight control problem (i.e., reduction and terminal problems) can be solved explicitly for type I tree reducible networks 1 , or tree reducible network for short. 1 Type II tree reducible network is shown Denition 17 in Section 17 88 4.6.1 Tree Reducible Network Denition 13 (Tree reducible network). A DC networkG = (V;E) with supply-demand vector p2 R V is called tree reducible (see also [44]) if there exists a sequence consisting of the following three operations through whichG can be reduced to a tree: 1. Degree-one reduction: delete a degree one vertex with p v = 0 and its incident edge. 2. Series reduction: delete a degree two vertex v 2 and its two incident edgesfv 1 ;v 2 g and fv 2 ;v 3 g, and add a new edgefv 1 ;v 3 g. 3. Parallel reduction: if a node pair has multiple, i.e., two or more, links between them, then remove one of those links. In particular, if the terminal network produced from the above three reduction operations contains only one link, then we call the original networkG link reducible. Similar to the denition of reducible network (cf. Denition 10 and Remark 10), the denition of a tree reducible network involves conditions on the graph topology as well as the locations of supply and demands nodes. For example, a network consisting of the graph in Fig. 4.4 is tree reducible if the supply and demand nodes only include v 1 and v 4 , while it is not tree reducible if v 1 and v 2 are the supply nodes and v 4 is the demand node. v1 v2 v3 v4 e1 e2 e3 e4 e5 Figure 4.4: A candidate graph topology for tree reducible network It is easy to see that, for a tree network, the link ows are independent of link weights. We next show that, for a tree reducible network, the link ow directions are independent of link weights. 89 Lemma 14. For a tree reducible network consisting of multigraphG = (V;E) and supply-demand vector p2 R V , sign (f i (w;p)) = sign (f i ( ~ w;p)) 8i2E;w; ~ w2 R E >0 Proof. It is clear that the above result holds for a tree, as a special case of tree reducible networks. For a general tree reducible network, the result follows from invariance of ow direction in the three operations in the denition of tree reducible networks. In degree-one reduction, the link removed has ow equal zero. In series reduction, sign (f v1v2 (w;p)) = sign (f v2v3 (w;p)) = sign (f v1v3 (w;p)) for all w > 0. In parallel reduction, the removed link has the same direction of ow as the remaining links. Remark 39. Lemma 14 implies that, for a tree reducible network with a given supply-demand vector p2 R V , one can choose direction convention for links such that f(w;p) 0 for all w> 0. We implicitly adopt this convention for the rest of this section 2 . Recall from Section 4.5.3 that a reduction problem in the multilevel formulation is (an equality constrained) weight control problem for a subnetwork of the original network. Since the original network is assumed to be tree reducible, this subnetwork is link reducible. Therefore, Remark 39 implies that the reduction problem for the network, i.e., a problem of the kind (4.20), can be simplied as C l (w eq ) = min z2R;w2R E z C u (w eq ) = max z2R;w2R E z subject to w l ww u subject to w l ww u H(w;G;v 1 ;v 2 ) =w eq H(w;G;v 1 ;v 2 ) =w eq zf(w;a v1v2 )c l zf(w;a v1v2 )c u (4.28) 2 We emphasize that the lower and upper capacities c l and c u , respectively, are dened with respect to chosen direction convention. 90 whereG = (V;E) is the network's underlying graph and w eq 2R(H(w;G;v 1 ;v 2 )). By setting z 0 :=z in the problem forC l (w eq ), it is straightforward to see that it is the same problem as that forC u (w eq ). Settingc :=c l forC l (w eq ), andc :=c u forC u (w eq ), the two problem instances in (4.28) can be uniformly written as follows. C(w eq ) = max z2R;w2R E z subject to w l i ww u i z c i (w i ) f i (w;a v1v2 ) 8i2E H(w;G;v 1 ;v 2 ) =w eq (4.29) We begin by focusing on solving the following simplied version of the reduction problem (4.29): g(w eq ) = max 2R;w2R E subject to w l i w i w u i 8i2E i (w i ) 8i2E H(w;G;v 1 ;v 2 ) =w eq (4.30) for given w eq 2R(H(w;G;v 1 ;v 2 )) and functions i : R >0 ! R >0 , i2E, representing the second set of inequalities in (4.29). Note that the second set of inequalities in (4.30) are separable across links, whereas they are not in (4.29). This simplication will be shown to be lossless. We shall then devise a methodology that sequentially uses solution to (4.30) for parallel and serial networks, to obtain an iterative scheme to solve (4.29). 91 4.6.2 Input-output Properties of the Simplied Version of the Reduction Problem In order to develop the sequential procedure, we interpret (4.30) to be dening an output function g(w eq ) with link level functions i (w i ), i2E as input. We next introduce a property which will be shown to be invariant from the input functions to the output function, and will be helpful to compute the function g(w eq ) specied by (4.30). Denition 14 (S 0 function). A function : [x l ;x u ] R! R is called aS 0 function if it is continuous, and there exist x2 [x l ;x u ] and x2 [ x;x u ] such that (x) is strictly increasing over [x l ; x], constant over [ x; x], and strictly decreasing over [ x;x u ]. We shall sometimes refer to x and x as rst and second transition points (w.r.t.S 0 property), respectively, of (x). Figure 4.5 provides an example of aS 0 function. It is easy to see that aS 0 function is also quasiconcave, but the converse is not true in general. (x) 0 x x l x x x u max Figure 4.5: A sampleS 0 function. Proposition 14. Consider a network consisting of graph topologyG = (V;E), lower and upper bounds on link weights w l 2 R E >0 and w u 2 R E >0 respectively and supply and demand node v 1 ;v 2 2V respectively. If the equivalent weight functionH(w;G;v 1 ;v 2 ) is strictly monotone with respect to w for this network, and i (w i ) is aS 0 function for all i2E, then the g(w eq ) function dened by (4.30) is also aS 0 function. 92 Proof. In general,H(w) is not one-to-one, i.e., there could exist ~ w; ~ ~ w2 [w l ;w u ], ~ w6= ~ ~ w, such that H( ~ w) =H( ~ ~ w). However, the strict monotonicity ofH(w) implies that the only feasible points of (4.30) for w l eq :=H(w l ) and w u eq :=H(w u ) are (min i2E i (w l i );w l ) and (min i2E i (w u i );w u ) respectively, and that w eq 2 [w l eq ;w u eq ] for all w 2 [w l ;w u ]. Hence g(w l eq ) = min i2E i (w l i ) and g(w u eq ) = min i2E i (w u i ). Let g max := max weq2[w l eq ;w u eq ] g(w eq ), then g(w l eq ) =:g l g max and g(w u eq ) =:g u g max . Motivated by this, and with the objective of ultimately provingS 0 property ofg(w eq ), we construct inverse functions of g(w eq ) over [g l ;g max ] and [g u ;g max ]. We denote these inverse functions as ^ g + : [g l ;g max ]! [w l eq ;w u eq ] and ^ g : [g u ;g max ]! [w l eq ;w u eq ], respectively. We construct these inverses as compositions: ^ g + (x) =H! + (x) ^ g (x) =H! (x) (4.31) whereH is the equivalent weight function from (4.18), and ! + : [g l ;g max ]! [w l ;w u ] and ! : [g u ;g max ]! [w l ;w u ] are dened as: for all i2E, ! + i (x) := 8 > > < > > : w l i if x i (w l i ) minfw i : i (w i ) =xg if x> i (w l i ) ! i (x) := 8 > > < > > : w u i if x i (w u i ) maxfw i : i (w i ) =xg if x> i (w u i ) (4.32) It is easy to see that g max = min i2E max wi2[w l i ;w u i ] i (w i ) (4.33) 93 Combining (4.33) with the fact that i is aS 0 function for all i2E, the denitions in (4.32) imply that, for all i2E, ! + i (x)2 [w l i ; w i ] [w l i ;w u i ] & x i (! + i (x)); 8x2 [g l ;g max ] ! i (x)2 [ w i ;w u i ] [w l i ;w u i ] & x i (! i (x)); 8x2 [g u ;g max ] (4.34) where we refer to Denition 14 for notations w and w. Moreover, since i (w i )2S 0 , for all i2E, ! + i is nondecreasing and ! i is nonincreasing, and, it is easy to see that, for every x2 [g l ;g max ], there exists at least one i2E such that ! + i (x) is strictly increasing, and that, for every x2 [g u ;g max ], there exists at least one i2E such that ! i (x) is strictly decreasing. This combined with the strictly increasing property ofH(w) implies that ^ g + : [g l ;g max ]! [w l eq ;w u eq ] and ^ g : [g u ;g max ]! [w l eq ;w u eq ] are strictly increasing and strictly decreasing bijections, respectively. Moreover, it is easy to see that w l eq ^ g + (g max ) ^ g (g max )w u eq , where the middle inequality follows from (4.31), (4.32), and the strict monotonicity ofH. In the remainder of the proof, our strategy for proving thatg(w eq ) is aS 0 function is as follows: we show that (i) ^ g + is the inverse of g(w eq ) over w eq 2 [w l eq ; ^ g + (g max )], (ii) ^ g is the inverse of g(w eq ) over w eq 2 [^ g (g max );w u eq ], and (iii) g(w eq ) g max over w eq 2 [^ g + (g max ); ^ g (g max )]. In particular, ^ g + (g max ) and ^ g (g max ) will play the role of x and x (cf. Denition 14) in proving that g(w eq ) is aS 0 function. The proof for (i) and (ii) are similar, and hence we provide details only for (i). In order to show that ^ g + is the inverse of g(w eq ) over w eq 2 [w l eq ; ^ g + (g max )], we show that g(^ g + (x)) = x for all x2 [g l ;g max ]. In order to show this, we show that, for all x2 [g l ;g max ], (x;! + (x)) is the unique optimizer for (4.30) corresponding to w eq = ^ g + (x). (4.31) and (4.34) readily imply that (x;! + (x)) is feasible for (4.30). Therefore, for all x2 [g l ;g max ], g(^ g + (x))x (4.35) 94 Consider an arbitrary ~ w2 [w l ;w u ] such that ~ w6= ! + (x) andH( ~ w) = ^ g + (x) =H(! + (x)). It is sucient to show that ~ <x for all ~ such that (~ ; ~ w) is feasible to (4.30). Forx =g l , by denition ! + (x) = w l andH(! + (x)) = w l eq . Strict monotonicity ofH implies that (x;! + (x)) is the only feasible point and hence the unique optimizer of (4.30). For allx2 (g l ;g max ] 3 , it is clear from the denition of g l and g max that the setfi2Ejx> i (w l i )g is not empty. Since ~ w k w l k =w + k (x) for allk2fi2Ejx i (w l i )g, if ~ w k ! + (x) for allk2fi2Ejx> i (w l i )g, strict monotonicity ofH impliesH( ~ w) > H(! + (x)). That is to say, in order to satisfy H( ~ w) = H(! + (x)) and ~ w6=! + (x), there is at least one k2fi2Ejx> i (w l i )g such that ~ w k <! + k (x) w k . Using this along with the fact that k is aS 0 function, and hence k is strictly increasing in [w l k ; w k ], we get k ( ~ w k ) < k (! + k (x)) = x, where the equality is due to the implication of k (w l k ) < x in (4.32). Therefore, the last inequality constraint in (4.30) implies that ~ < x for all feasible ~ . In other words, (g(w eq );! + (g(w eq ))) is the unique solution to (4.30) for all w eq 2 [w l eq ; ^ g + (g max )]. Similar result is true for w eq 2 [^ g (g max );w u eq ]. Recall that g max is the maximum value of g(w eq ) over all w eq . Therefore, in order to show that g(w eq ) g max for all w eq 2 [^ g + (g max ); ^ g (g max )], it suces to show that, for every w eq 2 [^ g + (g max ); ^ g (g max )], there exists a ~ w2 [w l ;w u ] such that (g max ; ~ w) is feasible for (4.30). Since, by denition in (4.31),H(! + (g max )) = ^ g + (g max ) andH(! (g max )) = ^ g (g max ), continuity and monotonicity of H implies that, for all w eq 2 [^ g + (g max ); ^ g (g max )], hence there exists ~ w 2 [! + (g max );! (g max )] satisfying H( ~ w) = w eq . Moreover, the S 0 property of i implies that g max i (w i ) for all w i 22 [! + i (g max );! i (g max )]. This shows that (g max ; ~ w) is feasible for (4.30). Finally, the continuity of g(w eq ) follows from the continuity of the inverse functions ^ g + and ^ g , which in turn follows from the continuity ofH from (4.18), and continuity of! + and! from (4.32) implied by the continuity of i 's beingS 0 functions. 3 It is possible that gmax =g l . In this case, considering the case x =g l is sucient. 95 The solution to (4.30) is not unique in general for an arbitrary w eq . However, it is unique for w eq within a certain range, as shown in the above proof and summarized in Remark 40. Remark 40. (a) (4.30) has unique solution (g(w eq );! + (g(w eq ))) and (g(w eq );! (g(w eq ))) for any w eq 2 [w l eq ; ^ g + (g max )] and w eq 2 [^ g (g max );w u eq ], respectively. (b) ! + (g(w eq )) is nondecreasing w.r.t. w eq for w eq 2 [w l eq ; ^ g + (g max )], since by denition ! + is nondecreasing function andS 0 property of g(w eq ) implies that g(w eq ) is strictly increasing for w eq 2 [w l eq ; ^ g + (g max )]. ! (g(w eq )) is nondecreasing w.r.t. w eq for w eq 2 [^ g (g max );w u eq ] due to similar reason. The proof of Proposition 14 implies that the solution to (4.30) is given by: g(w eq ) = 8 > > > > > > < > > > > > > : inv ^ g + (w eq ) w l eq w eq < ^ g + (g max ) g max ^ g + (g max )w eq ^ g (g max ) inv ^ g (w eq ) ^ g (g max )<w eq w u eq (4.36) where inv ^ g + and inv ^ g are the inverses of ^ g + and ^ g , respectively, as dened in (4.31), g max is dened in (4.33). Proposition 14 implies thatg is continuous. However, it may not be dierentiable in general. Let g 0 (w eq ) := lim 4weq"0 g(w eq +4w eq )g(w eq ) 4w eq ; g 0 (w + eq ) := lim 4weq#0 g(w eq +4w eq )g(w eq ) 4w eq (4.37) be the left and right derivatives, respectively. We provide derivation for explicit expressions of these derivatives in the appendix. These expressions are used in Sections 4.6.3 and 4.6.4 to provide an explicit solution for series and parallel networks. 96 4.6.3 Series Networks In a series network,jVj =jEj + 1. A series network consisting of three links is shown in Fig. 4.6. v1 v2 v3 v4 e1 e2 i3 Figure 4.6: A three link series network Consider a series network with n + 1 nodes numbered v 1 ;:::;v n+1 such that (v j ;v j+1 )2E for all j 2 [n], and link weights w 2 R n >0 . As already shown in Example 5, the equivalent weight function between v 1 and v n+1 is given byH(w) = P n i=1 (1=w i ) 1 . Moreover, the ow on any link i2 [n] is equal to one when a unit ow enters at node v 1 and leaves at v n+1 , i.e., f i (w;a v1vn ) = 1. Therefore, (4.29) can be simplied for a series network as (4.38), which gives the equivalent capacity function between nodes v 1 and v n+1 . C(w eq ) = max z2R;w2R n >0 z subject to w l i w i w u i zc i (w i ); i2 [n] n X i=1 1 w i ! 1 =w eq (4.38) For constant link capacities, i.e.,c i (w i )c i ,i2E, then it is easily to see thatC(w eq ) = min i2[n] c i . For weight-dependent capacities, we now establish a functional property ofC(w eq ), which is a stronger version of theS 0 property dened in Denition 14. Denition 15. A function : [x l ;x u ] R >0 ! R is called aS 1 function if it is aS 0 function (cf. Denition 14), and if there exists a x o 2 [x l ; x] such that min@ (x)> (x)=x for all x2 x l ;x o and min@ (x) = (x)=x for allx2 [x o ; x), where x is the rst transition point, w.r.t.S 0 property, @ (x) denotes the set of subgradients of (x). We shall sometimes refer to x o and x as rst and second transition points (w.r.t.S 1 property), respectively, of (x). 97 Remark 41. Note that, in Denition 15, we allow x o = x, in which case, the only requirement for aS 0 function to beS 1 is that min@ (x)> (x)=x for all x2 x l ; x . Denition 15 clearly implies that, if (x) is aS 1 function, then it is also aS 0 function. The next result extends theS 0 implication also to (x)=x. Lemma 15. If : [x l ;x u ] R >0 ! R >0 is aS 1 function, then (x)=x : [x l ;x u ] R >0 ! R >0 is aS 0 function. Proof. Let ~ (x) := (x)=x. The continuity of ~ (x) follows from that of (x). Then, the left and right derivative of ~ (x) are, respectively, given by: ~ 0 (x ) = 0 (x )x (x) x 2 ; ~ 0 (x + ) = 0 (x + )x (x) x 2 (4.39) Note that these two derivatives completely specify the set of subgradients of ~ (x). Since (x) is aS 1 function, we have min@ (x) > (x)=x for all x2 [x l ;x o ). Therefore, (4.39) implies that ~ 0 (x ) and ~ 0 (x + ) are both strictly positive, and hence ~ (x) is strictly increasing over [x l ;x o ). For x2 (x o ; x), (4.39) implies that ~ 0 (x ) = ~ 0 (x + ) = 0, i.e., ~ (x) is constant. Since (x) is also aS 0 function, 0 (x ) and 0 (x + ) are both nonpositive for x2 ( x;x u ]. Therefore, (4.39) implies that ~ 0 (x ) and ~ 0 (x + ) are both strictly negative, and hence ~ (x) is strictly decreasing over ( x;x u ]. Collecting these facts, we establish that ~ (x) is aS 0 function. We conclude the proof by emphasizing that the transition points required for theS 0 property of the ~ function are the points corresponding to x o and x used in specifying theS 1 property of (cf. Denition 15). Remark 42. The proof of Lemma 15 implies that the rst and second transition points, w.r.t.S 1 property, of (x) i.e., x o and x, are the rst and second transition points, w.r.t. S 0 property, of (x)=x, respectively. Lemma 16. Consider a DC network consisting of series graph topologyG = (V;E), whereV = fv 1 ;:::;v n+1 g andE =f(v 1 ;v 2 );:::; (v n ;v n+1 )g, and lower and upper bounds on link weights 98 w l 2 R n >0 and w u 2 R n >0 respectively. If the link capacity functions c i (w i ) areS 1 for all i2 [n], then the the equivalent capacity function between v 1 and v n+1 , as given by (4.38), is also aS 1 function. Proof. Sincec i areS 1 functions for alli2 [n], by denition, they are alsoS 0 functions. Therefore, Proposition 14 implies thatC(w eq ), as given by (4.38), is also aS 0 function. In order to prove that C(w eq ) is aS 1 function, we need to show that there existsw o eq 2 [w l eq ; w eq ] such that min@C(w eq )> C(w eq )=w eq for w eq 2 [w l eq ;w o eq ) and min@C(w eq ) =C(w eq )=w eq for w eq 2 [w o eq ; w eq ). We now show that there exist w o eq 2 [w l eq ; w eq ] such thatC 0 (w + eq ) >C(w eq )=w eq for w eq 2 [w l eq ;w o eq ) andC 0 (w + eq ) =C(w eq )=w eq for w eq 2 [w o eq ; w eq ). Similar results hold true forC 0 (w eq ). Since min@C(w eq ) = minfC 0 (w eq );C 0 (w + eq )g, this then completes the proof. Noting the expression for the equivalent weight function in (4.38), we get that @H(w) @w i = 1 w 2 i n X i=1 1 w i ! 2 = w 2 eq w 2 i : Substituting into (A.8), we get C 0 (w + eq ) = 0 @ X i2 ~ K + (C(weq)) w 2 eq c 0 i (w + i )w 2 i 1 A 1 w=! + (C(weq)) C(w eq ) w 2 eq 0 @ X i2 ~ K + (C(weq)) 1 w i 1 A 1 w=! + (C(weq)) C(w eq ) w 2 eq 1 w eq 1 = C(w eq ) w eq where the rst inequality follows from the fact that, sincec i (w i )2S 1 ,c 0 i (w + i )w i min@c i (w i )w i c i (w i ), and by denition, c i (! + i (C(w eq ))) =C(w eq ) for all i2 ~ K + (C(w eq )), and it is equality if and only if ! i (C(w eq )) w o i for all i2 ~ K + (C(w eq )). The second inequality is equality if and only if ~ K + (C(w eq )) = E. If for some ~ w eq 2 [w l eq ; w eq ], both inequalities are equalities, i.e., ! i (C(w eq )) w o i for all i2E and ~ K + (C(w eq )) =E, thenC 0 (w + eq ) =C(w eq )=w eq holds for all 99 w eq 2 [ ~ w eq ; w eq ]. This is because of the nondecreasing property of function !(C()) (cf. Remark 40(b)) and ~ K + (C()) (by denition). Therefore, there exists w o eq 2 [w l eq ; w eq ] such that the both the inequalities are strict for w eq 2 [w l eq ;w o eq ) and is equality for w eq 2 [w o eq ; w eq ). 4.6.4 Parallel Networks We now focus on networks with parallel graph topology, i.e., whenG = (V;E), whereV =fv 1 ;v 2 g, and all the links inE are from v 1 to v 2 . An example is shown in Fig. 4.7. v1 v2 e1 e2 Figure 4.7: A two link parallel network Consider a parallel network with n links from node v 1 to node v 2 , and link weights w2 R n >0 . As already shown in Example 5, the equivalent weight function between v 1 and v 2 is given by H(w) = P n i=1 w i . With unit supply and demand onv 1 andv 2 , the ow on linke i isf i =w i =w eq . Substituting f i = w i =w eq into (4.29) and letting ~ z = z=w eq , the equivalent capacity function between nodes v 1 and v 2 takes the following simple form: C(w eq ) w eq = max ~ z2R;w2R n >0 ~ z subject to w l i w i w u i ~ zc i (w i )=w i 8i2 [n] n X i=1 w i =w eq (4.40) The following result is the equivalent of Lemma 16 for parallel networks. Lemma 17. Consider a DC network consisting of parallel graph topology G = (V;E), where V =fv 1 ;v 2 g and all the links inE are from v 1 to v 2 , and lower and upper bounds on link weights 100 are w l 2 R n >0 and w u 2 R n >0 respectively. If the link capacity functions c i (w i ) areS 1 for all i2 [n], then the the equivalent capacity functionC(w eq ) between v 1 and v 2 , as given by (4.40), is also aS 1 function. Proof. Since c i (w i ) areS 1 functions for all i2 [n], Lemma 15 implies that c i (w i )=w i areS 0 functions and Remark 42 implies that the second transition point ofc i (w i )=w i w.r.t.S 0 property is the rst transition point w i ofc i (w i ) w.r.t.S 0 property and max w l i wiw u i c i (w i )=w i =c i ( w i )= w i . Proposition 14 and its proof then implies thatg(w eq ) :=C(w eq )=w eq is aS 0 function andg(w l eq ) = min i2E c i (w l i )=w l i , g(w u eq ) = min i2E c i (w u i )=w u i , and g max = min i2E c i ( w i )= w i . Let w eq and w eq denote the rst and second transition points, respectively, w.r.t.S 0 property, forg(w eq ). In order to establishS 1 property ofC(w eq ), we look at its left and right derivatives: C 0 (w + eq ) =g(w eq ) +w eq g 0 (w + eq ); C 0 (w eq ) =g(w eq ) +w eq g 0 (w eq ) (4.41) Therefore, combining (4.41) with S 0 property of g(w eq ), we get that: C 0 (w + eq ) > g(w eq ) = C(w eq )=w eq for all w eq 2 [w l eq ; w eq ); g 0 (w eq ) 0, and henceC 0 (w eq ) = g(w eq ) =C(w eq )=w eq , for w eq 2 ( w eq ; w eq ). Moreover, using (A.8), for w eq 2 ( w eq ;w u eq ], g 0 (w + eq ) = 0 @ X i2K (g(weq)) 1 (c i (w + i )=w i ) 0 1 A 1 w=! (g(weq)) = 0 @ X i2K (g(weq)) w i c 0 i (w + i )c i (w i )=w i 1 A 1 w=! (g(weq)) g(w eq ) 0 @ X i2K (g(weq)) w i 1 A 1 w=! (g(weq)) g(w eq ) n X i=1 w i ! 1 w=! (g(weq)) =g(w eq )=w eq (4.42) 101 where the rst inequality follows from the fact that, by denition ofK ,c i (! i (g(w eq )))=! i (g(w eq )) = g(w eq ) for all i2K (g(w eq )), and c 0 i (w + i ) w=! (g(weq)) 0 8w eq 2 ( w eq ;w u eq ] (4.43) (4.43) is because of the following. Due to theS 0 property of functiong(w eq ),g(w eq )2 [g(w u eq ;g max ) for w eq 2 ( w eq ;w u eq ]. Remark 42 implies that the second transition point, w.r.t. S 0 property, of c i (w i )=w i is equal to the rst transition point, w i , w.r.t. S 0 property, of c i (w i ). This, combined with the second equation in (4.34), further implies that! i (g(w eq )) w i forg(w eq )2 [g(w u eq ;g max ) i.e., w eq 2 ( w eq ;w u eq ]. Thereafter, theS 0 property of c i implies (4.43). Now we consider conditions for (4.43) taking equalities. The second inequality in (4.42) takes equality if and only ifK =E. ConsideringK =E, the rst inequality in (4.42) takes equality for w eq w eq ^ g (max i2E c i ( w i )= w i ). Furthermore,K (g()) is nonincreasing and Remark 40 (b) implies that ! i (g(w eq )) is nondecreasing for w eq 2 ( w eq ;w u eq ]. Therefore, there exists ~ w eq 2 [ w eq ;w u eq ] such that for w eq w eq ~ w eq , g 0 (w + eq ) =g(w eq )=w eq and henceC 0 (w + eq ) = 0 from (4.41); and for ~ w eq <w eq w u eq ,g 0 (w u eq )<g(w eq )=w eq and henceC 0 (w + eq )< 0 from (4.41). One can show similar properties also forC 0 (w eq ), thereby proving thatC(w eq ) is aS 1 function. We now provide a characterization of the equivalent capacity function for a parallel network whose links have constant, i.e., weight-independent, capacities, in Example 6. This example generalizes our earlier work [4], where we compute only the maximum of the equivalent capacity function for parallel networks as solution to an optimization problem. Example 6 (Equivalent capacity for parallel networks with weight-independent link capacities). Consider a parallel network withn links from nodev 1 to nodev 2 . Let the lower and upper bounds on link weights be w l 2 R n >0 and w u 2 R n >0 respectively, and let the link capacities be c i > 0, i2 [n]. Then, for every i2 [n], c i is aS 1 function, with w l i = w o i = w i . LetC(w eq ) be the equivalent capacity function and hence g(w eq ) :=C(w eq )=w eq is the solution to (4.40) for this 102 network. Lemma 17 implies thatC(w eq ) and g(w eq ) areS 1 andS 0 functions, respectively. Let w o eq , w eq and w eq be the rst and second transition points, w.r.t. S 1 property, and the second transition point, w.r.t.S 0 property, ofC(w eq ), respectively. Remark 42 implies that w o eq and w eq are the rst and second transition points, w.r.t.S 0 property, of g(w eq ), respectively. With w l eq = P n i=1 w l i and w u eq = P n i=1 w u i , it is easy to see that w o eq =w l eq , g max =g(w l eq ) = min i2[n] c i =w l i (4.44) and g(w u eq ) = min i2[n] c i =w u i . SinceH(w) = P n i=1 w i , the inverse function in (4.31) satises ^ g (x) = P n i=1 ! i (x) for all x2 min i2[n] c i =w u i ; min i2[n] c i =w l i . Indeed, ! i (x) can be explicitly written as ! i (x) = minfc i =x;w u i g. Therefore, ^ g (x) can be written as: w eq = ^ g (x) = 1 x X i:w u i >ci=x c i + X i:w u i ci=x w u i (4.45) Note ^ g (x) is decreasing. By denition, w eq = ^ g (g max )2 [w l eq ;weq u ]. It is straightforward that ^ g (g max ) w u eq , and (4.44) implies that c i =g max w l i for all i2 [n], and hence ^ g (g max ) P n i=1 w l i =w l eq . For w eq 2 w l eq ; w eq , g(w eq ) =g max . For w eq 2 w eq ;w u eq , the monotonicity of ^ g implies thatfijw u i >c i =g(w eq )g =fijw eq < ^ g (c i =w u i )g. Therefore, (4.45) implies that g(w eq ) = inv ^ g (g(w eq )) = P i:weq<^ g (ci=w u i ) c i w eq P i:weq^ g (ci=w u i ) w u i (4.46) Based on these calculations, the equivalent capacity function is characterized as follows: For w eq 2 w l eq ; w eq , C(w eq ) =w eq g max =w eq min i2[n] c i =w l i (4.47) which is a linear function with slope min i2[n] c i =w l i . 103 If w u i > c i =g max for all i2 [n], i.e., g max > max i2[n] c i =w u i , then (4.45) implies that ^ g (x) = P n i=1 c i =x for all x2 max i2[n] c i =w u i ;g max . Equivalently, for allw eq 2 w eq ; ^ g (max i2[n] c i =w u i ) , we getg(w eq ) = P n i=1 c i =w eq , and hence C(w eq ) =w eq g(w eq ) = n X i=1 c i (4.48) It is straightforward to see that w eq = maxf^ g (g max ); ^ g (max i2[n] c i =w u i )g2 [ w eq ;w u eq ]. Finally, for w eq 2 w eq ;w u eq C(w eq ) =w eq g(w eq ) = w eq w eq P i:weq^ g (ci=w u i ) w u i X i:weq<^ g (ci=w u i ) c i (4.49) In summary, (4.47), (4.48) and (4.49) completely characterize the equivalent capacity function for parallel networks, and an illustration is provided in Fig. 4.8. Every point in the curve in Fig. 4.8 (w eq ;C(w eq )) corresponds to an optimal solution of weight w to (4.29) for a parallel network with constant capacities on all the links and equivalent weight w eq . In general, this optimal solution is not unique. However, since w eq is the second transition point of function c i =w i w.r.t.S 0 property in this case, Remark 40 implies that the optimal solution is unique and nondecreasing for w eq 2 [ w eq ;w u eq ]. This is summarized in Remark 43. As shown in Fig. 4.8, , being the maximum of functionC(w eq ), can be computed explicitly, which in turn implies that the margin of robustness for parallel networks can be computed explicitly. Remark 43. For a parallel network with constant capacity on each link,! (g(w eq )) = minfc i =g(w eq );w u i g is the unique optimal solution to (4.40) and is nondecreasing for w eq 2 [ w eq ;w u eq ], whereg(w eq ) is shown in (4.46). 104 C(w eq ) 0 w eq w l eq w eq w eq w u eq Figure 4.8: Equivalent capacity function for a parallel network consisting of links with constant capacities. 4.6.5 Computing Margin of Robustness for Tree Reducible Networks Using Lemmas 16 and 17, and Denition 12, one sees that, for parallel and series networks, the equivalent capacity functions areS 1 functions. Indeed, one can use Lemmas 16 and 17 recursively to showS 1 property for the equivalent capacity function for a broader class of networks. In order to see this, consider the network illustrated in Figure 4.4 wherep v1 =p v4 > 0, andp v2 =p v3 = 0. Lemma 17 (and Example 6) imply that the capacity of an equivalent link, say e 4;5 corre- sponding to links e 4 and e 5 , is weight-dependent, and the capacity function for the equivalent link e 4;5 is aS 1 function. Lemma 16 then implies that the equivalent capacity function for the equivalent link e 2;4;5 corresponding to links e 2 and e 4;5 is also aS 1 function. The same property also holds true for equivalent linke 1;3 corresponding toe 1 ande 3 . Finally, the equivalent capacity function between nodes v 1 and v 4 corresponding to links e 1;3 and e 2;4;5 can also be shown to be S 1 by Lemma 17. Specic numerical examples are provided in Section 4.8.2. In summary, for the network in Figure 4.4, theS 1 property is invariant from the capacities at individual link to equivalent capacity functions associated with intermediate equivalent parallel and series reduc- tions, nally to the one associated with the equivalent link corresponding to the entire network. SinceS 1 impliesS 0 , (4.36) then gives a computationally ecient recursive procedure to compute the equivalent capacity function of the entire network in terms of capacities of individual links. Recalling that, for a given network, computing the equivalent capacity is the same as solving the 105 reduction problem, the above procedure can also be used to solve the reduction problem for tree reducible network. Theorem 4. Consider the reduction problem (4.29) on a link reducible networkG = (V;E), with lower and upper bounds on link weights asw l 2 R n >0 andw u 2 R n >0 respectively. Then its solution functionC(w eq ) is aS 1 function. Finally, the margin of robustness for a tree reducible network can be computed using the multilevel approach from Section 4.5 as follows. Recall from Section 4.5.3 that the multilevel formulation consists of multiple reduction problems, and a single terminal problem. Theorem 4 along with (4.36) provides an explicit solutions to the reduction problems. Since the original network is tree reducible, the terminal problem is over a tree. Even though this tree has weight dependent capacity functions on the links, Proposition 10 can be used to solve the terminal problem, and hence gives the margin of robustness. Specically, for a tree networkG = (V;E) with link capacity functionsC l i (w i ) andC u (w i ), i2E, one can use Proposition 10 with c l i := min wi C l (w i ) and c u i := max wi C u (w i ) to compute the margin of robustness. Remark 44. Proposition 10 illustrates how Theorem 4 is useful to provide analytical results on the margin of robustness for tree reducible networks. For general networks, on top of the computational gains from the multilevel formulation (cf. Remark 38(c)), Theorem 4 provides potentially additional computational gains. This is because, if the lower level problem involves series or parallel sub-networks, then Theorem 4 provides an analytical solution to that lower level problem. 106 4.7 Decentralized Control Policies From a feedback control point of view, we re-state the weight control problem as follows. The network responds to disturbances by changing the weights dynamically, which in turn also induces dynamics in the line ows due to (2.10). This dynamics can be written as: _ w i (t) =u i (W(t);F(t); ) (4.50) whereW(t) =fw() :2 [0;t]g, andF(t) =ff(w()) :2 [0;t]g are the historical values of line weights and ows, respectively, through time t. The weight control in (4.50) is required to satisfy the following constraints 0w l ww u (4.51) wherew l andw u are the lower and upper limits, respectively, for the operation range of the weight controller. The dynamical system (4.50) will be called feasible under a given disturbance and control policy u if (4.1) is satised asymptotically. In Sections 4.2-4.6, we described various approaches to compute the margin of robustness for a centralized control policy (which has information about link ows and weights, disturbances, as well as link ow capacities and operational range of weights), and we recall that this is an upper bound for any control policy. In this section, we analyze the robustness of decentralized policies, for parallel networks, that do not require information about the disturbance or link capacities, and moreover weight bounds information is private to each link. 4.7.1 Optimal Centralized Control for Parallel Networks Consider a parallel network consisting of n links from the supply node to the demand node. Let the magnitude of supply/demand be equal to 0. We rst specialize the margin of robustness computation to this setting. Since Remark 35 (d) implies that the margin of robustness for a 107 parallel network is related to the maximum of equivalent capacity over all feasible equivalent weights, Example 6 implies that the margin of robustness for a parallel network is given by: = max w l eq ww u eq C(w eq ) =g max ^ g (g max ) =g max X i:w u i <ci=gmax w u i + X i:w u i ci=gmax c i (4.52) where we recall g max = min i2[n] c i =w l i = 1=(max i w l i =c i ) and other notations used in (4.52) from Example 6. Moreover, an optimizer in (4.52) is w opt eq = ^ g (g max ) = P n i=1 minfc i =g max ;w u i g, with the corresponding link weights given by w opt i =w i (g max ) = minfc i max i w l i =c i ;w u i g: (4.53) Indeed, for a parallel network, since all disturbances are of multiplicative type, and the link ows for a parallel network are explicitly given by f i = w i =( P n j=1 w j ), it is easy to see from (4.8), as is also shown in [4, Section III-B], that the margin of robustness for a parallel network is equal to the following: max 2R;w2R n subject to w l ww u f i = w i P n j=1 w j 8i2f1;:::;ng 0fc (4.54) Remark 43 implies that w opt dened in (4.53) is the minimal optimal solution to (4.54), as summarized in Remark 45. Remark 45. For a n link parallel network with constant capacities, w opt dened in (4.53) is the minimal optimal solution to (4.54), chapter ~ w i w opt i for alli2 [n] and all optimal solution ~ w of (4.54). 108 The decentralized control policies considered in this chapter are partially inspired by the implication of Proposition 5 for a parallel network that, the decrease in the weight of a link leads to a decrease in ow on that link but an increase in ow on the parallel links connecting the same nodes. While this implication of Proposition 5 does not necessarily extend to the case when multiple links change weights simultaneously, we identify conditions under which the decentralized control policies considered in this chapter are provably robust, chapter their margin of robustness is equal to the quantity computed in (4.52), or equivalently the optimal value of (4.54). We now state two control policies and analyze their robustness within the dynamical framework of (4.50). 4.7.2 A Memoryless Controller Consider the following control policy: for all i2 [n] u 1 i (w i (t);f i (t)) = 8 > > < > > : i f i (t)>c i & w i (t)>w l i 0 otherwise (4.55) where i > 0 is an arbitrary constant denoting the rate of decrease of w i . Sincew(t) is nonincreasing underu 1 and is lower bounded byw l , the dynamics in (4.50) always converges to an equilibrium under u 1 . This is formally stated next. Lemma 18. Consider a DC networkG = (V;E) with lower and upper bounds on link weights as w l 2 R n 0 and w u 2 R n >0 , respectively. Then, for every 2 R E >0 , and w(0)2 [w l ;w u ], there exists w 2 [w l ;w(0)] [w l ;w u ], such that, under the dynamics in (4.50) with the controller u 1 in (4.55), lim t!+1 w(t) =w . 4 4 Notice that Lemma 18 is stated for a general, chapter not necessarily parallel, networks. 109 The ow f(w ) at the equilibrium w established in Lemma 18 may not necessarily satisfy f(w )2 [0;c] under all supply/demand . We next characterize the upper limit on this quantity and compare it with respect to the upper bound . Unless otherwise stated explicitly, in this section, we adopt the shorthand notation min i and max i to imply minimum and maximum, respectively, over [n]. Let r i :=w i (0)=c i ; i2 [n] (4.56) Without loss of generality, label the links in increasing order of r i , chapter r 1 r 2 ::: r n . Let r := max i w l i c i = 1 g max (4.57) Sincew(0)w l ,r n = max i w i (0)=c i max i w l i =c i and thereforer r n . Let k := minfj2 [n]jr j r g. This implies that r k1 <r r k (4.58) Consider the following functions: V k := 1 r k k1 X i=1 w i (0) + n X i=k c i ; k2 [n]; V := 1 r k1 X i=1 w i (0) + n X i= k c i (4.59) (4.59) implies that V 1 = P n i=1 c i and, when r r 1 , k = 1 and V = P n i=1 c i = V 1 . Since r k r k+1 , V k is nonincreasing in k: V k+1 = 1 r k+1 k X i=1 w i (0) + n X i=k+1 c i 1 r k k1 X i=1 w i (0) + w k (0) r k + n X i=k+1 c i =V k Similarly, we can show that V k V <V k1 (4.60) 110 Theorem 5. Consider a parallel DC network consisting of n links, with lower and upper bounds on link weights as w l 2 R n >0 and w u 2 R n >0 , respectively, link capacities c2 R n >0 , and sup- ply/demand with magnitude 0. Then, for every 2 R n >0 and w(0)2 [w l ;w u ], there exists w 2 [w l ;w(0)] [w l ;w u ], such that, under the dynamics in (4.50) with the controller u 1 in (4.55), w(t) monotonically converges to w 2 [w l ;w(0)] [w l ;w u ]. Moreover, (i) if 2 [0;V n ], then w =w(0) and f(w )2 [0;c]; (ii) if 2 (V n ;V ] then w i = 8 > > < > > : w i (0) 1i ^ k 1 ^ rc i ^ kin (4.61) where ^ k := minfj2 [n]jV j g, ^ r := 8 > > < > > : P^ k1 i=1 w i (0) P n i= ^ k c i <V 1 r 1 =V 1 =V (4.62) and f(w )2 [0;c]; (iii) if >V then f(w ) = 2 [0;c] where V j and V are as dened in (4.59). Proof. Monotonic convergence ofw(t) follows from Lemma 18. If2 [0;V n ], then the initial ow on link i2 [n] is given by: f i (0) = w i (0) P n j=1 w j (0) w i (0) P n j=1 w j (0) P n j=1 w j (0) r n c i chapter the system is feasible at t = 0. Therefore, if V n , then u 1 (t) 0, and hence the equilibrium is w =w(0). This establishes part (i) in the theorem. If>V 1 = P n i=1 c i , then it is trivially f(w) = 2 [0;c] for any w. Hence V 1 is considered in the following proof. Moreover, we emphasize that since V V 1 ,<V 1 is satised in case (ii) if 111 V <V 1 . The denition of ^ k implies thatV ^ k <V ^ k1 . This, combined with (4.62) and (4.59), implies that ^ r 0, and hence w 0, for all 2 (V n ;V 1 ). (4.59) and (4.56) imply that 1 r ^ k ^ k1 X i=1 w i (0) n X i= ^ k c i < 1 r ^ k1 ^ k1 X i=1 w i (0) Therefore, the denition of ^ r in (4.62) implies that, ^ rr ^ k ; 82 (V n ;V 1 ] (4.63) In writing (4.63), we used the fact that, when = V 1 , then ^ k = 1, and hence r ^ k = r 1 = ^ r. Additionally, ^ r>r ^ k1 ; 82 (V n ;V 1 ) (4.64) We now establish the following claims: with w as given in (4.61), (I) for V n <V 1 [w ;w(0)] is positively invariant under (4.50) with controller u 1 ; (II) for 2 (V n ;V ], (a) w 2 [w l ;w(0)], (b) w is the only equilibrium in [w ;w(0)], (c) f(w )2 [0;c] (III) for V <V 1 , f(w) = 2 [0;c] for all w2 [w ;w(0)]\ [w l ;w(0)]. (I) and (II) establish part (ii) of the theorem, whereas (I) and (III) establish part (iii). Proof of (I): Since w(t) w(0) for all t 0 under controller u 1 , it suces to show that w(t) w for all t 0 under u 1 . Assume by contradiction that this is not true. Continuity 112 of w(t) then implies that there exists t 1 > 0 and ^ i2 [n] such that w(t) w for all t2 [0;t 1 ], w ^ i (t 1 ) =w ^ i and _ w ^ i (t 1 )< 0. The latter implies that f ^ i (t 1 )>c ^ i . However, f ^ i (t 1 ) = w ^ i (t 1 ) P n j=1 w j (t 1 ) w ^ i P n j=1 w j (4.65) If <V 1 , then (4.61), (4.62) and (4.65) imply f ^ i (t 1 )w ^ i =^ r. For j2 [ ^ k 1], (4.56), (4.62), (4.63) and (4.64) imply w j =^ r = w j (0)=^ r = c j r j =^ r c j . For j2f ^ k;:::;ng, w j =^ r = c j . These together imply f ^ i (t 1 )c ^ i , giving a contradiction. If =V 1 , then ^ k = 1, and therefore (4.61) and (4.62) imply w =r 1 c. Using this with (4.65) implies f ^ i (t 1 )c ^ i =( P n j=1 c i ) =c ^ i , again giving a contradiction. Proof of (II-a): Following (4.61), we only need to show thatw i 2 [w l i ;w i (0)] fori2f ^ k;:::;ng. It is sucient to show that r ^ r r ^ k . This is because ^ r r combined with (4.57) implies that ^ r w l i =c i , and hence w i w l i , for all i2f ^ k;:::;ng; and ^ r r ^ k , which has already been established in (4.63), combined with the non-decreasing property of the sequencefr k g n k=1 implies ^ r r i , and hence w i w i (0) for all i2f ^ k;:::;ng, from (4.56). Since 2 (V n ;V ], (4.60) implies ^ k k. If ^ k = k, then P n i= ^ k c i V P n i= k c i = P k1 i=1 w i (0) =r . (4.62) then implies ^ rr . If ^ k> k, chapter ^ k 1 k, then the non-decreasing property offr k g n k=1 implies r ^ k1 r k , which when combined with (4.64) and (4.58) implies ^ r >r if <V 1 . On the other hand, if =V 1 , then ^ r =r =r 1 . This completes the proof for w 2 [w l ;w(0)] Combining this with the denition of w in (4.61) implies that w i (t)w i (0); i2 [ ^ k 1] (4.66) If = V 1 , then ^ k = 1, the set [ ^ k 1] is empty. However, in this case, w 1 = r 1 c 1 = w 1 (0). Therefore, w 1 (t)w 1 (0); 82 (V n ;V 1 ]: (4.67) 113 Proof of (II-b): By contradiction, suppose ~ w2 [w ;w(0)]nfw g is also an equilibrium. (4.66) and (4.67) imply there existsE 0 fmaxf2; ^ kg;:::;ng such that ~ w i > w i for all i2E 0 , and ~ w i =w i for i62E 0 (we have already proven w(t)w for all t 0). Therefore, X i2E 0 f i ( ~ w) = P i2E 0 ~ w i P j2E 0 ~ w j + P j62E 0w j > P i2E 0w i P n j=1 w j = X i2E 0 c i where the inequality is due to the fact that [n]nE 0 is nonempty, and the equality follows from the same argument used in the proof of (I): if < V 1 , then =( P n j=1 w j ) = 1=^ r, and hence ( P i2E 0w i )=( P n j=1 w j ) = P i2E 0w i =^ r = P i2E 0c i ; if = V 1 = P n i=1 c i , then ^ k = 1, w = r 1 c, and hence ( P i2E 0w i )=( P n j=1 w j ) = P i2E 0c i . Therefore, there exists at least one j2E 0 such that f j ( ~ w) > c j . This, combined with the fact that ~ w i > w i w l i for all i2E 0 , implies that ~ w can not be an equilibrium under u 1 . Proof of (II-c): For any i2 [n], f i (w i ) =w i =( P n j=1 w j ). Along the same argument used in the proof of (I), we have: If < V 1 , then =( P n j=1 w j ) = 1=^ r, and therefore f i (w i ) = w i =^ r. This is equal to c i for i2f ^ k;:::;ng, from (4.61). For i2 [ ^ k 1], since ^ r > r ^ k1 r i from (4.63) and non-decreasing property offr k g n k=1 , we have, f i (w ) =w i (0)=^ rw i (0)=r i =c i from (4.56). If =V 1 , then ^ k = 1, and hence w =r 1 c. Therefore, f i (w i ) =c i for all i2 [n]. Proof of (III): If 2 (V ;V 1 ), then 2 ^ k k. If =V 1 , then ^ k = 1, ^ r =r 1 and w =r 1 c. In particular, w 1 = r 1 c 1 = w 1 (0). Therefore, for convenience, we can set ^ k = 2 for = V 1 and (4.61) remains valid. In summary, we set the convention that k ^ k 2 for all 2 (V ;V 1 ]. Consequently, the setf1;:::; ^ k 1g is not empty, and, using similar argument as in the proof of (II-b), it can be shown that P n i= ^ k f i (w) > P n i= ^ k c i for any w2 [w ;w(0)]nfw g, chapter f(w) = 2 [0;c] for all w in [w ;w(0)] other than w . Furthermore, w = 2 [w l ;w(0)]. This is because of the following: Since r >r k1 from (4.58), maximum offw l i =c i g n i=1 occurs inf k;:::;ng. If ~ k denotes one such maximizer, then ~ k k ^ k. Therefore, w l ~ k >c ~ k ^ r =w ~ k , where the inequality is 114 due to r > ^ r, which can be shown using argument similar to the one in the proof of (II-a), and the equality is due to the denition of w ~ k for ~ k ^ k from (4.61). Theorem 5 implies that the margin of robustness ofu 1 is equal toV . The following proposition states sucient conditions foru 1 to be maximally robust, chapter sucient conditions forV = . Proposition 15. Consider a parallel network consisting of n links, with lower and upper bounds on link weights asw l 2 R E >0 andw u 2 R E >0 , respectively, link capacitiesc2 R n >0 . Then, for every 2 R n >0 and w(0)2 [w l ;w u ], we have R(u 1 ) = V = (cf. (4.52) and (4.59)) if and only if w(0)w opt , where w opt is an optimal solution to (4.54), as dened in (4.53). Proof. LetE 0 :=fi2 [n]jw u i =c i < r = max i w l i =c i g. (4.52) and (4.53) then imply that = P i2E0 w u i =r + P i= 2E0 c i and w opt i =w u i ; 8i2E 0 ; w opt i =c i r ; 8i2 [n]nE 0 (4.68) We need to show that w(0)w opt is necessary and sucient condition for: V = 1 r k1 X i=1 w i (0) + n X i= k c i = 1 r X i2E0 w u i + X i= 2E0 c i = (4.69) Sincew(0)w u , by denitionE 0 [ k1]. Moreover, by denitionw i (0)=r <c i fori k1. Therefore, it is straightforward to see that (4.69) is true if and only if: (i)E 0 =f1;:::; k 1g; and (ii) w i (0) =w u i for all i2E 0 . SinceE 0 f1;:::; k 1g, (i) is equivalent tof1;:::;ngnE 0 f k;:::;ng, which is further equivalent to w i (0) c i r for all i2f1;:::;ngnE 0 . Therefore, considering the denition of w opt from (4.68), conditions (i) and (ii) can be succinctly written as w(0)w opt . Remark 46. 115 (a) Since the controller u 1 only decreases weights, the initial weight must be greater than at least one optimal solution of weight in order for the controller u 1 to be maximally robust. Remark 45 implies that the condition w(0) w opt is not conservative because w opt is the minimal optimal solution. (b) Since w u w opt , Proposition 15 implies that u 1 is maximally robust for parallel networks if w(0) =w u . (c) Referring to Theorem 5, the weights on links in setf1;:::; maxf1; ^ k 1gg does not change under u 1 , whereas the weights on links in setfmaxf2; ^ kg;:::;ng potentially changes, and indeed these links become capacitated at the equilibrium w . (d) i in (4.55) can be arbitrary and time varying. 4.7.3 Controller with Memory We now present a control policy which augments u 1 by increasing weight on link i when the ow f i is increasing. The control policy is formally stated as follows: for all i2 [n], u 2 i (w i (t);f i (t)) = 8 > > > > > > > > > > < > > > > > > > > > > : i f i (t)>c i & w i (t)>w l i i f i (t)<c i & _ f i (t )> 0 & w i (t)<w u i 0 otherwise (4.70) where i > 0 is an arbitrary constant, and _ f i (t ) := lim 4t!0 (f i (t +4t)f i (t))=4t is the left derivative of f i (t). The control policy in (4.70) has a natural altruistic interpretation as follows: the controller on linki takes an action when either the ow on linki exceeds its capacity, or it sees an increase in the ow on link i. In particular, in the latter case, controller i increases weight on linki in order to further increase the ow on link i, and thereby possibly avoiding infeasibility on 116 other links. For parallel networks, if the disturbance att = 0 leads to increase in supply/demand, then it leads to increase in ows on all links. In such a case, under u 2 , _ f i (0 )> 0; 8i2 [n] (4.71) The maximal robustness of u 2 for n = 2 links is proven next. Proposition 16. Consider a parallel network consisting of 2 links, with lower and upper bounds on link weights asw l 2 R 2 >0 andw u 2 R 2 >0 , respectively, link capacitiesc2 R 2 >0 , and supply/demand with magnitude 0. Then, for every 2 R 2 >0 and w(0)2 [w l ;w u ], under the dynamics in (4.50) with the controller u 2 in (4.70), if < (cf. (4.52)), then lim t!+1 f(w(t))2 [0;c]. Proof. Assumption 3 implies that, if the disturbance decreases the supply/ demand , then the ow on each link decreases, and hence u 2 (t) 0. Therefore, the system is feasible. Hence, we only consider disturbances that increase , in which case (4.71) applies. Forn = 2, the optimal solution characterized in (4.52) and (4.53) can be explicitly written as shown in Table 4.1. Conguration (w opt 1 ;w opt 2 ) c 1 =c 2 <w l 1 =w u 2 (w l 1 ;w u 2 ) c 1 (1 +w u 2 =w l 1 ) w l 1 =w u 2 c 1 =c 2 w u 1 =w l 2 w 1 =w 2 =c 1 =c 2 c 1 +c 2 c 1 =c 2 >w u 1 =w l 2 (w u 1 ;w l 2 ) c 2 (1 +w u 1 =w l 2 ) Table 4.1: Explicit characterization of and w opt from (4.52) and (4.53), respectively, for n = 2. (I) If< (w 1 (0)+w 2 (0)) minfc 1 =w 1 (0);c 2 =w 2 (0)g, then it is straightforward to see thatf(0)< c. Note that, due to (4.71), this does not imply u 2 (0) = 0. Accordingly, we consider the following three cases. (I-A) If w(0) =w u , then u 2 (t) 0, and hence lim t!+1 f(w(t)) =f(0)2 [0;c]. (I-B) If w 1 (0) < w u 1 and w 2 (0) = w u 2 , then u 1 (0) = 1 > 0 and u 2 (0) = 0. w 1 (t) keeps increasing andw 2 (t) stays unchanged, and consequentlyf 1 (t) andf 2 (t) keep increasing 117 and decreasing, respectively, until either one of the following happens at some time t: w 1 ( t) =w u 1 or f 1 ( t) =c 1 . The weights do not change thereafter, and hence f 1 (t)c 1 and f 2 (t) < c 2 , for all t t. The argument for the other scenario w 1 (0) = w u 1 and w 2 (0)<w u 2 is symmetrical. (I-C) Ifw(0)<w u , thenu 2 (0) = and hence _ f 1 (0 + ) = _ f 2 (0 + ) = ( 1 w 2 (0) 2 w 1 (0))=(w 1 + w 2 ) 2 . (I-C-i) If w 1 (0))=w 2 (0) = 1 = 2 , then _ f 1 (0 + ) = _ f 2 (0 + ) = 0, and hence u 2 (t) 0. (I-C-ii) Ifw 1 (0))=w 2 (0)< 1 = 2 , then _ f 1 (0 + )> 0 and _ f 2 (0 + )< 0. This implies thatw 1 (t) keeps increasing and w 2 (t) stays unchanged at t = 0, and hence the asymptotic behavior is the same as in Case (I-B). Similar argument can be made for the other scenario when w 1 (0))=w 2 (0)> 1 = 2 . (II) If = (w 1 (0) +w 2 (0)) minfc 1 =w 1 (0);c 2 =w 2 (0)g), then f 1 (0) = c 1 or f 2 (0) = c 2 . Without loss of generality, assume f 2 (0) = c 2 and f 1 (0) < c 1 , in which case, w 1 (t) keeps increasing and w 2 (t) remains unchanged at t = 0, and the asymptotic behavior is the same as in Case (I-B). (III) If (w 1 (0) +w 2 (0)) minfc 1 =w 1 (0);c 2 =w 2 (0)g)< , then either f 1 (0)>c 1 and f 2 (0)< c 2 , or f 1 (0) < c 1 and f 2 (0) > c 2 . Without loss of generality, assume f 1 (0) > c 1 and f 2 (0) < c 2 . This implies c 1 =c 2 < w 1 (0)=w 2 (0) w u 1 =w l 2 . Then u 1 (0) = 1 < 0 and u 2 (0) = 2 > 0. Thereafter,w 1 (t) andf 1 (t) keep decreasing until either one of the following happens at t 1 : (e1) w 1 (t 1 ) = w l 1 or (e2) f 1 (t 1 ) = c 1 ; and w 2 (t) and f 2 (t) keep increasing until either one of the following happens at t 2 : (e3) w 2 (t 2 ) = w u 2 or (e4) f 2 (t 2 ) = c 2 . We now consider the two cases: t 1 <t 2 and t 2 <t 1 separately (ties are broken arbitrarily). (III-A) t 1 <t 2 : we consider two sub-cases depending on which of (e1) or (e2) happens rst. (e1) w 1 (t 1 ) = w l 1 , f 1 (t 1 ) c 1 , f 2 (t 1 ) < c 2 and w 2 (t 1 ) < w u 2 . In this case, w 1 stops decreasing at t 1 , but w 2 keeps increasing until t 2 when (e3) or (e4) happens. 118 If (e3) happens before (e4), thenf 1 (t 2 ) =w l 1 = w l 1 +w u 2 c 1 . Since (e4) has not occurred, then f 2 (t 2 ) c 2 , and since the weights are at the boundary at t 2 , they do not change thereafter. If (e4) happens before (e3), then f 2 (t 2 ) = c 2 , and therefore, f 1 (t 2 ) = f 2 (t 2 )c 1 +c 2 f 2 (t 2 )c 1 . (e2) f 1 (t 1 ) = c 1 , w 1 (t 1 ) w l 1 , f 2 (t 1 ) < c 2 and w 2 (t 1 ) < w u 2 . In this case, w 1 stops decreasing at t 1 , but w 2 keeps increasing until t 2 when (e3) or (e4) happens. It is straightforward to see that the system is feasible under both of these scenarios. (III-B) t 2 <t 1 : we consider two sub-cases depending on which of (e3) or (e4) happens rst. (e3) w 2 (t 2 ) = w u 2 , w 1 (t 2 ) > w l 1 , f 1 (t 2 ) > c 1 and f 2 (t 2 ) < c 2 . In this case, w 2 stops increasing at t 2 , but w 1 keeps decreasing until t 1 when (e1) or (e2) happens. In- deed, in this case, (e2) always precedes (e1). This is because, implies that, in the all the relevant (i.e., , rst and second) congurations in Table 4.1, w l 1 = w l 1 +w u 2 c 1 . When (e2) happens, f 1 (t 1 ) =c 1 , and f 2 (t 1 ) =f 1 (t 1 ) c 1 +c 2 f 1 (t 1 )c 2 . (e4) This is not possible because (e4) never precedes (e2). This is because, by contra- diction, if it does, then f 2 (t 2 ) =c 2 andf 1 (t 2 )>c 1 implying =f 1 (t 2 ) +f 2 (t 2 )> c 1 +c 2 . Remark 47. (a) Proposition 16 implies thatu 2 is maximally robust for parallel networks with 2 links. More- over, this maximal robustness property of u 2 , unlike u 1 , does not require extra conditions on w(0). (b) For parallel networks with 2 links, the action of the controller u 2 can be shown to be a descent algorithm to solve (4.11). 119 (c) Note that the proof of Proposition 16 implies that, under any disturbances, the asymptotic link weights under u 2 are on the boundary in many scenarios. It is possible to address this feature by proper selection of i ,i2 [n], and by extending the criterion for a link to increase the weight. Robustness analysis under such extensions to general parallel networks will be reported in future. Remark 48. For both u 1 and u 2 , it can be shown that the results of Theorem 5, Proposition 15 and Proposition 16 hold true for the case when w l = 0. 4.8 Simulations All the simulations were performed using Matlab 2015b on a desktop with the following congu- rations: Intel(R) Core(TM) i7-6700K CPU 4.00GHz and 16GB RAM. 4.8.1 Margin of Robustness Estimates 39 2 3 17 8 5 6 14 4 Figure 4.9: The IEEE 39 bus system (left) and its terminal network (right). Consider the IEEE 39 bus system shown in Figure 4.9 with the supply-demand vector p 0 chosen to be such that p 0;39 = 1,p 0;4 =1 andp 0;v = 0 for every other node v. The correspond- ing terminal network obtained by the multilevel formulation is shown in Figure 4.9. The ow 120 capacities were chosen to be symmetrical: c u =c l = 2:600 1. w u was selected to be the value of susceptances for this network provided by [93]. Without weight control, i.e., , when w l =w u , = 7:450. For weight control, we choose w l = 0:95w u . Expanding upon the relationship to conventional network ow problems, as described in the proof of Proposition 10, one can show that, for a network with general graph, the margin of robustness is upper bounded by the minimum cut residual capacity, which in this case evaluates to 8:400. We compared solutions to (4.4)-(4.5), restricted to nongenerative disturbances, computed by exhaustive search method for the original (i.e., (4.4)-(4.5)) as well as the multilevel formulation described in Section 4.5. For exhaustive search, [w l ;w u ] is discretized with resolution 0:5, and the cost function is evaluated at each of the discrete points according to a natural lexicographical order. The average time for evaluation of a single feasible point for the original and the multilevel formulation was 1:24 10 4 and 7:68 10 5 seconds respectively, illustrating the computational gains per evaluation from the network reduction procedure underlying the multilevel formulation. For the original formulation, due to the large number of feasible discrete points, it was found to be impractical to exhaustively evaluate the cost function at each of these discrete points. However, for the multilevel formulation, the exhaustive search method terminated in about 59:3 hours yielding 7:612. This illustrates computational advantage of the multilevel formulation. 4.8.2 Equivalent Capacities for Link Reducible Networks Consider the network shown in Fig. 4.4 with nodes v 1 and v 4 being the supply and demand nodes, respectively, and the weights bounds and link capacities are selected as follows: w l = [4 3 4 1 2] T , w u = [9 10 18 5 8] T , and c = [16 18 20 10 10] T . The equivalent capacities for sub-networks formed during the sequential reduction process described in Section 4.6, are illustrated in Figure 4.10. Note that each of the equivalent capacity functions isS 1 . 121 0 2 4 6 8 10 12 14 Equivalent Weight 10 15 20 25 30 35 Equivalent Capacity i 4,5 i 2,4,5 i 1,3 i 1,2,3,4,5 Figure 4.10: The equivalent capacity functions in the process of reduction for the network shown in Fig. 4.4. 4.9 Conclusions and Future Work In this paper, we developed a multilevel programming approach to compute margin of robustness for DC networks under control policies that change link weights in response to disturbances on supply-demand vector. In developing this approach, we identied a class of, possibly non- convex, optimization problems which can be equivalently converted into a bilevel problem. This methodology, as well as the network reduction, in terms of weight and capacity, underlying the lower level problem could be of independent interest. This study opens up several directions for future research. It is of interest to extend the proposed robustness framework to advanced notions of ow feasibility, e.g., by letting line ca- pacity depend on the line weight. It is also of interest to investigate extensions of the proposed methodologies, possibly under suitable approximations, when key assumptions in this paper are relaxed. This includes generalization to the case of additive disturbances and to networks which are not reducible. Moreover, designing distributed control policies for non-parallel networks with provable robustness guarantees is also an important direction of research. Finally, as DC power 122 networks are considered, we plan to extend our analysis to other approximations of AC power ow, e.g., [28], to get better estimate of margin of robustness for active power ow. 123 Chapter 5 Computing Optimal Control of Cascading Failure in DC Networks 5.1 Introduction Cascading failure in physical networks can be modeled via discrete-time dynamics, where the time epochs correspond to component failures. The map of the dynamical system is described in terms of composition of a failure rule with a control policy. A common failure rule is permanent removal of a link from the network if its physical ow exceeds capacity. Analysis of such dynamics under a given control policy has attracted considerable attention, primarily through simulations, e.g., see [11,29,33,71,88]. However, control design is relatively less well understood, e.g., see [18] and our previous work in [79] for few such examples. In this chapter, we consider such an optimal control problem for power networks. The network state is described in terms of active links, i.e., links which have not been removed so far, and the external power withdrawal/injection, also referred to as demand-supply, at the nodes. Under the failure rule, at a given network state, links are permanently removed if their power ow, as determined by the DC model, exceeds thermal capacity constraint. The control actions correspond to changing demand-supply at the nodes. A network state is called feasible if it is invariant under the failure rule, and is called infeasible otherwise. We are interested in 124 designing control actions to steer the network from an arbitrary initial state to a terminal feasible state within a given nite time horizon. In this chapter, admissible control actions are those under which the supply-demand at the nodes is non-increasing, and we consider the setting in which there is no running cost and the cost associated with a terminal feasible state is equal to the negative of cumulative demand-supply associated with that state. The optimal control problem studied in this chapter was formulated in [18, 19], where the focus was primarily on low-complexity control policies. To the best of our knowledge, a for- mal framework for computing optimal control beyond these low-complexity policies is lacking in the literature. The objective of this chapter is to develop rigorous approaches to address this shortcoming. Specically, we provide two distinct approaches. The rst approach is geared towards type II tree reducible networks, i.e., networks which can be reduced to a tree by recursively replacing subnetworks between two nodes, and containing no supply or demand nodes in the interior, with single links. For such networks, we decompose the (global) optimal control problem into a system of coupled local problems associated with nodes in the tree corresponding to the reduced network. This system of coupled problems can be solved to optimality in two iterations. In the rst iteration, from leaf nodes to the root, every node computes an optimal solution to the associated local problem as a function of the local coupling variable. This coupling variable corresponds to the sequence of out ows from the nodes. In the second iteration, in the reverse order from the root to the leaf nodes, the local optimal solutions are instantiated with specic values of the coupling variables. When restricted to constant control actions, i.e., when the supply-demand at the nodes are decided at t = 0 and remain constant thereafter, the local problems, in spite of non-convexity, possess a piecewise linear property with respect to the coupling variables, which facilitates analytical solution. The second approach computes optimal control by searching for an optimal feasible terminal state among the states reachable from the initial condition. This search is made possible by show- ing that the reachable set admits an equivalent nite representation. The key is that the one-step 125 reachable set from any network state can be partitioned into a nite number of aggregated states, with each corresponding to the same reachable active link set. These partitions are determined by admissibility constraints for control actions (to maintain monotonicity of supply-demand at the nodes), and the link failure rules. Taking into account the linearity of these constraints, we leverage and extend tools from the domain of arrangement of hyperplanes e.g., see [38] [85, Chap- ter 24], and convex polytopes, e.g., see [92] [49], to construct these partitions. Each element of the partition, at every network state appearing in the search, is a polytope. We use the notion of incidence graph to represent them, and provide an incremental algorithm to compute their incidence graphs. The latter is based on a novel transformation between polytopes. Among the key determinants of the time complexity for constructing arrangement are (i) the dimension of the control action space, which is one less than the number of non-transmission nodes in a connected network, and (ii) the computation of ow in DC model as a function of supply- demands at nodes under dierent active link sets. We address (i) by developing a projection-based approximation. An extreme case is projection on to a one dimensional space, to which belongs the scaling-based, or proportional, control policies in [19, Section 6.1.1]. Note that these are the only control policies reported in the literature, to the best of our knowledge, for the problem considered in this chapter. We address (ii) by using Proposition 8 in Section 3.5 to incrementally compute ow redistribution under link failure. In summary, the chapter makes several important contributions in computing optimal control of cascading failure in DC networks. First, it provides a formal description of the control problem and a general search-based solution strategy that could be implemented using, e.g., a variety of sampling or other approximation techniques. Second, the network decomposition approach illustrates how certain properties of the network and the admissible control action set translate into tractable computational complexity. Third, the algorithmic procedure for constructing equivalent nite representation of the reachable set is, to the best of our knowledge, the rst instance of the application of the elegant computational geometric tools related to arrangement of hyperplanes 126 to an engineering application, beyond path planning and related topics in robotics, e.g., see [50]. Overall, these contributions bring a much needed formalism to the domain of resilient operation of power networks faced with the prospect of cascading failures. The rest of the chapter is organized as follows. A formal description of the cascading failure dynamics and the optimal control problem, as well as an outline of the search based approach for computing optimal control, are provided in Section 5.2. The decomposition based approach for computing optimal control for type II tree reducible networks is provided in Section 5.3. The equivalent nite representation of the reachable set to enable the search based approach is described in Section 5.4. Section 5.5 contains an algorithmic procedure based on arrangement of hyperplanes to eciently construct the nite representation of the reachable set. Discussions on time complexity of the algorithmic procedure is in Section 5.6. Section 5.7 contains the projection-based approximation and illustrative simulations on the benchmark IEEE 39 network, and concluding remarks are provided in Section 5.8. 5.2 Problem Setup The problem is formulated within the same simplications as in [19] on modeling of power system failures, that is, link failures and slow-moving cascade processes are considered; and fast scale dynamics is not explicitly modeled and the standard linearized (DC) approximation to power ow is used. This is to say, DC networks are considered. Furthermore, we consider symmetric link capacities, denoted by c2 R E >0 . Dierent from Chapter 3 and Chapter 4, link weights w are assumed to be xed in this chapter, and the supply-demand vector p is considered as the control variable, whose value can be changed through load shedding. Nevertheless, link failure is allowed, 127 which is equivalent to saying that the weight of a link can take value zero. For convenience, we dene the following DC ow function, where we show explicit dependence on the active link set: f(E;p) :=WA T L y (E)p (5.1) It is straightforward to see that (5.1) is the same as (2.10). Additionally, we recall thatB E is the set of balanced supply-demand vector for an active link setE (cf. (2.5)). 5.2.1 Cascading Failure Dynamics LetE 0 be the initial link set and letp 0 be the initial supply-demand vector satisfying the balance condition in (2.5). The corresponding link ow f is uniquely determined by (2.9) or (5.1). We associate with each link i2E 0 a thermal capacity c i > 0. If the magnitude of ow on a link i2E 0 exceeds its thermal capacity, i.e.,jf i j>c i , then linki fails and is removed from the network irreversibly. This changes the topology of the network, causing ow redistribution, which might lead to more link failures, and so on. Such continuing link failures constitute the uncontrolled cascading failure dynamics. Note that we consider a link failure rule which is deterministic and which depends solely on the instantaneous ow. This is to be contrasted with other deterministic outage rules based on moving average of successive ows, or stochastic line outage rules, e.g., see [13,18]. Our objective in this chapter is to stop cascading failure through appropriate control actions. While shedding all load att = 0 achieves this objective trivially, we desire to take control actions that are optimal in a certain sense. Consider the following description of controlled cascading failure dynamics in discrete-time. Each time epoch corresponds to failure of some links. The node set remain the same. Let (E t ;p t ) be the state of the network at time t, withE t 2 E 0 and p t 2 R V denoting the active link set and supply-demand vector at time t, respectively. We consider load shedding as the control and, for convenience, employ control variable u2 R V to 128 be supply-demand vector after load shedding. The controlled cascading failure dynamics, for t = 0; 1;:::, and starting from the initial state (E 0 ;p 0 ), is given by: E t+1 ;p t+1 =F E t ;p t ;u t ; u t 2U(E t ;p t ) (5.2) where F E (E;p;u)F E (E;u) :=fi2Ejc i f i (E;u)c i g F p (E;p;u)F p (u) :=u (5.3) where the functionsF E andF p are for the maps forE t+1 and p t+1 , respectively. As dened in (5.3),F E is the set of feasible links inE under supply-demand vector u; and the control input u t at time t becomes the next state supply-demand vector p t+1 . Remark 49. 1. Before the system reaches a steady state, the time epoch corresponds to link failure in (5.2). After that, the time epochs are chosen arbitrarily, e.g., at xed intervals. 2. Even though its form appears to be specic to DC power networks, (5.2)-(5.3) can be ex- tended to other dynamical systems in the situations where the underlying physical dynamics are on a considerable faster time scale compared with the time scale for cascading failure dynamics. In that case, the system is assumed to reach an physical equilibrium at each time epoch in (5.3) and u is used to denote the controllable settings that is capable of choosing the physical equilibrium state in the way described by functionF p . In order forF E (E;u) in (5.3) to be well-dened,u must be balanced with respect to the active link setE. This is ensured by the following denition of state-dependent control space U(E;p): U(E;p) = cube(p)\B E (5.4) 129 where cube(p) characterizes the load shedding property, and is dened as: cube(p) := n u2 R V j 0u v p v for p v 0; p v u v 0 for p v < 0 o (5.5) U(E;p) includes all admissible load shedding controls at state (E;p). In particular, if all the supply and demand nodes are disconnected from each other at state (E;p), thenB E =f0g, and in this case U(E;p) =f0g. Remark 50. The number of connected network components may increase in (5.2). When this happens at a network sate (E;p), it is possible that p62B E . The denition of U(E;p) in (5.4) ensures the the balance condition is satised by the controls over all components. Figure 5.1 illustrates repeating sequence of ow redistribution, control action, and link failures, under the cascading dynamics proposed in (5.2)-(5.3). If the initial ow f(E 0 ;p 0 ) is infeasible, and if the control action is u 0 2U(E 0 ;p 0 ), then the next state supply-demand vector is p 1 =u 0 . If the resulting ow is within the capacity, i.e.,jf(E 0 ;p 1 )j c, then there are no link failures, and henceF(E 0 ;p 0 ;u 0 ) = (E 1 ;p 1 ) = (E 0 ;p 1 ). In order to maintain the network state, one would choose subsequent control actions as u t =p 1 for all t 1. On the other hand, ifjf i (E 0 ;p 0 )j > c i for some links i, then those links will fail, and hence E 1 =F E (E 0 ;u 0 )6=E 0 and p 1 = u 0 . Therefore, the new link ows are f(E 1 ;p 1 ). Thereafter, control action u 1 is chosen, possibly equal to p 1 ifjf(E 1 ;p 1 )jc E 1, and the process repeats. (E 0 ;p 0 ) f(E 0 ;p 0 ) p 1 =u 0 f(E 0 ;p 1 ) FE (E 0 ;p 1 ) (E 1 ;p 1 ) N Figure 5.1: The sequence of ow redistribution, control action, and link fail- ures, under the proposed cascading dynamics. 130 5.2.2 Problem Formulation Let S :=f(E;p)jp2B E ;c i f i (E;p)c i ;8i2Eg (5.6) denote the set of feasible states. SetS is invariant under the uncontrolled cascading dynamics. Note that (E; 0)2S for everyE2 2 E 0 . SinceE t ;t 0, is non-increasing sequence, andE 0 is nite, the dynamics converges to a steady state within 2 E 0 time epochs. Our objective is to choose control actions to steer the network from an arbitrary given initial state (E 0 ;p 0 ) to a feasible state (E N ;p N )2S within a given nite horizon N, while optimizing a certain performance criterion. The control horizon N is typically much smaller than 2 E 0 . Let a generic sequence of control actions over the control horizon be denoted by u := (u 0 ;:::;u N1 ). In this chapter, we wish to solve the following optimal control problem: sup u2D(E 0 ;p 0 ;N) s T p N (5.7) where s2f1; 0;1g V is a constant dened as: s v := 1 for v2V + , s v :=1 for v2V , and s v := 0 otherwise, and the set of feasible control actions is dened as: D(E 0 ;p 0 ;N) := (u 0 ;:::;u N1 )ju t 2U(E t ;p t ) for t = 0;:::;N 1; (E N1 ;u N1 )2S; (E t ;p t ) t2[N] satises (5.2) (5.3) (5.8) For brevity, we shall not show the dependence ofD onE 0 , p 0 and N when clear from the context. Remark 51. 1. (5.8) implies that, when checking feasibility of a given u, one has to check an additional condition foru N1 in comparison to that foru t fort = 0;:::;N 2. In addition 131 to checkingu N1 2U(E N1 ;p N1 ), one also needs to check if the resulting (E N ;p N ) belongs toS. 2. In (5.7), we use supremum rather than maximum becauseD is not closed in general, as illustrated in Example 7 below. This matter is addressed in Section 5.4.2 and it could be ignored before that. The computational complexity of characterizingD, and hence of solving (5.7) is attributed to the cascading dynamics in (5.2). Example 7. Consider the optimal control problem in (5.7) for the network shown in Fig. 5.2a, for N = 3. Node 1 is the supply node and nodes 2 and 3 are the demand nodes. The initial supply-demand vector is p 0 = [30;10;20] T . The link weights are w = [2; 1; 1; 1] T and the link capacities are c = [6; 7; 14; 5] T . Consider u 0 = [21 + 2=k;7 1=k;14 1=k] T for some k 1. The resulting ow is f(E 0 ;u 0 ) = [8 + 6=(7k); 4 + 3=(7k); 9 + 5=(7k); 5 + 2=(7k)] T . Consequently, links e 1 and e 4 fail due to ow exceeding capacity, and the resultingE 1 is shown in Fig. 5.2b. For u 1 = [21;7;14] T , f 2 (E 1 ;u 1 ) = 7 c 2 , f 3 (E 1 ;u 1 ) = 14 c 3 , and therefore there are no more link failures. This implies that u [N] (k) := (u 0 (k);u 1 (k);u 2 (k)) = ([21 + 2=k;7 1=k;14 1=k] T ; [21;7;14] T ; [21;7;14] T )2D for everyk 1. However, ^ u = (^ u 0 ; ^ u 1 ; ^ u 2 ) = lim k!1 u(k) = ([21;7;14] T ; [21;7;14] T ; [21;7;14] T )62D. This is because f(E 0 ; ^ u 0 ) is such that only link e 1 fails, and the resulting ^ E 1 is shown in Fig. 5.2c. Furthermore, f( ^ E 1 ; ^ u 1 ) = [null; 28=3; 35=3; 7=3] T , which implies thate 2 fails. Thereafter, ^ E 2 =fe 3 ;e 4 g,f 3 ( ^ E 2 ; ^ u 2 ) = 21>c 3 and f 4 ( ^ E 2 ; ^ u 2 ) = 7 > c 4 . All links fail under ^ u and u62D. This demonstrates thatD is not closed for the given choice of network parameters. 1 2 3 e1 e2 e3 e4 (a) 1 2 3 e2 e3 (b) 1 2 3 e2 e3 e4 (c) Figure 5.2: The graph topology for the network used in Example 7 to illustrate that the feasible control action setD is not necessarily closed. 132 Remark 52. In writing the optimal control problem in (5.7), we only consider the terminal cost s T p N , in addition to imposing feasibility condition on the terminal state. s T p N is the remaining cumulative supply and demand once the cascading failure stops, and hence is a natural choice for the objective function in (5.7). Extension to including running cost is discussed in Remark 65. 5.2.3 Solution by Optimal Tree Search A generic approach to solving (5.7) is by performing an optimal search on a directed tree composed of the states reachable from the initial state (E 0 ;p 0 ) in at most N time steps, e.g., see [77, Chap 3]. In other words, the tree is rooted at (E 0 ;p 0 ), and has depth N. Each node of the tree corresponds to a state (E;p) which is reachable in one time step from its parent node under a control action which is associated with the incoming arc to that node; see Fig. 5.3 for an illustration. When considering one time step reachable set from a given node (E;p), one only considers control actions belonging to U(E;p). The set of goal states for the search isS and we associate every state (E;p)2S with a utility s T p. The objective is to search for a state inS with maximal utility. :::::: :::::: :::::: :::::: :::::: :::::: :::::: :::::: :::::: :::::: :::::: :::::: :::::: feasible state infeasible state :::::: :::::: :::::: :::::: :::::: :::::: t = 0 t = 1 t = 2 t =N 1 t =N Figure 5.3: The tree composed of states reachable from (E 0 ;p 0 ) in at most N time steps. 133 Let J t (E;p) be the maximum among utilities of all the states that can be reached in at most t time steps starting from (E;p). Solving (5.7) is equivalent to computing J N (E 0 ;p 0 ). This computation can be done as follows: J 1 (E;p) = max u2U(E;p) s T u s.t. c E f(E;u)c E (5.9a) J t (E;p) = sup u2U(E;p) J t1 (F E (E;u);u); t = 2;:::;N (5.9b) where (5.9a) uses the ow capacity constraint to account for the additional constraint to be satised by u N1 , as commented on in Remark 51. (5.9a) is a linear program and commonly referred to as LP power redispatch, e.g., see [25]. (5.9b) uses supremum becauseF E (E;u), and hence J t1 (F E (E;u);u), is not continuous w.r.t. u. It is straightforward to see that J 1 (E;p) J t (E;p)s T p for all (E;p) and t2 [N]. Remark 53. Though (5.9) is similar to the value iteration in dynamic programming, we note that in this chapter, a search algorithm in forward direction over the state tree is preferred over the value iteration in backward direction, because the reachable set can be considerably smaller than the state space. Comparison of time complexity of these two algorithms can be found in Section 5.6.2. Moreover, we shall be implicitly referring to reachable set by state space hereafter, as only the former is relevant for searching. Executing optimal tree search, or equivalently implementing (5.9), is not directly amenable to a computational procedure, since the number of one-step reachable states from (E;p), or equivalently the set of feasible control actions U(E;p), is a continuum in general. A natural strategy is to discretize U(E;p), at the expense of getting less scalable algorithms and approximate solutions. In this chapter, we propose the following two approaches for better computational eciency: (I) (semi-)analytic solution for a certain class of networks, or for optimal solution within a certain subset of feasible control actions (Section 5.3); and 134 (II) an algorithmic procedure to construct an equivalent nite abstraction of the set of feasi- ble control actions, such that computing optimal solution over this nite abstraction gives solution to (5.7) (Section 5.4.1). Approach (II) essentially relies on developing a nite abstraction for the one time step reach- able set from an arbitrary (E;p), where the number of abstractions is related to the number of active link sets reachable from (E;p). Following (5.1) and (5.3), this involves computation of ow functions associated with several active link sets that are subsets ofE. Doing such a computation from scratch for each subset could overall prove to be a computationally expensive procedure, par- ticularly for largeE 0 or for large N. This challenge is addressed by Proposition 8 in Section 3.5 with an incremental approach for computing ow redistributions. 5.3 Analytical Solution In this section, we present (semi-)analytical solution to (5.7) in some special cases. 5.3.1 Parallel Networks 1 2 e 1 e 2 Figure 5.4: A parallel network with two links. A parallel network (f1; 2g;E) consists of two nodes that are connected by multiple parallel links, e.g., see Figure 5.4. We set the convention that the links are directed from node 1 (supply) to node 2 (demand). Since p 2 =p 1 , following (2.9), the link ows are given by: f i = w i P j2E w j p 1 ; i2E (5.10) 135 The following monotonicity result is a straightforward consequence of (5.10), whose proof is omitted. Lemma 19. Consider two arbitrary parallel networks (f1; 2g;E) and (f1; 2g;E 0 ) such thatEE 0 and two arbitrary supply-demand vectorsp =p 1 [11] T andp 0 =p 0 1 [11] T such that 0<p 1 p 0 1 . Then the following are true: (i)F E (E;p 0 )F E (E;p); and (ii)F E (E;p)F E (E 0 ;p). Remark 54. For a parallel network (f1; 2g;E) and a natural numberN 1, letu = (u 0 ;:::;u N1 ) and ~ u = (~ u 0 ;:::; ~ u N1 ) be two sequences of control actions and (E 0 ;:::;E N ) and ( ~ E 0 ;:::; ~ E N ) be the topology sequences under the two controls. Lemma 19 implies that if u t 1 ~ u t 1 for all 0tN 1, thenE t ~ E t for all 0tN 1. Furthermore, according to (5.10), f i =c i = p 1 = P k2E w k (w i =c i ) for all i2E. Noting the common factor p 1 = P k2E w k among all links, we label links in the increasing order of w i =c i , i.e., w i =c i w j =c j for i j, i;j2E. The chronological order of link failures according to (5.2) is expected to be aligned with the reverse labeling of the links, and is not aected by dierent load shedding actions, as implied by the following result. Lemma 20. Consider the cascading dynamics (5.2) for a parallel network (f1; 2g;E). Iff j (t)c j for some j2E and t2 [N], then f i (t)c i for all ij. Proof. Since f j (t) =p 1 (t)w j = P k2E w k c j , then, for i<j, we have f i (t) =p 1 (t)w i = P k2E w k = w i f j (t)=w j w i c j =w j c i . The last inequality is due to w i =c i w j =c j for all ij. Remark 55. Lemma 20 implies that for allt2 [N], there existsj2E such thatE t = [j]. Because E t is non-increasing, at mostjEj + 1 number of distinct network topologies can occur in the cascading dynamics. The monotonicity properties shown in Lemma 19 and the tight characterization of the reach- able set of topologies, as implied by Remark 55, allows optimal control synthesis relatively easily. Specically, we show that a one-shot control dened next is optimal within all control policies for parallel networks. 136 Denition 16. For an initial supply-demand vector p 0 2 R V , a N stage control sequence (u 0 ;:::;u N1 ) is called one-shot control if there exists 0 t 1 N 1 such that u t = p 0 for all t < t 1 , u t1 2 cubep 0 , and u t = u t1 for all t t 1 . Moreover, if t 1 = 0, then it is also called constant control. In order to describe the analytical expression of an optimal one-shot control for a parallel network (f1; 2g;E), we rst introduce several notations. Let R i := (c i =w i ) P i j=1 w j for all i2E. In general, R i is neither decreasing nor increasing with respect to i. The following remark is straightforward. Remark 56. R i is the maximum supply or demand the network can support when only the rst i links are active. ([i];p 0 )2S if and only if p 0 1 R i for all i2E. Let o 1 := max argmax i2E R i , and let o j+1 := max argmax i>oj R i if o j <jEj. Let end be the maximum number such that o end is dened. It is straightforward to see that o 1 < o 2 < ::: < o end =jEj andR o1 >R o2 >:::>R o end =R jEj . For a given initial balanced supply-demand vector p 0 2 R 2 , an optimal control depends on the value of N. A bigger N provides more exibility for control design. A small N forces to shed big portion of loads to ensure network feasibility at small time instants. For example, for N = 1, suciently large amount of load needs to be shed at t = 0 to ensure that all links become feasible. Next we dene N j (p 0 ) for every balanced p 0 2 R 2 and j2 [end]. They are used in specifying optimal controls. Let (E 0 un ;:::;E N un ) be the non-increasing topology sequence of the uncontrolled cascading dynamics (5.2) (that is, u t = p 0 for all t). Let R o0 :=1, R o end+1 = 0 andE 1 un E 0 un for convenience. For 0 j end, if R oj+1 <p 0 1 R oj , then let N k (p 0 ) := 1 + min t2f0;:::;Ngj (E t un ;p 0 )2S for 1kj and N k be such thatE N k (p 0 )1 un [o k ]E N k (p 0 )2 un for all j + 1 k end. The above denition implies thatjEjN 1 (p 0 ):::N end (p 0 ) = 1 for all p 0 . In addition, let N 0 (p 0 ) :=1 for all p 0 for convenience. 137 Proposition 17. Consider a parallel network (f1; 2g;E) with link weights w2 R E >0 , ow ca- pacities c2 R E >0 and initial supply demand vector p 0 and let R i , i2E, o j , j2 [end], N k (p 0 ), k2 [end][f0g be as dened above. If N j (p 0 ) N < N j1 (p 0 ), then an optimal control ac- tion is as follows: u t; = p 0 for all 0 t < N j (p 0 ) 2 and u t; = minfR oj ;p 0 1 g[1 1] for all maxfN j (p 0 ) 2; 0gtN 1. Proof. We consider the following cases: 1) p 0 1 R jEj . Remark 56 implies that (E;p 0 )2S. The optimal control for every N 1 would be shedding no load. In this case, by denition N j (p 0 ) = 1 for all j2 [end]. Every N 1 satises N 1 (p 0 ) N < N 0 (p 0 ). Hence, u ;t = minfR o1 ;p 0 1 g[1 1] for t 0. Followed by the fact that p 0 1 R jEj R 1 , u is optimal. 2) R o k+1 <p 0 1 R o k for some 0k end1. By denition,N 1 (p 0 ) =N 2 (p 0 ) =::: =N k (p 0 ) and N k+1 (p 0 ):::N end1 (p 0 ) 2>N end (p 0 ) = 1. We have the following cases. a) N = 1. Since N end (p 0 ) 1N end1 (p 0 ), u = minfR o end ;p 0 1 g[1 1] =R jEj [1 1] is optimal, where the second equality follows from p 0 1 >R o k +1 R o end =R jEj . b) NN k . In this case, N 1 (p 0 )N <N 0 (p 0 ). Since p 0 1 R o k R o1 , u t; =p 0 for all t. u is optimal if feasible. The latter is a straightforward result from the denition of N k (p 0 ). c) 2 N < N k . In this case, N j (p 0 ) N < N j1 (p 0 ) for some k + 1 j end 1. Therefore, p 0 1 >R o k+1 R oj . u ;t =p 0 for 0t<N j (p 0 ) 2 and u t; =R oj [1 1] for N j (p 0 ) 2 t N 1. We rst show that u 2D(E;p 0 ;N). Let (E 0 ;:::;E N ) be the topology sequence under u . It is straightforward that (E 0 ;:::;E Nj (p 0 )2 ) = (E 0 un ;:::;E Nj (p 0 )2 un ). By denition of N j (p 0 ), [o j ]E Nj (p 0 )2 un =E Nj (p 0 )2 . In addi- tion, Remark 56 implies ([o j ];R oj [1 1])2S and, plus the denition of o j , futher implies that ([l];p 0 )62S for all l > o j . Therefore,E t = [o j ] for all t N j (p 0 ) 1 and u 2D(E;p 0 ;N). We then show optimality of u through contradiction. Suppose 138 there exists a control ~ u2D(E;p 0 ;N) such that R oj < ~ u N1 1 p 0 1 . Let ( ~ E 0 ;:::; ~ E N1 ) be the topology sequence under control ~ u. Remark 54 implies thatE N1 un ~ E N1 . At the same time, since N <N j1 (p 0 ),E N1 un E Nj1(p 0 )2 un [o j1 ]. Note the inclusion is strict. Therefore, [o j1 ] ~ E N1 . Remark 56, combined with the denition ofo j and the assumption that ~ u N1 1 > R oj , implies that ( ~ E N1 ; ~ u N1 )62S. This contradicts with ~ u being feasible. Remark 57. While Proposition 17 gives the explicit expression of a one-shot control that is optimal for parallel networks, the fact that a one-shot control being optimal is proved in more general settings in [3]. The proof of Proposition 17 implies the following. Corollary 4. For a parallel network (f1; 2g;E) with link weightsw2 R E >0 , ow capacitiesc2 R E >0 and initial supply demand vector p 0 , for N jEjo 1 , the following constant control u is an optimal control: u t; = [1 1] minfp 0 1 ;R o1 g for all 0tN 1. One one hand, Proposition 17 and Corollary 4 justify the study of optimal control within special classes of control policies. On the other hand, while a one-shot control action can be optimal for non-parallel networks, it is not true in general. This is illustrated in Example 8. In fact, while it is straightforward to see that the set of feasible constant control actions is connected for parallel networks, this is not the case for a general network, even if it admits a one-shot control action that is optimal. Example 8. Consider the network illustrated in Fig. 5.5, containing a single supply node 1 and a single demand node 3, having link weights w = [1; 1; 1; 1; 1], and with initial supply- demand vector p 0 = [3; 0;3]. Consider two scenarios corresponding to link capacities c 1 = [0:8; 0:5; 0:6; 0:25; 1:5] and c 2 = [0:8; 0:5; 0:7; 0:25; 1:5], where note that the two scenarios dif- fer only in the capacity of link e 3 . We consider the optimal control problem for N = 2. Let 139 u t =z t [1; 0;1], i2f0; 1g, 0z 1 z 0 be the control actions. The ow under u t for relevant network topologies are: f([5];u t ) = 1 4 z t [1; 1; 1; 1; 2] T ,f(f1; 2; 3; 5g;u t ) = 1 5 z t [1; 1; 2; null; 3] T , f(f1; 3; 5g;u t ) = 1 3 z t [1; null; 1; null; 2] T andf 5 (f5g;u t ) =z t . The maximal value ofz t the can be supported by these networks are, respectively, 1; 1:5; 1:8; 1:5 in the rst scenario and 1; 1:75; 2:1; 1:5 in the second scenario. It is straightforward to see that the network would get disconnected in both scenarios if no load shedding is implemented. By considering all possible topology sequences that can occur under a control policy, we obtain the following: (i) The optimal one-shot control that sheds loads at t = 1 isz 0 = 3,z 1 = 1:5 in both scenarios. (ii) The optimal constant controls are: z 0 =z 1 = 1:5 in the rst scenario and z 0 =z 1 = 2:1 in the second scenario. (iii) An optimal control is z 0 = 2:1 andz 2 = 1:8 in the rst scenario and is z 0 =z 1 = 2:1 in the second scenario. We can see in both cases, constant controls perform no worse than one-shot control, and while the best constant control is not optimal over all controls in the rst scenario, it is optimal in the second scenario. Furthermore, in the second scenario, a feasible constant control has to satisfy z 0 =z 1 2 [0; 1]\ (2; 2:1]. This set is neither connected nor closed. 1 2 3 e 1 e 2 e 3 e 4 e 5 Figure 5.5: The graph topology for the network used in Example 8 to illustrate that the set of feasible one-shot control actions is neither connected nor closed. 5.3.2 A Decomposition Approach for Type II Tree Reducible Networks In this section, we develop a decomposition approach to compute optimal control for type II tree reducible networks as dened in Denition 17. Type II tree reducible networks extend the 140 concept of tree reducible networks in Denition 13. The latter only allows for network reduction over parallel and series subnetworks. Denition 17 (Type II Tree Reducible Network). A networkG = (V;E) with supply-demand vectorp is called type II tree reducible if it is a tree, or it can be reduced to a tree 1 T = (V T ;E T ) by recursively applying network reduction described in Denition 10. In this case,T is called the reduced tree ofG. (a) 0 1 2 3 4 5 6 7 8 (b) Figure 5.6: (a) a tree reducible networkG; and (b) a reduced treeT ofG. Fig. 5.6 provides an example of a tree reducible network, where each sub-network (denoted by G 1 ;:::;G 8 ) in Fig. 5.6a represents a reducible component and corresponds to a link in the reduced treeT = (V T ;E T ) shown in Fig. 5.6b. We assign directions for the links inE T as follows 2 . Pick an arbitrary node inV T , and call it the root node. The directions for all the links incident to the root node are chosen to be incoming to the root node. The directions for the remaining links are similarly chosen to be directed towards the root node; see Figure 5.6b for an example. For the resulting directed tree, we x a reverse topological ordering 3 of the nodes (0; 1;:::;jV T j 1), with 0 being the root node. Figure 5.6b illustrates such an ordering. In order to minimize notations, we use the same label for a link and its tail node. For example, the link (5; 2) in Figure 5.6b is labeled as link 5. An in-neighbor (resp., out-neighbor) of a given node is called its child (resp., parent) node. For node i2V T , letC i denote the set of its children nodes, and let C i denote the 1 A tree is an undirected graph in which any two nodes are connected by at most one path. 2 The results presented in the current Section 5.3.2 do not depend on the particular choice of directions for links inE T , as selected here (see also Remark 59). 3 That is, for every directed link (i;j), we have i>j. 141 set of nodes consisting of the descendants of i and the node i itself. For example, in Fig. 5.6b, C 2 =f5; 6g and C 2 =f2; 5; 6; 7; 8g. With this denition,V T C 0 . Node i is called a leaf if it has no child node, i.e., ifC i =;. Let us start with the simple case when the entire network consists of a single reducible com- ponent, sayG i , so that the reduced tree isT = (f0; 1g; 1). Let the nodes 0 and 1 correspond, respectively, to nodes v 0 and v 1 inV i . Recall from Denition 10 that v 0 and v 1 are the only supply-demand nodes inG i . Let a i 2f1; 0;1g Vi be such that a v is equal to 1 if v =v 0 , is equal to1 if v = v 1 , and is equal to zero otherwise. We do not x the individual identities of v 0 and v 1 as supply or demand nodes, and the choice of the signs of entries of a i is merely to set some convention. We let z t i a i with z t i 2 R denote the supply-demand vector inG i for t2 [N], or equivalently, the control sequence for t2f0;:::;N 1g. If ~ p i a i , for ~ p i 2 R, is the initial supply demand vector, then the set of feasible control sequences as per (5.8) isD(N;E 0 i ; ~ p i a i ), whereE 0 i is the initial active link set inG i at t = 0. Recall thatD(N;E 0 i ; ~ p i ) captures the constraint that the terminal state at t = N is feasible, as well as the monotonicity constraint implied by (5.4). We split these two constraints asD(E 0 i ; ~ p i a i ;N) = ~ D i (N)\ ^ D i (N), where ~ D i (N) captures feasibility of terminal state, while relaxing monotonicity, and ^ D i (N) captures monotonicity while relaxing feasibility of the terminal state. These two sets are formally dened as: ~ D i (N) := n (z 0 i ;:::;z N1 i )2 R N j (E N1 i ;z N1 i a i )2S;E t i =F E (E t1 i ;z t1 i a i );8t2 [N 1] o (5.11) ^ D i (N) := n (z 0 i ;:::;z N1 )2 R N jz 0 i 2 cube ~ p i ; z t i 2 cubez t1 i ; 8t2 [N 1] o (5.12) When clear from the context, we shall not show the explicit dependence of ~ D i and ^ D i on N. Remark 58. 1. Note that ~ D i includes control actions that cause loss of connectivity inG i . In this case, since we have only one supply and demand, the constraint that the terminal 142 state (E N1 i ;z N1 i a i ) is feasible, implies that z N1 i = 0. In addition, since all links have symmetrical capacities, ~ D i = ~ D i . 2. ^ D i is a polytope. However, ~ D i is non-convex in general, as indicated by the disconnected feasible set of constant control actions in Example 8. Nevertheless, without considering the trivial case that (;; 0)2S, ~ D i is bounded as the link capacities are assumed to be nite. The explicit computation of ~ D i follows from the discussion in Section 5.4.3 (cf. Remark 68). The exibility aorded by splitting the control constraints into (5.11) and (5.12) for an isolated reducible sub-network allows to translate capacity constraints from individual links in a general tree reducible networkG into equivalent constraints for the equivalent links inE T as follows. The controlu t i 4 at nodei2V T at timet2f0;:::;N 1g is split asu t i =z t i P j2Ci z t j . For example, referring to Figure 5.6, u t 5 = z t 5 z t 7 z t 8 . In this case, the out ow from link i inE T , z t i , is interpreted asG i 's share of control input u t i . u i := (u 0 i ;:::;u N1 i ), i2V T , is constrained to satisfy (5.12) for ~ p i =p 0 i , and z i := (z 0 i ;:::;z N1 i ), i2E T , is constrained to satisfy (5.11). Consider the following optimization problem that will inform the decomposition approach. For i2E T , given z i 2 ~ D i , let: g i (z i ) := sup z k 2 ~ D k 8k2 Cini u k 2 ^ D k 8k2 Ci P k2 Ci s k u N1 k s:t: z k =u k + P j2C k z j ; 8k2 C i (5.13) (5.13) can be interpreted as maximizing a certain utility function over the subtree rooted at node i2V T , given that the out ow sequence from node i is z i 2 R N . (5.13) is a generalization of (5.7), in the sense thatg 0 (0) is equal to the optimal value of (5.7). Since the objective function of (5.13) is linear and separable and the decision variables are coupled with only equality constraints of a simple form, standard distributed algorithm, for example, ADMM [21], can be used to solve (5.13) if sets ~ D k are convex. However, ~ D k are non-convex in general (cf. Remark 58). In order to 4 With a slight abuse of notation, we use u to denote control inputs for the original network as well as the reduced network. 143 reduce the complexity due to this non-convexity, we decompose (5.13) into the following nested form: g i (z i ) = sup zj2Zj8j2Ci ui2 ^ Di s i u N1 i + P j2Ci g j (z j ) s:t: z i =u i + P j2Ci z j (5.14) whereZ j := ~ D j \ ^ D j + P k2Cj Z k combines both the constraintsz j 2 ~ D j andz j =u j + P k2Cj z k , for allj2E T . The equivalence between (5.13) and (5.14) can be seen via induction. In particular, if i is a leaf node, thenC i =;. Both (5.13) and (5.14) reduce to g i (z i ) = max zi=ui2 ^ Di s i u N1 i . The optimal value function is g i (z i ) =s i z N1 i if z i 2 ~ D i \ ^ D i =Z i , and is1 otherwise due to infeasibility. Ifi is not a leaf node, then (5.14) can be interpreted as being associated with a local star subnetwork inT . Fig. 5.7 shows all such local (star) sub-networks for the network shown in Fig. 5.6b. 5 z 5 7 8 (a) 1 z 1 3 4 (b) 2 5 z 5 z 2 6 (c) 0 1 z 1 0 2 z 2 (d) Figure 5.7: The four local (star) sub-networks corresponding to Figure 5.6b. The solution to (5.13) is obtained by solving sub-problems in (5.14) over two iterations: (I) Compute g i :Z i ! R via (5.14) for every i2V T in the reverse topological order; (II) Set z 0 = 0. Following the topological order, for every i2V T , compute an optimal solution (u i ;fz j ;j2C i g) to (5.14) corresponding to g i (z i ). It is easy to check that, for all i2V T , we have 02 ^ D i and 02 ~ D i , and hence 02Z i . Therefore, iteration (II) is well-posed. 144 Remark 59. The optimal solution to (5.7) as computed by the decomposition approach is invariant with respect to the choice of root node, directions of the links inE T , and the topological ordering used for labeling nodes inV T . While the decomposition approach reduces complexity of solving (5.7) for general type II tree reducible networks, the bottleneck is still non-convexity of the local problems in (5.14). In Section 5.3.4, we show that, for constant controls, the two iterations involving solutions to the local problems in (5.14) admit closed-form expressions. The required machinery is rst developed in the next subsection in a general setting. 5.3.3 Input-output Properties of the Subproblem for Constant Control The original problem (5.13) and the sub-problem (5.14) is simplied to a great extent if only constant controls are considered: for all i2V T , the decision variables z i and u i reduce to one dimensional real numbers; and the sets ^ D i and ~ D i reduce to cubep 0 i and the collection of multiple one dimensional line intervals, respectively. Remark 68 explains how to obtain ~ D i in this case. As illustrated in Example 8, the line intervals for ~ D i can be half open, which can make (5.13) unsolvable. Nevertheless, because the objective function in (5.13) is linear and hence continuous, we use the closure of ~ D i in (5.13) and hence closure ofZ i in (5.14) for simplicity. An interior feasible point that is arbitrarily close to the solution to the problem over the closure can be obtained, as shown in Proposition 20. Therefore, we obtain the following problem, which generalizes (5.14) in the case of constant controls. g out (z) = f(g in j ;X j )g j2[n] := max x2R n P n j=1 g in j (x j ) s:t: 1 T x =z; x j 2X j ;8j2 [n] (5.15) whereX j R is the union of nite number of disjoint closed intervals for all j2 [n]. Operator maps from n input functions g in j with restricted domain X j to a single output function g out with 145 domain P n j=1 X j , where the domain ofg out is not explicitly written in (5.15). Due to the possible disconnected feasible set, (5.15) is in general non-convex. Remark 60. (5.15) becomes (5.14) with the following substitutions: i = 1,C i = [n]nf1g,g in 1 (x) = s i x, X 1 = cubep 0 i , g in j (x) =g j (x) and X j =Z j for all j2 [n]nf1g. We now show that a special class of functions are invariant through . This class of functions relates to the following construct. For a given point = ( 1 ; 2 )2 R 2 , dene a (continuous) piecewise linear function: : R! R as: (x) := 8 > > < > > : x 1 + 2 x 1 x + 1 + 2 x> 1 (5.16) As shown in Fig 5.8, function contains two rays joining at , which is referred to as the top point of . It is straightforward that intersects the vertical axis at (0; 2 j 1 j). (x) 0 x 2 j 1 j Figure 5.8: Illustration of (x) dened in (5.16). It is often useful to have a function dened over a restricted domain. When the restricted domain is a closed interval, Lemma 21 can be used to translate the top point into the domain, if not already inside, without changing function values. Lemma 21. Consider a point = ( 1 ; 2 )2 R 2 and a closed interval [b 1 ;b 2 ] R such that 1 62 [b 1 ;b 2 ], let 0 := (b 2 ;b 2 1 + 2 ) if 1 > b 2 and 0 := (b 1 ;b 1 + 1 + 2 ) if 1 < b 1 . Then (x) = 0(x) for all x2 [b 1 ;b 2 ]. 146 We are now ready to present the results on operator . We rst consider the case when every X j ,j2 [n], is a closed connected subset of R. In this case, the feasible set of (5.15) is convex and hence (5.15) is convex if input functions g in j are concave. The next result shows that functions are invariant through the operator if X j is a single piece of interval for all j2 [n]. Lemma 22. If g in j is a function (cf. (5.16)) with top point j = ( 1 j ; 2 j ) and X j = [q l j ;q u j ] (q l j 1 j q u i ) for all j2 [n], then the following hold true for dened in (5.15): (i) g out = f(g in j ;X j )g j2[n] is a function with top point 1 T 1 ; 1 T 2 and domain [1 T q l ; 1 T q u ]. (ii) the set of maximizers of (5.15) is x 2 [q l ; 1 ]j 1 T x =z if 1 T q l z < 1 T 1 ,f 1 g if z = 1 T 1 , and is x 2 [ 1 ;q u ]j 1 T x =z if 1 T 1 <z 1 T q u . where 1 := 1 [n] , 2 := 2 [n] , q l :=q l [n] and q u :=q u [n] are n dimensional vectors. Proof. First of all, it is clear that (5.15) is feasible only for z2 [1 T q l ; 1 T q u ]. Secondly, since g in j is concave and X j is convex, (5.15) is convex and strong duality holds. Therefore, it is sucient to consider the dual problem in order to solve (5.15). Let 2 R be the Lagrange multiplier associated with the constraint z = 1 T x. The dual function is then given by: () =z + max q l xq u n X j=1 j (x j ) +x j =z + n X j=1 max q l j xjq u j j (x j ) +x j = 0 @ n X j=1 x j ()z 1 A + n X j=1 j (x j ()) where x j ()2X j () := argmax q l v xjq u v j (x j ) +x j , for all j2 [n]. For 2 [1; 1], j (x j ) +x j is piecewise linear: nondecreasing with slope (1 + ) over (1; 1 j ] and nonincreasing with slope ( 1) over ( 1 j ; +1). Therefore, 1 j 2X j () for all 2 [1; 1] and j2 [n]. This implies that () is linear over [1; 1], for every z. In particular, X j (1) = [q l j ; 1 j ] andX j (1) = [ 1 j ;q u j ] for all j 2 [n]. For > 1, j (x j ) +x j is strictly increasing, and henceX j () =fq u j g. Since z 1 T q u , () is linear and non-decreasing over 147 (1; +1). Similarly, by considering <1, we haveX j () =fq l j g and () is linear and non- increasing over (1;1). Collecting these facts gives that, for every z2 [1 T q l ; 1 T q u ], the dual function () is convex and piecewise linear with possible break points at =1 and = 1. Therefore, g out (z) = min 2R () = minf(1);(1)g. As 1 j 2X j () for2 [1; 1], we have(1) =z1 T 1 +1 T 2 and(1) =z+1 T 1 +1 T 2 . (1)(1) for z2 [1 T q l ; 1 T 1 ], and (1)(1) for z2 [1 T 1 ; 1 T q u ]. Therefore, g out (z) = 8 > > < > > : z 1 T 2 + 1 T 2 1 T q l z 1 T 1 z + 1 T 1 + 1 T 2 1 T 1 z 1 T q u Comparing with (5.16) establishes (i). (ii) follows from the fact that, for a given optimal dual solution , x j ( )2X j ( ), j2 [n], is an optimal primal solution if and only if the constraint z = 1 T x is satised. Remark 61. 1. Lemma 21 implies that the condition 1 j 2X j in Lemma 22 is without loss of generality. In addition, the top point of g out is inside its domain, that is, 1 T 1 2 [1 T q l ; 1 T q u ]. 2. Lemma 22 implies the following. If g in j , j2 [n], are linear functions with identical slope 1 (and1, respectively), then g out is linear with the same slope 1 ( and1, respectively). For a set S [n], if g in j is a linear function with slope 1 (and1, respectively) for all j2S (and j2 [n]nS, respectively), then g out contains two linear pieces with slope 1 over domain [1 T q l ; 1 T 1 ] and slope1 over domain [1 T 1 ; 1 T q u ]. Moreover, forz2 [1 T 1 ; 1 T q u ]: x j =q u j for all j2S, and for z2 [1 T q l ; 1 T 1 ]: x j =q l j for all j2 [n]nS. 3. It is straightforward to see from the proof of Lemma 22 that the same results hold even if there exists j2 [n] such that X j is unbounded, that is q l j =1 and/or q u j =1. We now consider the case when X j is the union of a nite number of disjoint intervals. For all j2 [n], assume X j contains m j pieces of intervals and denote each piece by X k j , then 148 X j =[ mj k=1 X k j . Let := n j=1 [m j ] R n . The way to solve the nonconvex problem (5.15) is to decompose it into multiple convex subproblems. Each subproblem is associated with a com- bination 2 of intervals X j j and is denoted by () := f(g in j ;X j j )g j2[n] . The following is straightforward. f(g in j ;X j )g j2[n] = max 2 () (5.17) If the restriction of every input functiong in j overX k j is a function for allk2 [m j ], then Lemma 22 can be used to solve the subproblem () for all 2 . This is illustrated by the next example. Example 9. Consider the case shown in Fig. 5.9a: n = 2, m 1 = m 2 = 3, X 1 1 = [3;2:5], X 2 1 = [1; 1], X 3 1 = [2:5; 3], X 1 2 = [3;2], X 2 2 = [1:5; 1:5], X 3 2 = [2; 3]. convX 1 = convX 2 = [3; 3]. Functiong in 1 : [3; 3]! R andg in 2 : [3; 3]! R are functions with top point (2:5; 6) and (1:5; 5), respectively. Since the input functions are functions over the convexied domains, the output function g out cvx := fg in j ; [3; 3]g 2 j=1 from the convexied problem is a function with top point (1; 11), as shown using dotted line in Fig 5.9b. For every2f1; 2; 3gf1; 2; 3g, the output functiong out := () from the subproblem is a function upper bounded byg out cvx , as shown using solid line in Fig 5.9b. More specically, for2f(1; 1); (2; 1)g and2f(3; 2); (3; 3)g,g out is linear and coincides with the left and right wings of g out cvx , respectively, and for 2f(1; 1); (2; 1)g, g out is function that contains two pieces and is smaller than ~ g on at least one piece. As a result, the output function g out := f(g in j ;X j )g j2[n] consists of two pieces. Each piece is a function. Denition 18. A single variable real valued function g : X R! R is called a piecewise function if the domain X is the union of multiple disjoint intervals, and over each of these intervals, g is a restricted function as dened in (5.16). Remark 62. The domainX of a piecewise function, as dened in Denition 18, is not necessarily connected. 149 3 2 1 0 1 2 3 1 2 3 4 5 x g in (a) 6 4 2 0 2 4 6 2 4 6 8 10 z g out g out cvx g out (b) Figure 5.9: (a) Input functions to ; (b) Output function from each subprob- lem. Example 9 shows that the output function from is a piecewise function if the input functions are all function. The next result generalizes this observation and shows that piecewise functions are invariant through operator . Proposition 18. If, for allj2 [n],X j R is the union of disjoint and closed intervals (including rays) and g in j :X j ! R is a piecewise function, then fg in j ;X j g n j=1 is a piecewise function. Proof. The proof follows straightforwardly from (5.17), Lemma 22, Remark 61 and the fact the point-wise maximum of multiple functions is a piecewise function. While the point-wise maximum of multiple functions is not necessarily a function, as demonstrated by Example 9, the next result gives the necessary and sucient conditions for the output function from to be a function given that all the input functions are functions. The proof is provided in Appendix A.6. Proposition 19. For all j 2 [n], let X j R be the union of disjoint and closed intervals (including rays) and let g in j : convX j ! R be a function with top point j = ( 1 j ; 2 j ). Then f(g in j ;X j )g j2[n] = f(g in j ; convX j )g j2[n] if and only if (1) 1 j 2X j for all j2 [n]; and (2) both P n j=1 X j \ (1; 1 j ] and P n j=1 X j \ [ 1 j ;1) are connected sets. 150 The second condition in Proposition 19 is related to the relative lengths of the intervals and the gaps between them, and the number sets in summation, as illustrated by Lemma 23 and 24. Lemma 23. Let X 1 be a closed interval and X j be the union of multiple disjoint and closed intervals. LetjX j j := maxX j minX j and j be the maximal length of the gaps between intervals of X j . Assume X j is labeled such that j j+1 for all j, then P j X j is connected if j+1 P j k=1 jX j j for all j. Remark 63. Lemma 23 implies that the second condition in Proposition 19 is satised if X j = R for some j2 [n]. Lemma 24. Considerm disjoint closed intervals denoted byX j ,j2 [m], and labeled in increasing order, that is, maxX j < minX j+1 for allj2 [m1]. LetjX j j := maxX j minX j for allj2 [m], j = minX j+1 maxX j for all j2 [m 1], and X =[ m j=1 X j . Then P n i=1 X is connected for n 1 + max m1 i=1 j = minfjX j j;jX j+1 jg. Proof. We rst prove for m = 2. Let X 1 =: [x 1 ;x 2 ] and X 2 =: [x 3 ;x 4 ]. Then n X i=1 X =[ n k=0 [(nk)x 1 +kx 3 ; (nk)x 2 +kx 4 ] The necessary and sucient condition for P n i=1 X to be connected in this case is (nk)x 1 +kx 3 (nk+1)x 2 +(k1)x 4 for allk2 [n], which is equivalent tox 3 x 2 (n1) minfx 2 x 1 ;x 4 x 3 g. The condition can be equivalently written as n 1 + j = minfjX 1 j;jX 2 jg. Then the assertion for m> 2 follows by considering each adjacent pair of intervals separately. 5.3.4 Optimal Constant Control Action for Type II Tree Reducible Networks Lemma 22, Proposition 18 and Proposition 19 can be used in the two iteration algorithm proposed in Section 5.3.2 to obtain an optimal load shedding control action within the class of constant 151 control actions (cf. Denition 17) for type II tree reducible networks. We provide the explicit algorithm in Algorithm 1 for the case when the conditions in Proposition 19 are satised. The algorithm for the general case can be extrapolated from this straightforwardly. With Remark 60 showing the correspondence between (5.14) and (5.15), we use the following additional notation in Algorithm 1. Let i = ( 1 i ; 2 i ) be the top point and [z l i ;z u i ] :=Z i be the domain of functiong i (z) for alli2V; the domain and the top point of functions i z i are cubep 0 i and (p 0 i ;jp 0 i j), respectively. For a real numberx, let [x] + := maxfx; 0g and [x] := minfx; 0g. Hence, cubep 0 i [p 0 i ] ; [p 0 i ] + . Furthermore, letq i > 0 be the maximal value of ~ D i for the case of constant control for all i2E T . Remark 58 implies thatq i is the minimal value of ~ D i . Algorithm 1 shows the two iterations using two for loops. The rst loop computes i ,z l i andz u i for alli2V in reverse topological order according to Lemma 22, where (p 0 i ;jp 0 i j) on line 3 is a top point for functions i z i . The second loop extracts a particular maximizer from the solution set shown in Lemma 22. Remark 64. In Algorithm 1, the specic value of i is chosen to ensure that the constraint z i = u i + P j2Ci z i is satised. While one can choose other values of i , this particular choice ensures that the resulting maximizer (u i ;fz j g j2Ci ) on lines 12 and 15 to lie in cube 1 at every iteration. 5.4 An equivalent state aggregation approach The computational approach proposed in Section 5.3 has provable guarantees only for type II tree reducible networks. In this section, we return to the approach outlined in Section 5.2.3, and (5.9) in particular, for a general network topology. We recall that, in its current form, (5.9) is not amenable to implementations because the underlying state space is uncountably innite. In this section, we develop an equivalent nite abstraction of the state space through state aggregation, and correspondingly develop an aggregated version of (5.9). 152 Algorithm 1: Optimal constant control for tree reducible network. input : reduced treeT = (V;E), initial supply-demand vector p 0 2 R V , q l 2 R E , q u 2 R E output: optimal constant load shedding control action u2 R V 1 for i2V in reverse topological order do 2 z l i = max n q l i ; [p 0 i ] + P j2Ci z l j o , z u i = min n q u i ; [p 0 i ] + + P j2Ci z u j o ; 3 1 i =p 0 i + P j2Ci 1 j , 2 i =jp 0 i j + P j2Ci 2 j ; 4 update ( 1 i ; 2 i ) according to Lemma 21 if 1 i 62 [q l i ;q u i ] 5 end 6 z 0 = 0; 7 for i2V in topological order do 8 if z i = 1 i then 9 u i =p 0 i , z j = 1 i for all j2C i ; 10 else if 0z i < 1 i then 11 i = z i P j2Ci [ 1 j ] [p 0 i ] = P j2Ci [ 1 j ] + + [p 0 i ] + ; 12 u i = minfp 0 i ; i p 0 i g, z j = minf 1 i ; i 1 i g for all j2C i ; 13 else if 1 i <z i < 0 then 14 i = z i P j2Ci [ 1 j ] + [p 0 i ] + = P j2Ci [ 1 j ] + [p 0 i ] ; 15 u i = maxfp 0 i ; i p 0 i g, z j = maxf 1 i ; i 1 i g for all j2C i ; 16 end 17 end 18 return u 5.4.1 A State Aggregation Approach The key idea is to develop a nite consistent partition of one time step reachable sets. We recall a few standard terminologies. A cover of a setS is a collection of nonempty subsetsfS i g i2I ofS such that S =[ i2I S i and a partition is a cover with pairwise disjoint elements. We call a cover or partition nite if it contains nitely many elements. Furthermore, for a network state (E;p), a partitionfS i g i2I of set SB E is said to be consistent if,F E (E;u) =F E (E; ~ u) for all u; ~ u2 S i , i2I. Consistency implies that it is valid to writeF E (E;u)F E (E;S i ) for all u2S i and i2I. Note here that the set S i is not necessarily the set of feasible control actions U(E;p), and can be any arbitrary set of balanced supply-demand vectors. We extend the notion of the set of feasible control actions (5.4) as follows: for a link setE and a set PB E , U(E;P ) :=[ p2P U(E;p) =B E \ cubeP (5.18) 153 where cubeP :=[ p2P cubep. A nite consistent partition of the set of control actions induces a natural nite cover of the state space at each stage in the cascading dynamics. At t = 0, the state spacef(E 0 ;p 0 )g is a sin- gleton, and thereforef(E 0 ;P 0 )g forms a trivial partition withP 0 :=fp 0 g. LetfU 0 i g i2I1 be a nite consistent partition ofU(E 0 ;P 0 ). Then att = 1, the reachable state space (F E (E 0 ;u);u)ju2U(E 0 ;p 0 ) is covered byf(E 1 i ;P 1 i )g i2I1 =f(F E (E 0 ;U 0 i );U 0 i )g i2I1 . LetfU 1 j g j2I i 2 be a nite consistent par- tition of U(E 1 i ;P 1 i ), for all i 2 I 1 . Then at t = 2, the state space reachable from (E 1 i ;P 1 i ), (F E (E 1 i ;u);u)ju2U(E 1 i ;P 1 i ) , is covered byf(E 2 j ;P 2 j )g j2I i 2 =f(F E (E 1 i ;U 1 j );U 1 j )g j2I i 2 , for all i2I 1 . Repeated application of this procedure to all the subsequent stages then gives the desired nite representation. Let (E t ;P t ) denote an arbitrary element of the cover at time t. A natural extension of (5.2) to dynamics over aggregated states is as follows: E t+1 ;P t+1 =F E t ;P t ;U t ; U t 2U(E t ;P t ) (5.19) where F E (E;P;U)F E (E;U) :=fi2Ejc i f i (E;u)c i for all u2Ug F P (E;P;U)F P (U) :=U (5.20) U(E t ;P t ) is dened by the particular choice of consistent partition of U(E t ;P t ), and serves as the set of feasible aggregated control actions at state (E t ;P t ). We associateE with a vector 2f1; 0;1g E . i is used to denote whether the ow capacity constraint of link i2E is satised or not: the ow stays within capacities for i = 0 and exceeds the upper and lower capacities for i = 1 and i =1, respectively. The consistent partition used in this paper is: U(E;P ) := U(E;P; j )jj2I(E;P ) (5.21) 154 where I(E;P ) := jjU(E;P; j )6=; U(E;P;) :=fu2U(E;P )jf i (E;u)<c i for i =1;c i f i (E;u)c i for i = 0; f i (E;u)>c i for i = 1;8i2Eg (5.22) It is straightforward to see thatU(E;P ) dened in (5.21)-(5.22) is a consistent partition ofU(E;P ). IfU(E;P ) is a polytope, then each member ofU(E;P ) is also a polytope, with possibly half open boundary due to the strict inequality in U(E;P;). An algorithmic procedure to compute and store the partition U dened in (5.21)-(5.22) will be presented in Section 5.5. State aggregation gives a nite tree (cf. Section 5.2.3), whose nodes are aggregated states and arcs are aggregated control actions. The set of aggregated goal states isS :=f(E;P )j (E;p)2S;8p2Pg and the utility associated with an aggregated state (E;P )2S is sup p2P s T p = max p2cl(P) s T p, where cl(P ) denotes the closure of P . The optimal search over the aggregated tree can be performed through the following calculations, which are adaptations of (5.9): J 1 (E;P ) = max u2U(E;cl(P)) s T u s.t. c E f(E;u)c E (5.23a) J t (E;P ) = max U2U(E;P) J t1 (F E (E;U);U); t = 2;:::;N (5.23b) where J t (E;P ) is the maximum among values of all the aggregated states that can be reached in at most t time steps starting from (E;P ). Similar to (5.9a), the ow constraint is imposed in (5.23a) to ensure the additional constraint on u N1 (cf. Remark 51). This implies that the unique (and optimal) aggregated control action associated with J 1 (E;P ) is U N1; =U(E;P; 0). This is to be contrasted with (5.23b) for t 2 that the optimal aggregated control action U t; is not trivial to obtain. (5.23a) maximizes a linear function over a bounded closed set and (5.23b) 155 maximizes over the nite setU(E;P ). Therefore, the optimal value is achievable on every iteration in (5.23). The next result shows that the iterations in (5.23) give the same value as that in (5.9). Theorem 6. Consider a network with initial state (E 0 ;p 0 ), link weights w 2 R E 0 0 and link capacities c2 R E 0 0 . For every aggregated state (E;P ) obtained from the consistent partition in (5.21)-(5.22), the computations in (5.9) and (5.23) satisfy the following for t2 [N]: J t (E;P ) = sup p2P J t (E;p) Proof. We prove by induction. For t = 1: sup p2P J 1 (E;p) = sup p2P max u2U(E;p) s T u s.t. c E f(E;u)c E which is equal to J 1 (E;P ) as shown in (5.23a). Suppose the claim is true for t2 [k]. J k+1 (E;P ) = max U2U(E;P) J k (F E (E;U);U) = max U2U(E;P) sup u2U J k (F E (E;u);u) = sup u2U(E;P) J k (F E (E;u);u) = sup p2P sup u2U(E;p) J k (F E (E;u);u) = sup p2P J k+1 (E;p) where the rst equality is due to (5.23b); the second equality is due to the induction assumption; the third and forth equalities are due to[ U2U(E;P) U =U(E;P ) =[ p2P U(E;p), as implied by the denitions of U(E;P ) and U(E;P ); and the last equality is due to (5.9). We make the following remarks on the state aggregation and aggregated tree search. Remark 65. 1. Every path of length N in the aggregated search tree corresponds to a, possibly dierent, topology sequence that can occur in the cascading dynamics. The state aggregation provides 156 a way to quantify the complexity of a general optimal control problem. Details are provided in Section 5.6.1. 2. The aggregated tree search based on (5.23) can be interpreted as a systematic way to decom- pose the nonconvex feasible setD R NjVj in (5.7)-(5.8) into a nite number of subsets. Each subset is a polytope and corresponds to a topology sequence and aggregated control action sequence. For example, the subset corresponding to (U 0 ;:::;U N1 ) is N1 t=0 U t . Sim- ilar to (5.17), the optimal value of (5.7) is equal to the maximum among the optimal values of multiple subproblems associated with the subsets. Each subproblem is a linear program of the form sup u2 N1 t=0 U t s T u N1 and indeed coincides with (5.23a). 3. The state aggregation approach allows to include running cost into the problem formulation. In that case, instead of a linear program, a dynamic program with constraint u2 (u 0 ;:::;u N1 )2 N1 t=0 U t ju t 2 cubeu t1 ; 1tN 1 is to be solved for the path (U 0 ;:::;U N1 ). 5.4.2 Optimal Control Synthesis: From Aggregated to the Original State Space The numerical implementation of (5.23) is shown in Section 5.5. Given such a procedure to compute U = (U 0; ;:::;U N1; ), we next present a result to derive u = (u 0; ;:::;u N1; ), i.e., control actions for the cascading failure dynamics in (5.2)-(5.3) over the unaggregated state space. However, since the set of feasible control action sequencesD is not necessarily closed (cf. Remark 51(ii)), nding u whose cost is exactly the same as that of U may not be possible. It is however possible to nd u whose cost is arbitrarily close to that of U . 157 Proposition 20. For a network with initial state (E 0 ;p 0 ), link weights w2 R E 0 0 , and link ca- pacities c2 R E 0 0 , consider J N (E 0 ;fp 0 g) computed by (5.23). For every > 0, there exists ~ u2D such that J N (E 0 ;fp 0 g)s T ~ u N1 J N (E 0 ;fp 0 g). Proof. For brevity, in this proof, we let J N (E 0 ;fp 0 g) J N (E 0 ;p 0 ). Theorem 6 implies that J N (E 0 ;p 0 ) =J N (E 0 ;p 0 )s T u N1 for allu2D. Therefore, we only prove the second inequality. Let U be an optimal aggregated control sequence associated with computing J N (E 0 ;p 0 ) in (5.23), and letE be the induced active link set sequence (recallU N1; =U(E N1; ;U N2; ; 0)). Let u N1; be a maximizer to (5.23a) for J 1 (E N1; ;U N2; ). Then, u N1; 2 cl(U N1; ) and J N (E 0 ;p 0 ) = s T u N1; . We now show that, for arbitrary > 0, there exists ~ u2D such that s T ~ u N1 s T u N1; . Let M(u;) be the open ball centered at u2 R V with radius . Since u N1; 2 cl(U N1; ), U N1; \M(u N1; ;=jV l j)6=; for every > 0. It is then possible to pick ~ u N1 2 U N1; \ M(u N1; ;=jV l j) such that ~ u N1 6=u N1; and s T ~ u N1 >s T u N1; . It is now sucient to show that there exist ~ u 0 ; ~ u 1 ;:::; ~ u N2 such that ~ u t+1 2 cube ~ u t and ~ u t 2U t; for all 0tN2. We provide details for ~ u N2 ; the reasoning for ~ u N3 ;:::; ~ u 0 follows along the same lines. Since u N1; 2U(E N1; ; cl(U N2; )), there exists u N2; 2 cl(U N2; ) such that u N1; 2 cubeu N2; . Hence, we can pick ~ u N2 2 U N2; \ M(u N2; ;ku N1; ~ u N1 k 2 ) so that ~ u N2 6= u N2; and ~ u N1 2 cube ~ u N2 , where the special choice of ~ u N1 6= u N1; ensures that M(u N2; ;ku N1; ~ u N1 k 2 ) has positive radius. Proposition 20 implies that, in order to nd u , it is sucient to solve for U and u N1; from (5.23). Because the value ofJ 1 (E;P ) depends only on cl(P ) in (5.23a), the value ofJ t (E;P ) depends only on cl(U) and cl(P ) in (5.23b) for every t 2. Therefore, in order to nd U , for 158 sake of numerical implementation, we use (E; cl(P )) and cl(U) in place of (E;P ) and U, without introducing error in computation of U . Equivalently, we use the following variant of (5.22): U(E;P;) :=fu2U(E;P )jf i (E;u)c i if i =1;c i f i (E;u)c i if i = 0; f i (E;u)c i if i = 1;8i2Eg (5.24) Therefore, without stating explicitly, hereafter we use (5.24) in place of (5.22). In order to nd u N1; , the next result implies that (5.23a) is a linear program by showing that, for every (E;P ), the set U(E;P ) is a polytope. Lemma 25. Consider a network with initial state (E 0 ;p 0 ), link weights w2 R E 0 0 and link ca- pacities c2 R E 0 0 . For every aggregated state (E t ;P t ), t2 [N][f0g, induced by the consistent partition in (5.21) and (5.24), both P t and U(E t ;P t ) are polytopes. Proof. The claim is proved by induction. It is easy to see that P 0 =fp 0 g and U(E 0 ;P 0 ) = U(E 0 ;p 0 ) are polytopes. Suppose P t and U(E t ;P t ) are polytopes for some t2 [N 1][f0g. It is sucient to show that, for arbitrary aggregated control action U2U(E t ;P t ), the resulting P t+1 and U(E t+1 ;P t+1 ) are polytopes. P t+1 = U t = U(E t ;P t ;) for some 2f1; 0; +1g E t (cf. (5.21)-(5.22)). Combining this with the induction assumption that U(E t ;P t ) is a polytope, we get that P t+1 is a polytope. It follows from the denition in (5.18) that U(E t+1 ;P t+1 ) is a polytope as well. Remark 66. Combined with the denition of cubeP , the proof of Lemma 25 also implies that both P and U(E;P ) are contained in a closed orthant of R V l for every (E;P ). 5.4.3 Ecient Aggregated Tree Search With (5.19) and (5.21) specifying how to expand nodes of aggregated network state and (5.23) directing the goal of search, one can then employ any classical tree search algorithm, e.g., the ones in [77, Chap 3], to solve the problem. Regarding the implementation of tree search algorithms, 159 the following remarks are in order. Firstly, the following relationship can be used for tree pruning in a standard branch and bound algorithm framework. J 1 (E;P )J t (E;P ) max p2cl(P) s T p 8 (E;P );8t2 [N] Secondly, iterative deepening depth-rst search algorithm presents several advantages for the optimal control problem. On one hand, it achieves a good balance between time and space complexities. This is of particular importance because, as would shown in Section 5.6.1, the number of aggregated states in the optimal control problem can be quite large. On the other hand, the search can be stopped anytime in the process of computation while producing a feasible control action with reasonable performance. In fact, the search over the rstt<N layer provides an optimalt-stage load shedding scheme. At the same time, the upper bound provides an estimate of performance gap from the optimal value when search is terminated early. Finally, while the detailed search algorithm including pruning is standard and omitted here, its implementation with set objects, that are, the aggregated control action U2U(E;P ) and the set U(E;P ), require additional tools. Section 5.5 is devoted to this particular problem. 5.5 Computing Aggregation Through Arrangement of Hyperplanes The numerical implementation of (5.23) relies critically on proper representation of the set of fea- sible aggregated control actionsU(E;P ) and its partitionU(E;P ). While Lemma 25 characterizes an important property of these objects, in this section, we provide an algorithmic procedure for their representation. Our machinery relies on and extends tools from the domain of arrangement of hyperplanes e.g., see [38] [85, Chapter 24], and convex polytopes, e.g., see [92] [49]. 160 1 3 2 4 e1 e2 e3 e4 (a) u 2 o u 1 d e r a b c f 4 f 2 f 3 U 1 (b) o d e r (c) a b c (d) Figure 5.10: (a) A network (V;E 0 ) withV s =f1; 2g,V d =f4g,w = 1,c = [10; 3; 3; 6] T , p 0 = [5;5; 0; 10] T ; (b) Projection of U(E 0 ;p 0 ) and u2B E jf i (E 0 ;u) =c i , i 2 f2; 3; 4g, on u 1 u 2 plane; (c) Incidence graph of U(E 0 ;p 0 ); (d) incidence graph of U 1 . 5.5.1 Arrangement of Hyperplanes, Polytope and Incidence Graph We start with the simple illustrative example shown in Fig. 5.10a. Referring to (5.5), there are three non-trivial components of p 0 . Taking into account the additional constraint imposed by B E , referring to (5.4), the set of feasible control actions U(E 0 ;p 0 ) can be completely understood in terms of its two dimensional projection, say on the u 1 u 2 plane. In Fig. 5.10b, the box oder and the point e corresponding to the projections of U(E 0 ;p 0 ) andp 0 respectively. The solid lines labeled by f 2 ;f 3 ;f 4 , or (projections of) hyperplanes, correspond to the capacity constraints associated with the three links. The ow capacity constraint for e 1 is ignored here because it is satised by all u2 U(E 0 ;p 0 ) and hence irrelevant for the problem. Line f 2 , f 3 and f 4 dissect the box oder into seven pieces. These seven pieces constitute the partition U(E 0 ;fp 0 g). Each piece, e.g., the triangle abc denoted by U 1 , is an aggregated control action, e.g., U 1 = U(E 0 ;p 0 ; [0; 0; 0; 1] T ). In general, a nite collectionH of hyperplanes in R d dissects R d into nitely many connected pieces of various dimensions. The collection of these pieces is called the arrangement, denoted byA(H), induced byH , and each piece is called a face, denoted by , of the arrangement 5 . 5 In some literature, cells are used to refer to the connected pieces. In that context, face is used exclusively for the 2-face. In this chapter, we adopt the terminology convention in [38] and use cell to denote d-face exclusively inR d . 161 The dimension of a face is the dimension of its ane hull; a k dimensional face is called a k- face, denoted by k . For convenience, 0-face, 1-face, (d 2)-face, (d 1)-face and d-face are, respectively, referred to as vertex, edge, ridge, facet and cell. We call two faces incident if one is contained in the boundary of the other and if the dierence in their dimensions is one. In a pair of incident faces, the lower (or higher) dimensional face is called the subface (or superface) of the other. In Fig. 5.10b, the three solid lines (i.e., hyperplanes) dissect box oder into seven cells, nine edges (or facets) and three vertices (or ridges). As indicated by this example, in the setting of this chapter, for every state (E;P ), the capacity constraints, balanced condition (captured by B E ), and load shedding requirement (captured by cube(P )) form the collection of hyperplanes. We are interested in the substructure of the arrangement of these hyperplanes inside U(E;P ), since the closure of each facet in this arrangement corresponds to an aggregated control action, and U(E;P ) corresponds to the collection of these facets 6 . The closure of a bounded face in the arrangement is a convex polytope, or polytope for short. We use the same letter P , as in the aggregated state, to denote a general polytope for simplicity, because every aggregated state is a polytope, as shown in Lemma 25. Formally, a polytope is a point setP R d that can be presented either as a convex hull of a nite number of points in R d or the bounded intersection of nite number of closed half spaces in R d [92]. The same notion of face, as in an arrangement, is used for a polytopeP R d 7 and furthermore, a face ofP is can be described as =P\ n x2 R d j T x = 0 o , where the linear inequality T x 0 must be satised for all x2 P , that is, the hyperplane n x2 R d j T x = 0 o must contain P on one of its closed sides. We call P R d full dimensional if its dimension is d. For a full dimensional polytope, the ane hull of its facet d1 i is a hyperplane, denoted by H i = n x2 R d j ( i ) T x = i 0 o and referred as the dening hyperplane of facet d1 i . As a convention, the direction of i for H i is 6 Fig. 5.10b shows the projection of the arrangement inR 3 onto the u 1 u 2 plane. Each cell in Fig. 5.10b is the projection of a facet of the arrangement. 7 The notion of face is slightly dierent for an arrangement and a polytope. While the former considers a face as an open set, the latter treat a face as as a closed set. This dierence does not aect the results in this section and ignored. 162 chosen to point outwards from P . Everyk-face k ofP (k< dimP ) is the convex hull of exactly k + 1 vertices and every ridge d2 is contained in exactly two facets [49, Chapter 3]. The geometry of the arrangement of hyperplanes and polytopes is dicult to comprehend, especially in high dimensions. Because of this, they are represented using the incidence graph (sometimes called the facial lattice or face lattice). The incidence graph of an arrangement or a polytope contains the incidence relationship between various faces. It is a layered (undirected) graph whose nodes have a one-to-one correspondence with faces of the arrangement or polytope, and an (undirected) edge exists between two nodes if and only if the corresponding faces are incident. All the nodes corresponding to faces of the same dimension constitute a layer. We place the layer corresponding to vertices at the bottom, and the layer corresponding to cells on the top. For example. Fig. 5.10c shows the incidence graph of the polytope oder (or action set U(E 0 ;p 0 )). The single node at the top layer corresponds to oder itself, the four nodes in the middle layer correspond to the four edges or, re, ed and do, and the four nodes in the bottom layer correspond to the four vertices o, r, e and d. The edges between these layers correspond to the incidence relation between the faces, as shown in Figure 5.10b. Similarly, 5.10d shows the incidence graph of the polytope acb (or aggregated control action U 1 ). Furthermore, we have the following remark on the auxiliary information required for storing the incidence graph of an arrangement or a polytope. Remark 67. When implemented, the incidence graph is usually associated with some auxiliary information that enables numerical algorithms with geometric operations such as determining if a hyperplane intersecting with a face and nding the intersection of hyperplane and edge. Details of this can be found in textbooks, e.g., [36] and [38]. While other choice of auxiliary information is possible, what we use in this chapter includes the following: the analytical expression of hyper- planes, coordinate of vertices and the mean of the coordinates of all contained vertices of a face. We shall not explicitly mention these auxiliary information hereafter. 163 5.5.2 On Construction of the Incidence Graph of U(E;P) and U(E;P) We recall from Section 5.4.3 that the implementation of (5.23) relies on an ecient procedure to construct representation of the one-time aggregated reachable set from an arbitrary aggregated state (E;P ). It is sucient to get such a representation for U(E;P ), and its partition U(E;P ) consisting of aggregated control actions. Ideally, such a representation for U(E;P ) and U(E;P ) should build upon the representation for P . With regards to U(E;P ), it is sucient to focus on cubeP , since U(E;P ) is the intersection of cubeP with hyperplanes corresponding toB E (cf. (5.18)). Upon constructing the incidence graph of cubeP , one can add hyperplanes corresponding to balance conditions to get the arrangement forU(E;P ). The incidence graph of the arrangement, associated withU(E;P ), is then obtained by adding to the incidence graph ofU(E;P ), one by one, the hyperplanes corresponding to the ow capacity constraints while maintaining the representa- tion of arrangement of the objects added so far. The key step for updating the incidence graph after adding a hyperplane is adopted from the well-known algorithm in [39] and [38, Chapter 7] for constructing arrangement of an arbitrary setH of hyperplanes in R d . Therefore, in this chapter, we focus our eorts on constructing the incidence graph of cubeP , given the incidence graph of P , and implicitly assume that one can adopt the standard algorithm to complete the construction of incidence graph of U(E;P ), and then the arrangement associated with U(E;P ). Thereafter, the aggregated tree search in Section 5.4.3 can be implemented by constructing alternatively the incidence graphs of U(E;P ) and U(E;P ) throughout the search process. Remark 68. (1) This algorithm in [39] and [38, Chapter 7] has optimal time complexity (jHj d ) for con- structing a general positioned arrangement in R d . If the arrangement is contained in a k dimensional ane space, the time complexity is O(jHj dk ). Note for a connected net- work with active link setE, U(E;P ) is contained in thejV l j 1 dimensional ane space B E \ n p2 R V jp i = 0;8i2VnV l o . 164 (2) For a network with single supply and single demand, the hyperplanes are single points and control actions are associated with line intervals that can be represented using two real numbers. The relevant incidence graph at a state (E;P ) can be computed in linear time with respect to the number of infeasible links. Furthermore, in this case cube(P ) can be computed in constant time: forP = (p l ;p u ], cube(P ) = [0;p u ] ifp u 0 and cube(P ) = [p l ; 0] if p l 0. (3) The set ~ D i used in (5.13) and (5.14) can be obtained similarly except that U(E;P ) needs to be replaced with the interval from origin to the minimum cut capacity of subnetworkG i , at the step of constructing the incidence graph of U(E;P ) (also see Remark 58). (4) The restricted set of ~ D i for constant control used in Section 5.3.3 is obtained in the same way as that for ~ D i . The dierence is that U(E;P ) is to be replaced with interval P . 5.5.3 Constructing the Incidence Graph of cubeP from the Incidence Graph of P Remark 66 implies that it is sucient to focus onP2 R d 0 . We rst consider cubep 0 forp 0 2 R d 0 . Since the dimensions corresponding to p 0 i = 0 can be ignored, we assume p 0 2 R d >0 without loss of generality. In this case, cubep 0 R d 0 is a hypercube and its incidence graph can be obtained straightforwardly by Lemma 26, whose proof is omitted. Fig. 5.10c shows an example in R 2 . Lemma 26. For p 0 2 R d >0 , let : cubep 0 !f1; 0; 1g d be dened as: i (x) =1 if x i = 0, i (x) = 0 if x i 2 (0;p 0 i ), and i (x) = 1 if x i =p 0 i . Then, (i) every ~ 2f1; 0; 1g d is associated with a (dj~ j 1 )-face (~ ) := cl( x2 cubep 0 j(x) = ~ ) of cubep 0 ; (ii) two faces ( 1 ) and ( 2 ) of cubep 0 are incident if 1 and 2 are equal except for one component which equals zero in one among 1 and 2 . 165 Remark 69. Lemma 26 (i) describes a procedure to enumerate all the nodes in the incidence graph of cubep 0 , in terms of all vectors inf1; 0; 1g d , whereas (ii) species how to add edges to the incidence graph. For a general polytopeP R d 0 , we present a sequential procedure to construct the incidence graph of cubeP from that ofP . For this purpose, we dene the projection and sweep of a polytope P R d 0 in direction 1 k , k2 [d]: proj k (P ) :=fpp k 1 k jp2Pg (5.25) sweep k (P ) :=fp k p k 1 k jp2P; k 2 [0; 1]g (5.26) It is straightforward that P sweep k (P ) and proj k P sweep k (P ). In fact, sweep k (P ) is the trace ofP projecting to proj k P in the direction of 1 k and therefore, it is also a polytope in R d 0 . One can again apply sweep on sweep k (P ) along 1 i for some i6=k and get sweep i (sweep k (P )) = p 1 p k 1 k 2 p i 1 i jp2P; ( 1 ; 2 )2 [0; 1] 2 . This motivates us to dene sweep for an index set I [d] as sweep I (P ) := ( p X k2I k p k 1 k jp2P; k 2 [0; 1]8k2I ) Similarly, proj I (P ) := p P k2I p k 1 k jp2P . With this denition, sweep ; (P ) = P and sweep [d] (P ) = cubeP . sweep [d] (P ) can be obtained by recursively applying sweep on P , e.g., sweep [d] (P ) = sweep 1 (sweep 2 (::: sweep d (P ))) Therefore, in order to obtain cubeP , it is sucient to focus on constructing sweep k (P ) from P for a given k2 [d]. Let H := n x2 R d jx k = 0 o and H + := n x2 R d jx k 0 o . 8 For a given polytope P H + and k2 [d], sweep k (P ) relates to projection between two ane spaces: aP R d and 8 We do not show subscript k for brevity in notation. 166 a proj k P H. We dierentiate between the following two scenarios based on the dierence in dimensions of these two anes spaces: (I) dimP = dim(proj k P ); and (II) dimP = dim(proj k P )+ 1. Scenario I occurs when P H := n x2 R d j T x = 0 o with k 6= 0, i.e., when H is not perpendicular to H. In this case, there is a one-to-one correspondence between the points in P and proj k P . On the other hand, scenario II occurs when either dimP =d, or every hyperplane containing P is orthogonal to H. The two scenarios are illustrated in Figure 5.11 for R 2 and k = 1. x 2 0 x 1 P proj 2 P sweep 2 P (a) x 2 0 x 1 P proj 2 P sweep 2 P (b) x 2 0 x 1 P proj 2 (P) sweep 2 (P) (c) x 2 0 x 1 P proj 2 P sweep 2 P (d) Figure 5.11: Two possible scenarios for projection: (a) and (b) show the projection onto a space of the same dimension; and (c) and (d) show projection onto a space of lower dimension. Scenario I In this scenario, proj k (P ) is anely isomorphic 9 to P . Therefore, its the incidence graph is identical to that of P . The following results relates the incidence graph of sweep k (P ) to that of P and proj k (P ). Proposition 21. Consider an n dimensional polytope P n x2 R d jx k > 0 o , 0 n d 1. If there exists a hyperplane containing P which is not perpendicular to H, then: (i) sweep k (P ) is an (n + 1) dimensional polytope; (ii) an l-face of sweep k (P ) is either a l-face of P or of proj k P , or it is sweep k l1 for some (l 1)-face l1 of P ; and (iii) each face of P and of proj k P is a face of sweep k (P ), and sweep k is a face of sweep k P for every face of P . 9 Two polytopes P 1 R d 1 and P 2 R d 2 are anely isomorphic to each other if there exists an ane and bijection map between them. 167 Proof. Let ^ := 1:2 max x2P x k . SinceP is contained in a hyperplane that is not perpendicular to H, the line segment [0; ^ 1 k ] is not parallel to aP . Let ^ P :=P [0; ^ 1 k ]. We have the following facts on ^ P [49, Chapter 4.4]: ^ P is an (n + 1) dimensional polytope; anl-face of ^ P is either al-face of P or of Pf^ 1 k g, or it is the vector-sum off^ 1 k g with some (l 1)-face of P ; each face of P and of Pf^ 1 k g is a face of ^ P and the vector-sum off^ 1 k g with any face of P is a face of ^ P . It is then sucient to show that sweep k P and ^ P are combinatorial isomorphic 10 . By denition, sweep k (P ) = ^ P\ H + . The facets of ^ P includeP H + ,P [0; ^ 1 k ] H := n x2 R d jx k < 0 o and n1 [0; ^ 1 k ] for all ridge n1 ofP . The facet n1 [0; ^ 1 k ] is orthogonal to and intersects with H on proj k n1 for every ridge n1 ofP . It is then straightforward to see that sweep k (P ) and ^ P are combinatorial isomorphic, with facet P of sweep k P corresponding to facet P of ^ P , facet Pf^ 1 k g corresponding to proj k (P ), and l1 f^ 1 k g corresponding to proj k ( n1 ) for all ridges n1 of P . Remark 70. For P H, it is straightforward that sweep k (P ) =P = proj k (P ). For P6 H and P H + , one can consider a disturbed version ~ P of P such that ~ P n x2 R d jx k > 0 o and employ Proposition 21 to obtain the incidence graph of ~ P . The incidence graph of P can then be obtained by merging the \close" faces (within disturbances) of ~ P while keeping their incidence ralationships. Example 10 illustrates how to use Proposition 21 to obtain the incidence graph of sweep k (P ) in R 2 . Example 10. Consider the polytopeP corresponding to the line segment between pointsa andb in Figure 5.12a. The gure also shows the corresponding ^ P , proj 2 (P ) and sweep 2 (P ). The subgraph shown in solid black in Figure 5.12b is the incidence graph of P , whereas the subgraph shown in 10 Two polytopesP 1 andP 2 are combinatorial isomorphic to each other if there exists a one-to-one correspondence ' between the set of faces in P 1 and the set of faces in P 2 , such that ' is inclusion-preserving, i.e., for two faces 1 and 2 of P 1 , 1 2 if and only if '( 1 )'( 2 ). 168 gray, which is identical to the solid black one, is the incidence graph of ^ P , where a 0 = proj 2 (a), b 0 = proj 2 (b). The incidence graph of sweep 2 (P ) is constructed from Proposition 21 as follows: 1. sweep 2 (P ) is a two dimensional polytope; 2. The vertices (0-faces) of sweep 2 (P ) contain only the vertices ofP , that area andb, and the vertices of proj 2 (P ), that are a 0 and b 0 ; 3. The edges (1-faces) of P contain both the edge of P , that is ab, the edge of proj 2 (P ), that is a 0 b 0 , and the edges formed by sweep of vertices, that are aa 0 and bb 0 ; 4. Edgesaa 0 andbb 0 are incident toa,a 0 andb,b 0 , respectively, because the edges, as a results of sweeping, contain the corresponding vertices and their dimensions dier by 1; 5. sweep 2 (P ) contains itself as a 2-face and is incident to all the edges, due to the same reason. x 2 0 x 1 a b P ^ a a 0 ^ b b 0 ^ P sweep 2 (P ) proj 2 (P ) (a) a b b 0 a 0 (b) Figure 5.12: Illustration of sweep in scenario I: (a) the geometrical graph; and (b) the incidence graph. Scenario II For the second scenario, we rst identify the faces of P that do play roles in the projection and then resort to Proposition 21 for construction of sweep k (P ). It is sucient to consider P R d 0 to be full dimensional, that is, dimP =d. One can work in the ane space aP otherwise and the same results hold. Let d1 i be a facet of P H + and H i = n x2 R d j ( i ) T x = i 0 o be its 169 dening hyperplane; recall that the direction of i is pointed outwards the polytope. A direction vector 2 R d classies the facets of P into three types [24] according to the value of T i : facet for T i = 0, bottom for T i < 0 and top for T i > 0. With this denition, a facet i belongs to 1 k top if i k > 0, 1 k facet if i k = 0 and 1 k bottom if i k < 0. This is illustrated in Figure 5.13. 1 k -top 1 k -bottom 1 k -facet 1 k Figure 5.13: Dierent facets according to the direction 1 k The 1 k -top and 1 k -bottom facets can be described as the facets that are \visible from the direction1 k " and \visible from the direction 1 k ", respectively. For the projection concerned in sweep k (P ), only points in 1 k top play a role. This is straightforward to see in R 2 and R 3 . For example, in Fig. 5.11c and 5.11c, the top vertex of the vertical edge and the top edge of the triangular face shade all other points and fully determines the sweeps. In general, Proposition 22 shows the same is true for R d , where we say pointx2 R d is shaded by point ^ x2 R d in direction 1 k if ^ x k x k and ^ x i =x i for alli2 [d]nfkg. It is clear from the denition that a point plays no role in the projection along 1 k if it is shaded by another point of P in 1 k . The ridges in 1 k top of P falls into two types: one in the intersection between a 1 k top facet and a 1 k bottom facet or 1 k facet and one in the intersection between two 1 k top facets. Proposition 22 implies that the rst type of ridges determines the boundary of sweep k (P ). They are hence called boundary ridges of P in direction 1 k . The following lemma, which is also referred to as the geometric version of Farkas Lemma [78], will be used in the next result. 170 Lemma 27 ( [92, Sect. 1.4]). Consider a full dimensional polytopeP R d with facets d1 i whose dening hyperplanes are H i := n x2 R d j ( i ) T x = i 0 o , and let \ i2S d1 i be a nonempty face of P for some index set S. Then, = argmax x2P T x for some 2 R d if and only there exists 2 R S >0 such that = P i2S i i . Proposition 22. For a full dimensional polytope P H + and k2 [d], the following are true: (i) every point in P is shaded in direction 1 k by a point in a 1 k top facet of P ; (ii) every 1 k -top facet is a facet of sweep k (P ); and (iii) for a ridge d2 ofP , sweep k ( d2 ) is a facet of sweep k (P ) if and only if d2 is a boundary ridge and d2 6 H. Proof. For each point x2 P , let (x) := argmax ^ x f^ x k j ^ x j =x j ;8j6=k; ^ x2Pg, and let P t := [ x2P f(x)g. It follows from the denition that every x2P is shaded by (x)2P t in direction 1 k . We now show that P t is included in the 1 k top facets of P . For each point ~ x 2 P t , P\ (f~ xg + (0; +1)1 k ) =;, where the setf~ xg + (0; +1)1 k is the half open ray starting from ~ x and pointing in the 1 k direction. The separating hyperplane theorem, e.g., see [22], then implies that there exist 2 R d and 0 2 R such that T x 0 for all x2 P and T x > 0 for all x2f~ xg + (0; +1)1 k . In order for the latter to hold, it must hold that k = T 1 k > 0. We now show that ~ x is included in a 1 k top facet of P . If ~ x is in the interior of some facet d1 i , then, sinceH i is the unique separating hyperplane, we get i k > 0. Hence d1 is a 1 k top facet. Ifx is in a lower dimensional face, then consider a non-empty setJ such thatx2\ i2J i . By contradiction, from Lemma 27, there exists a j2J such that j k > 0. This implies that ~ x belongs to the facet d1 i , a 1 k top facet. This establishes (i). For (ii), pick an arbitrary point ^ x from an arbitrary 1 k -top facet d1 i . By denition of a facet, ^ x2 argmax x2P ( i ) T x. This implies that (^ x) = ^ x, i.e., ^ x2P t . It is then straightforward to see that d1 i remains to be a facet in sweep k (P ), since the hyperplaneH i contains sweep k (P ) on one side. 171 We now prove (iii). Let d2 62 H be an arbitrary ridge of P and d1 i and d1 j be the two incident facets of d2 . We rst prove the conditions to be necessary by considering the following cases: (a) If d2 H, then sweep k ( d2 ) = d2 is of dimension (d 2) and it is trivial that sweep k ( d2 ) is not a facet of sweep k (P ). (b) If neither d1 i and d1 j are 1 k -top facets, all points in d2 are shaded by some other points in P and sweep k ( d2 ) is not a facet of sweep k (P ). (c) If both d1 i and d1 j are 1 k -top facets. As shown in previous paragraph, both d1 i and d1 j remains to be facets of sweep k (P ). Because a ridge is contained in exactly two facets, sweep k ( d2 ) can not be a facet of sweep k (P ). We then prove the condition is sucient. Let d2 be a boundary ridge and, without loss of generality, d1 i be the 1 k -top facet and d1 j be either a 1 k -facet or a 1 k -bottom facet. Hence i k > 0 and j k 0. Proposition 21 implies that sweep k ( d2 ) is of dimension (d 1). Therefore, sweep k ( d2 ) is a facet if it is a face of sweep k (P ). We now construct the dening hyperplane H of sweep k ( d2 ). Let := i k =( i k j k )2 (0; 1], := (1) i + j and 0 := (1) i 0 + j 0 . It is straightforward that k = 0. DeneH := n x2 R d j T x = 0 o . It is sucient to show that sweep k ( d2 )H andH contains sweep k (P ) on one side. Since k = 0, d2 H if and only if sweep k ( d2 )H. The latter is straightforward from the denition of H. In order to show that H contains sweep k (P ) on one side, we consider the following two cases. In the rst case, u j k < 0 and hence 2 (0; 1). Lemma 27 implies the claim. In the second case, u j k = 0 and hence = 1. H =H j , that is, the dening hyperplane of d1 j . The claim follows trivially. With Proposition 21 and Proposition 22, cubeP of a full dimensional polytope P R d 0 can be constructed as follows: set k = 1; while kd, do the following: (I) nd 1 k -top facets of P and remove all faces of P that are not contained in anyone of them; 172 (II) nd the boundary ridges and construct their sweep along 1 k according to Proposition 21 and Remark 70; (III) add a facet in H that is incident to the projections of all boundary ridges along 1 k and a cell, corresponding to sweep k (P ), that is incident to all the facets; (IV) set P = sweep k (P ), k =k + 1, and repeat; where the second step is possible because every boundary ridge belongs to a 1 k -top facet that is not perpendicular to H. The above four-step procedure is illustrated in Example 11. Example 11. ConsiderP =U 1 R 2 0 shown in Fig. 5.10b, in this case cubeU 1 = sweep 1 (sweep 2 (U 1 )). These two sweep operations are shown in Fig. 5.14a and 5.14a, respectively. In particular, Fig. 5.14b shows the incidence graph of sweep 2 (U 1 ), where the black substructure is inherited from the incidence graph of U 1 . As can be seen, the 1 2 -top facets, edges ac and ab, and the boundary ridges, verticesb andc, play critical role in construction of sweep 2 (U 1 ). The 1 2 -bottom facet cb is removed and the projection b 0 and c 0 and sweep bb 0 and cc 0 of boundary ridges are added as ridges and facets of sweep 2 (U 1 ). u 2 o u 1 a b c b 0 c 0 proj 2 (P ) sweep 2 (P ) P (a) a b c b 0 c 0 (b) u 2 o u 1 cube(P ) sweep 2 (P ) (c) Figure 5.14: Illustration of sweep: (a) sweep of U 1 ; (b) the incidence graph of sweep 2 (U 1 ); and (c) cubeU 1 as sweep of sweep 2 (U 1 ). Finally, we consider the case when the network gets disconnected. In this case, each connected network component is associated with a subspace of balanced supply-demand vectors that are supported on the network component. One rst needs to compute for each of the connected 173 network components the projection of P onto the associated subspace, and then perform the sweep operation within the subspace. The additional step of computing the projection of P can be achieved recursively in the same fashion as that for sweep of P , that is, proj fi;jg P = proj i (proj j P ). The implementation of the projection is straightforward by using Proposition 21 and 22. 5.6 Complexity Analysis This section includes discussion on the complexity of the optimal control problem and the relevant algorithms, as well as the extension to more general problem settings. A connected network G = (V;E) is considered without loss of generality. If the network consists of multiple connected components, the same analysis can be done on every component. In addition, we assume d := jV l j 1 to be a constant. 5.6.1 Complexity of Optimal Control of Cascading Failure For a connected networkG = (V;E), an initial supply demand vector p and a time constant N, we dene the complexity, denoted by (E;p;N), of the optimal control problem (5.7) to be the number of possible topology sequences that can occur in (5.2) under all possible controls u2D(E;p;N) (cf. (5.8)). The rationale of the denition is as follows. One one hand, Remark 65 implies in order to solve (5.7), it is sucient to solve(E;p;N) number of linear programs. On the other hand, for a general network, due to non-monotonicity of power ow [3, 5], the only way to determine an ordering, in terms of residual load, between two topology sequences is through direct computation. Therefore, in principle all the (E;p;N) topology sequences need to be considered in order to solve (5.7) exactly. Consider the optimal control problem for a tuple (E;p;N). For an arbitrary aggregated state (E 0 ;P 0 ) (cf. (5.19)-(5.21)) and a link i2E 0 , if n u2 R V jf i (E 0 ;u)<c i o \U(E 0 ;P 0 )6=; (or 174 n u2 R V jf i (E 0 ;u)>c i o \U(E 0 ;P 0 )6=;, respectively), then we call the constraintf i (E;u)c i (or f i (E;u) c i , respectively) an intersecting constraint and link i an infeasible link (in both case). Let n(E;p;N) be the maximal number of intersecting constraints over all aggregated states. It is clear thatn(E;p;N) is upper bounded by the maximal number of infeasible links over all aggregated states. We then have the following result on (E;p;N). Proposition 23. For a connected networkG = (V;E), an initial supply demand vector p and a time constant N, (E;p;N) = O(n d(N1) ), where n := n(E;p;N) is the maximal number of intersecting constraints, d :=jV l (p)j 1,jV l (p)j is the number of non-transmission nodes inG. Furthermore, there exists a choice ofG = (V;E), p and N such that the bound is tight, that is, (E;p;N) = (n d(N1) ). In these bounds the constant of proportionality depends on d. Proof. We rst show that(E;p;N) =O(n d(N1) ). It is clear that from the denition in (5.21), at an aggregated state (E 0 ;P ), the number of possible topologies at the next stage is no larger than jU(E;P )j. From Section 5.5.2, we know that U(E;P ) contains a subset of cells, the ones within U(E;P ), of the associated arrangement, that is, the arrangement of the hyperplanes associated with the ^ n intersecting capacity constraints at (E;P ). According to [85, Chapter 24],jU(E;P )j is upper bounded by P d i=0 ^ n d . For d ^ n=2, P d i=0 ^ n d ^ nd ^ n2d ^ n d = O(^ n d ). For d > ^ n=2, P d i=0 ^ n d 2 ^ n = O(2 d ) 11 . Considering the nontrivial case that ^ n 2, it is then clear that jU(E;P )j = O(n d ) for all aggregated state (E;P ). In other words, the branching factor of the aggregated search tree is O(n d ). Nevertheless, the additional constraints on the control at t = N 1, as commented in Remark 51, implies thatjU(E;P )j = 1 for all aggregated state (E;P ) at t =N 1. It is equivalent to consider the depth of the search tree to be N 1. Therefore, the total number of topology sequences is (E;p;N) =O(n d(N1) ). Furthermore, the network in Fig. 5.10a, with N = 2, provides an example to show that the bound can be tight. In this example, n = 3, d = 3 and the number of topology sequences is P 3 i=0 3 i = 7. 11 n k is dened to be 0 for n<k. 175 5.6.2 Time Complexity of the Solution Methods Table 5.1: Comparison of time complexity of several algorithms. Method Time Complexity Value iteration O N2 jEj n 2d 0 T Search with discretization O n dN 0 T Aggregated search with arrangement O n dN +n d(N1) jVj 2 Two iterations algorithm O jVj 2 ~ n N1 +jV T jn ^ dN 0 n0: number of discretized grid points along one dimension; n =n(E;p 0 ;N); T: time complexity for evaluating f(E 0 ;p 0 ) at a given state (E 0 ;p 0 ); ^ d is the maximal degree in the reduced tree; ~ n = max i2E T n(Ei;ai;N). Table 5.1 shows the time complexities of solving the optimal control problem for a connected networkG = (V;E) with initial supply-demand vector p 0 by using dierent methods. In all these methods, discretization, whenever used, is assumed to produce same number, denoted by n 0 , of grid points on every dimension, andT is used to denote the time to evaluate the value off(E 0 ;p 0 ) at a given state (E 0 ;p 0 ). The algorithms considered are as follows: (I) Value iteration: The state space is considered to be 2 E B E . AsB E is a linear subspace of dimension d, the number of grid points for the state space is 2 jEj n d 0 . At each grid point (E 0 ;p 0 ) , computing the state transition requires evaluating function value f(E 0 ;u) for all u2U(E 0 ;p 0 ). It takesO(n d 0 T ). Therefore, the overall time forN time steps isO(N2 jEj n 2d 0 T ). (II) Search with discretization: The control space U(E 0 ;p 0 ) for state (E 0 ;p 0 ) at each time step is discretized byn d 0 number of grid points. It is the branching factor of the search tree of depth equal toN. Hence the total number of nodes in the tree isO(n dN 0 ). In addition, computing ow and hence time T is required for state transition to each node. Therefore, the overall time complexity is O(n dN 0 T ). It can be seen that the search method is more ecient than the value iteration ifjEjdN lg 2 n 0 . (III) Aggregated search with arrangement: the proof of proposition 23 implies that the number of nodes in the aggregated tree is O(n d(N1) ). At an aggregated state (E 0 ;P 0 ), Remark 68 176 implies that U(E;P ) can be computed in time O(n d ) through constructing the associated arrangements by using the incremental algorithm in Section 5.5.2. The overall time for constructing the aggregated search tree is O(n d(N1) ). In addition, every node (E 0 ;P 0 ) needs to conduct two computational tasks. The rst task is to update the pseudo-inverse of Laplacian matrix according to (3.28) and (3.29). This takes O(jVj 2 ). The second task is to compute cubeP 0 by using Proposition 21 and Proposition 22. Its time complexity is determined by the time complexity of a single sweep operation. The latter is bounded by the number of faces in P 0 and hence is O(n d ). Therefore, the overall time complexity for the aggregated tree search is O n dN +n d(N1) jVj 2 . (IV) Two iteration algorithm: it works only for tree reducible network. Let the reduced tree be T = (V T ;E T ). The time complexity of the algorithm consists of two parts: computing ~ D i for alli2E T and solving (5.14) through discretization for all i2V T . This rst part amount to constructing the aggregated search tree for a network that contains a single supply demand pair. Since the sweep operation in one dimensional space takes constant time, the above analysis implies that it takes time O jV i j~ n N1 to compute ~ D i forG i = (V i ;E i ), where ~ n = max i2E T n(E i ;a i ;N) as dened in Section 5.6.1 and a i is as shown in (5.11). It is straightforward to see thatn(E i ;a i ;N) is much smaller thann(E;p;N) for alli2E T . Hence the total time for computing ~ D i all i2E T is O(jVj 2 ~ n N1 ), since in most cases P i2E T jV i j 2 is smaller thanjVj 2 . The second part is to solve sub-problems (5.14) of a dimension no larger than N ^ d, where ^ d is the maximal node degree of the reduced tree. Using the same discretization parameter n 0 , a single subproblem takes time O(n ^ dN 0 ) and the total time complexity of the recursive procedure is the summation of these subproblems over all nodes inT , that is, O jV T jn ^ dN 0 . Therefore, the overall time complexity for the algorithm is O jVj 2 ~ n N1 +jV T jn ^ dN 0 . 177 Remark 71. The time complexity for aggregated tree search may not be tight due to several reasons: The arrangement problem at a network state could be highly degenerated, that is, two hyperplanes do not necessarily intersect and more than d hyperplanes can intersect at a vertex, in which case the corresponding branching factor is much smaller than n d . When the network gets disconnected due to link failures, the arrangement problem is solved for each connected component independently. Therefore, an accurate estimate ofd is related to the number of non-transmission nodes in a component, as opposed to the entire network. Since the number of connected components is non-decreasing along a path in the search tree, the pertinent value of d may be much less thanjV l j 1 away from the initial state (E 0 ;fp 0 g). As shown in Section 5.5, lower and upper bounds at each aggregated state can be used to prune the aggregated search tree. This avoids searching through the entire aggregated state space without compromising the performance, and therefore improves the eciency of the aggregated tree search algorithm. 5.7 Approximation Algorithm and Simulations 5.7.1 Approximation Algorithm via Projection The exponential dependence of the time complexity on d =jV l j 1, as discussed in Section 5.6, may be prohibitive for networks that contains large number of non-transmission nodes. We now outline a strategy to project the (aggregated) supply-demand vectors onto a lower dimensional space. Aggregation and search in the lower dimensional space then gives an approximation. For a networkG = (V;E) with supply-demand vector p. For the sake of presentation in this section, assume, without loss of generality thatV =V l , i.e., every node is a non-transmission 178 node; if this is not the case, then one can focus only on the subspace of control actions corre- sponding to the non-transmission nodes. Let = [ 1 ;:::; jV l j ]2 R jV l jjV l j be an orthonormal (transformation) matrix, and let B [jV l j] be an index set. The approximation strategy, which is parametrized by (;B), considers aggregated set of feasible control actions in the sub-space U(E;P )\R( B ) = u2U(E;P )j T i u = 0; i62B , i.e., in the subspace of control actions which can be expressed as a linear combination of i , i2B. Remark 68 implies that this reduces the dimension, and hence correspondingly the time complexity, fromjV l j to B. Moreover, since the constraints T i u = 0, i62B, are hyperplanes, they can be easily integrated into the construction of arrangements to get U(E;P )\R( B ). In fact, by setting i = 1 p jV l j for some i62 B, the constraint T i u = 0 is the balance constraint for a connected network (cf. (2.5)). Example 12. Consider a network with initial supply-demand vector p 0 . By choosing B =f1g and 1 =p 0 =kp 0 k 2 , we get proportional control policies [19, Section 6.1.1], i.e., a class of control policies whose action set at state (E;p) isfpj 0 1g. The one-dimensional search space resulting from proportional control policy, as shown in Ex- ample 12, is favorable for computational purposes. However, the projection-based approximation strategy implies that one could possibly nd better control actions, using the comparable compu- tational budget, by using dierent projections. This is illustrated using simulations on IEEE 39 benchmark system in Section 5.7.2. 5.7.2 Simulations We conducted numerical experiments on the IEEE 39 bus system illustrated in Figure 5.15. Node 39 is selected to the only supply; nodes 4 and 16 are selected as loads; all the other nodes are transmission nodes. This particular choice of supply-demand nodes is consistent with the fact that the actual supply and demand on these nodes, as reported in [93], have relatively large values. p 0 was chosen to be proportional to the actual values reported in [93] for nodes 4, 16 and 39: 179 (a) 39 1 2 9 3 8 4 5 6 7 31 14 13 11 32 12 10 18 17 15 16 25 26 27 30 37 28 29 38 24 23 36 21 22 35 19 33 20 34 e 1 e 2 e 3 e 5 e 4 e 6 e 7 e 8 e 9 e 10 e 11 e 12 e 13 e 14 e 15 e 16 e 17 e 18 e 19 e 20 e 21 e 22 e 23 e 24 e 25 e 27 e 28 e 29 e 26 e 30 e 31 e 32 e 33 e 34 e 35 e 36 e 37 e 39 e 38 e 40 e 41 e 42 e 43 e 44 e 45 e 46 (b) Figure 5.15: The IEEE 39 bus network. p 0 4 =p 0 16 =5,p 0 39 = 10 andp 0 i = 0 for alli = 2f4; 16; 39g. Link susceptancesw are from [93]. The link capacities were selected as follows: c 8 = 0:5, c 9 = 1, c i = 2:5 for i2f13; 21; 22; 23g; c i = 3:0 for i2f3; 28; 29; 35; 36; 38g; c i = 3:5 for i2f16; 17g; c i = 4:0 for i2f7; 26; 30g; c i = 4:5 for i2f1; 2; 4; 24; 25; 31; 39; 40; 42g and c i = 2:0 for the remaining links, with respect to link labels in [93]. Table 5.2: Optimal residual load under (5.7) and under the projection-based approximations in (5.27) N 1 2 3 4 5 1-D approximation () 0.0 3.716 9.860 10.000 10.000 10.000 0.1 3.502 9.806 11.112 11.112 11.112 0.2 3.310 9.750 11.090 11.090 11.090 0.3 3.140 9.696 11.028 11.028 11.028 0.4 2.984 8.974 10.000 10.000 10.000 0.5 2.844 9.000 9.000 9.000 9.000 0.6 2.718 7.334 7.334 7.334 7.334 0.7 2.600 6.742 6.742 6.742 6.742 0.8 2.494 4.578 5.000 5.000 5.000 0.9 2.396 4.078 4.444 4.444 4.444 1.0 2.304 4.000 4.000 4.000 4.000 Optimal control 3.716 9.860 11.150 11.150 11.150 180 For the above network parameters, (E 0 ;p 0 ) is infeasible. Furthermore, under no load shedding control action, i.e.,u T p 0 for allt, the only supply node 39 would get disconnected att = 1 from the load nodes 4 and 16. However, using the control formulation of this paper, such a scenario can be prevented while minimizing the amount of load to be shed. Table 5.2 (last column) shows the values of residual load, i.e., the optimal solution to (5.7), computed by the techniques in Section 5.5, for dierent control horizons N. The residual load is expectedly is nondecreasing with N. This conrms that multi-round control does lead to increase in the residual load, or equivalently decrease in cumulative load shed, in comparison to the single round (N = 1) control underlying power re-dispatch. However, there are no gains in residual load beyond N 3. This is because the network in Figure 5.15 contains a very few cycles; and once the active link set becomes a tree, the optimal load shedding action is to ensure feasible of all the links in this case. Table 5.2 also shows optimal residual load within the class of control policies obtained by projection onto a one-dimensional space, as described in Section 5.7.1. Specically, we chose B =f1g; 1 = p 1 + (1) p 2 ; 2 [0; 1] (5.27) where p 1 2 R 39 has 1 and -1 on node 39 and 4, respectively, and 0 elsewhere; p 2 2 R 39 has 1 and -1 on node 39 and 16, respectively, and 0 elsewhere. Recalling Example 12, it is easy to see that the set of proportional control policies [19, Section 6.1.1] corresponds to = 0:5. Table 5.2 contains values for optimal residual load under such an approximation for dierent values of and N. These values show that, similar to the optimal control actions, for every , the optimal residual load is nondecreasing in N and stays constant for N 3. While there is no general monotone relationship in (uniformly for all N), the best control actions for N 3 correspond to = 0:1. The control actions corresponding to = 0:1 perform uniformly better than the proportional control policy ( = 0:5) which requires comparable computational cost, and give 181 fairly similar performance as the optimal control actions which are obtained under considerable computational costs (Section 5.5). 5.8 Conclusions and Future Work The phenomenon of cascading failure in physical networks has attracted a lot of interest, and yet formal approaches for its control are relatively very few. This chapter builds upon a previ- ously proposed formulation for optimal control of cascading failure in power networks under DC approximation, and provides approaches for computing optimal control within this setting. The cascading dynamic model can be readily extended to other large scale dynamical systems whose constituent components are allowed to fail and underlying physical dynamics are on a consid- erable faster time scale than the cascades of failure. The connections to network optimization and computational geometry underlying these approaches are novel, and are suggestive of several avenues for future work. We plan to extend the network decomposition approach beyond the specic settings of this chapter. Specically, we are interested in conditions for non tree-reducible networks under which there exists a decomposition into coupled local problems which can be solved to optimality asymp- totically through distributed optimization techniques. With regards to the search-based approach, we plan to explore approximation techniques beyond the projection approach in this chapter. In particular, we are interested in techniques for pruning the search tree while giving provable ap- proximation guarantees. Finally, the equivalent nite representation paradigm in this chapter is reminiscent of the literature on nite abstractions for hybrid systems, which has been done pri- marily for stability analysis. We plan to explore connections between the setting of this chapter, possibly extended to continuous time domain, and recent work on symbolic optimal control, e.g., see [76]. 182 Chapter 6 On Distributed Computation of Optimal Control of Trac Flow over Networks 6.1 Introduction The projected instrumentation of urban transportation infrastructure with sensing and actuation mechanisms will provide great opportunities for dynamic control, e.g., to respond eciently to trac incidents. The large scale of the problem, coupled with the time-critical nature of some of such applications necessitates fast computation of optimal control, possibly under a parallel, or peer-to-peer (distributed) computation, architecture. In this chapter, we adapt the Alternating Direction Method of Multipliers (ADMM) to develop a distributed algorithm for computing nite horizon optimal control for trac ow over networks. We model trac ow dynamics over networks, in continuous time, using the Cell Transmission Model (CTM) [34,35]. Every cell is endowed with a demand and a supply function, representing the maximum out ow and the maximum in ow on the cell given its density, respectively. In ows at on-ramps are modeled as exogenous. On the other hand, ows between contiguous cells, in particular at junctions, are determined by merging and splitting rules within constraints imposed by the cells' demand and supply functions as well as turning ratios. 183 The proposed optimal control problem is non-convex in general. Recently, an exact convex relaxation has been proposed in [30]. Motivated by this nding, we use the equivalent convex relaxation for adapting the ADMM. An equivalent nite dimensional representation for the con- straints is developed when the external in ows and control are piecewise constant. It is desired to have the following properties in an algorithm to solve the optimal control problem: (i) the iterations converge to an optimal value; and (ii) iterations can be implemented in a distributed manner. Achieving these objectives via usage of ADMM requires careful selection and assignment of variables among two blocks for the primal update. We propose to use a time shifted copy of the density vector and two exact copies of the ow vector as auxiliary variables. Rather than assigning these variables to two blocks in their entirety, we split the odd and even time components of the variables in order to achieve the desired properties of convergence and distributed implementation. In particular, we show that, if the cost function is proper, closed, convex and separable over density and ow variables, and over links, and if the initial density on every link is strictly positive and strictly less than its jam density, then the primal variables converge to an optimal value. The optimal control formulation in this chapter is reminiscent of the dynamic trac assignment problem [68, 69, 73]. While originally proposed mainly for planning purposes, it is also being increasingly used as a framework to compute optimal control for trac ow over freeway networks, e.g., see [48,72]. While the ADMM-based distributed algorithms have been proposed recently for dynamic network trac assignment in [74], there are key dierences with respect to our work. The algorithm in [74] does not provide rigorous convergence analysis for the primal variables. The convergence issue was settled for the trac equilibrium selection problem in our previous work [6]. The main contributions of this study are as follows. First, we formulate an equivalent - nite dimensional representation of the constraints under piecewise constant control and external in ow. Second, we adapt the alternating direction method of multipliers to solve the optimal 184 control problem, paying particular attention to the choice of variables that guarantee distributed implementation. 6.2 Problem Formulation 6.2.1 Network Flow Dynamics We describe the topology of the transportation network as a directed multi-graphG = (V;E) with linksi2E representing cells 1 and nodes representing either junctions (being them of merge, diverge, or mixed type) or interfaces between consecutive cells (brie y referred to as ordinary junctions). One particular node w 2 V represents the external world, with cells i such that i =w representing sources (identiable with on-ramps in freeway networks) and cellsi such that i = w representing sinks (representing o-ramps in freeway networks). The sets of sources and sinks are denoted byR andS, respectively. The dynamic state of the network is described by a time-varying vectorx(t)2R E whose entries x i (t) represent the trac volume in the cells i2E at time t 0. The inputs to the network are the exogenous in ows i (t) 0 at the sources i2R. Conventionally, we set i (t) 0 for all non-source cellsi2EnR and i (t) 0 for all non-sink cellsi2EnS, and stack up all the in ows in a vector (t)2 R E and all the out ows in a vector (t)2 R E . We consider a nonnegative, possibly time-varying,EE routing matrixR =R(t) satisfying the network topology constraints R ij = 0; i 6= j ; (6.1) and such that X j2E R ij = 1; i2EnS: (6.2) 1 We use cell and link interchangeably in the chapter. 185 The matrix R is to be interpreted as describing the drivers' route choices, with its entries R ij , sometimes referred to as turning ratios, representing the fractions of ow leaving cell i that is directed towards cell j. Equation (6.2) then guarantees that all the out ow from the non-sink cells is split among other cells in the network, while equation (6.1) guarantees that the out ow from celli is split among adjacent downstream cells only. We denote the total out ow from a cell i by z i (t). Therefore, the trac ow from a cell i to an adjacent cell j is equal to R ij z i . The trac dynamics is then described by _ x i (t) = i (t) + X j2E R ji (t)z j (t)z i (t); i2E (6.3) Equation (6.3) states that the time-derivative of the trac volumex i on a cell equals the imbalance between its in ow and its out ow. We now describe the model for the link out ows. We assume a concave Fundamental Diagram to model the relationship between trac volume and ows. Following Daganzo's Cell Transmission Model, we introduce demand and supply functions, which return the maximum out ow from and, respectively, the maximum in ow in a cell as a function of its current trac volume and, possibly, additional control parameters. In turn, they can be interpreted as representing the rising and, respectively, descending parts of the Fundamental Diagram. In particular, the demand functions are assumed to take the form d i (x i ; i ) = i d i (x i ); i2EnR; (6.4) d i (x i ; i ) =d i (x i ); i2R; (6.5) where: d i (x) is a continuous, non-decreasing, and concave function of the trac volume such that d i (0) = 0; i = i (t)2 [0; 1] is a possibly time-varying demand control parameter actuated via 186 speed limit control on the non-source cellsi2EnR and ramp-metering on the sourcesi2R. On the other hand, the supply functions s i (x i ) of every non-source cell i2EnR are assumed to be continuous, non-increasing, concave, and such thats i (0)> 0, withx jam i = inffx i > 0 : s i (x i ) = 0g denoting cell i's jam trac volume. Conventionally, s i (x i ) +1 at all sources i2R. The functional dependence of the out ow on trac volume and control parameters is then given by: z i =d i (x i ; i ); i2S; (6.6) z i = F i d i (x i ; i ); i = 2S; (6.7) where for all i2E, F i is the supremum of in [0; 1] such that max k2E: i= k X h2E R hk d h (x h ; h )s k (x k ) ! 0 (6.8) For a given assignment x i (0) =x 0 i ; i2E; (6.9) of initial trac volumes in the cells and dynamics of the in ows (t), routing matrix R(t), and demand control parameters (t), the evolution of the trac volume vector x(t) for t 0 is uniquely determined by equations (6.3) and (6.6)-(6.7). 6.2.2 Problem Statement Within the setting of Section 6.2.1, we consider an optimal control problem consisting in the minimization of the integral of a running cost (x;z) that is a function of vector of trac volume and out ows over the time interval [0;T ], where T > 0 is a given time horizon. 187 Given a (possibly multi-sink) transportation network, a running cost (x;z), a nite time- horizon T > 0, initial cell trac volumes x 0 i 0, in ows i (t) 0 for 0 t T at the source cells i 2 R, and a routing matrix satisfying (6.1)-(6.2), the optimal control problem for the continuous-time cell-based dynamic trac model can be formulated as follows: min (t);R(t): (6.1)(6.9) Z T 0 (x(t);z(t))dt (6.10) The optimal control problem in (6.10) is non-convex in general. It is shown in [30, Proposi- tion 1] that this problem can be transformed into an equivalent convex problem by relaxing the constraints involved in the denition of F i in (6.8), and then using control to enforce feasibility of the solution to the relaxed problem. The convex relaxation is given by: minimize x(t)0;z(t)0 Z T 0 (x(t);z(t))dt subject to x(0) =x 0 _ x(t) =(t) +R T z(t)z(t) R T z(t)s(x(t)) z(t)d(x(t)) (6.11) In the rest of the chapter, we focus on developing a distributed algorithm to solve (6.11). We make the following standing assumption on the cost functions throughout the chapter. Assumption 4. (x;z) is separable inx andz and over the links, i.e., (x;z) = P i2E ( x i (x i ) + z i (z i )), and x i and z i are proper, closed and convex for all i2E. Some results require the cost functions x i and z i to also be strictly convex, in which case such an assumption will be mentioned explicitly. 188 6.3 The Alternating Direction Method of Multipliers (ADMM) In this section, we adapt the ADMM method, e.g., see [21], to provide a numerical method to solve the optimal control problem stated in (6.11). The rst step is to convert (6.11) into a nite- dimensional optimization problem. A naive time discretization leads to an optimal solution that only satises the constraints on the nitely many discretization points, while possibly violating constraints between two discretization time steps. Instead, if one restricts the optimization in (6.11) over the class of piecewise constant controls, then one can obtain an equivalent nite- dimensional representation of the constraints in (6.11) under the assumption that is piecewise constant. On top of this, the cost function in (6.11) is then approximated using Trapezoidal rule for integration. Thus, the solution obtained by solving the nite-dimensional optimization problem is always feasible. 6.3.1 Piecewise Constant Control Let [0;T ] be discretized into N sub-intervals, each of length4, i.e.,4 =T=N. We assume (t) is piecewise constant for the discretization. Assumption 5. For the given discretization step size4 of [0;T ], (t) k for all t2 [k4; (k + 1)4) and k2f0; 1;:::;N 1g. A piecewise constant control policy z(t) is given by z(t) =z k ; t2 [k4; (k + 1)4); k2f0; 1;:::;N 1g for a given (z 0 ;z 1 ;:::;z N1 ). The linearity of the dynamics in (6.11) then implies that the state x(t) is piecewise linear, where the state values at timesf4;:::;N4g are related as follows (k2f0; 1;:::;N 1g): x k+1 =x k +4 k +4(R T I)z k (6.12) 189 where x k :=x(k4) for k2 [N], and x 0 =x(0). The piecewise linearity of x(t), combined with monotonicity of demand and supply functions, implies that the inequality constraints in (6.11) overt2 [0;T ] can be equivalently replaced by the following nite constraints: for k2f0; 1;:::;N 1g, R T z k s(x k );R T z k s(x k+1 ) z k d(x k );z k d(x k+1 ) The equivalence is due to that minfs(x k );s(x k+1 )g = min t2[k4;(k+1)4) s(x(t)) and minfd(x k );d(x k+1 )g = min t2[k4;(k+1)4) d(x(t)) for all k2f0; 1;:::;N 1g. Following the piecewise constant and piecewise linear properties of z(t) and x(t) respectively, the cost functions can be written as: Z (k+1)4 k4 X i2E z i (z i (t)) dt = X i2E z i (z k i )4 Z (k+1)4 k4 X i2E x i (x i (t)) dt X i2E x i (x k i ) + x i (x k+1 i ) 4 2 where the equality of the cost onz(t) is clear sincez(t) is constant fort2 [k4; (k + 1)4) and the approximation of the cost on x(t) is obtained using Trapezoidal rule for integration. With this special class of control action and in ow , the optimal control problem (6.11) with innitely many constraints can be approximated by a convex optimization problem with nitely 190 many number of constraints 2 . The convex problem is shown in (6.13), where (x k+1 ;z k ) are the decision variables (k = 0; 1;:::;N 1) and x 0 is the given initial density of the network. minimize x2R E 0 ;z2R E 0 N1 X k=0 X i2E x i (x k+1 i ) + z i (z k i ) 4 subject to x k+1 =x k +4 k +4(R T I)z k R T z k s(x k ); R T z k s(x k+1 ) z k d(x k ); z k d(x k+1 ) (6.13) The convexity of (6.13) follows from the fact that functions x i (x i ) and zi i (z) are convex and functions s(x) and d(x) are concave. 6.3.2 ADMM Iterations In this section, a distributed algorithm based on ADMM is proposed to solve (6.13). The algorithm involves introducing three sets of auxiliary variables, namely y, w and v. y k =x k+1 ; z k =u k ; z k =w k k = 0; 1;:::;N 1 With the auxiliary variables, the dynamic equation and supply and demand constraints can be written as follows. For each time step k2f0; 1;:::;N 1g and node v2V, y k i =x k i +4 k i +4 X j2E v R ji u k j 4w k i 8i2E + v X j2E v R ji u k j s i (x k i ); X j2E v R ji u k j s i (y k i ) 8i2E + v w k i d i (x k i ); w k i d i (y k i ) 8i2E + v (6.14) 2 To be precise, the cost function associated with a linki should be ( x i (x 0 i )+ x i (x N i ))4=2+4 P N1 k=2 x i (x k i )+ 4 P N k=1 z i (z k i ). Since x 0 is considered as given and the value of x i at x N i can be redened as x i (x N i ) := x i (x N i )=2, we write the cost function as in (6.13). 191 3 y 4 w 3 w 4 y 1 w 2 w 1 y 2 y 1 u 2 u 3 z 3 u 4 z 4 u 2 z 1 z 3 x 4 x 1 x 2 x v q v Figure 6.1: Illustration of variable separation For node v at time k, dene variable q k v := (x k i ;y k i ;u k j ;w k i ) with all i2E + v and j2E v . Fig. 6.1 is an illustration of the variables associated with a node v which has two incoming links, i.e., link 1 and link 2, and two outgoing links, i.e., link 3 and link 4. The superscriptsk of the variables are ignored for simplicity of representation. It is straightforward to see that for a given node v and time step k (6.14) only involves variable q k v . This is a result of a specic choice of auxiliary variables. Let D k v :=fq k v 0jq k v satises (6:14)g Thus, (6.13) can be rewritten as: minimize q k v 2D k v ;z T1 X k=0 X i2E x i (y k i ) + z i (u k i ) subject to y k i =x k+1 i ; z k i =u k i ; z k i =w k i (6.15) where k = 0; 1;:::; (N 1) and i2E in the constraints and4 is constant and hence removed from the cost function. Note that due to the constraint z k i = u k i , the cost function z i (z k i ) is 192 replaced with z i (u k i ) and the positiveness constraint on z is removed. This helps to simplify the iterations on variable z later. The augmented Lagrangian is L = T1 X k=0 X i2E x i (y k i ) + z i (u k i ) +p k 1i (x k+1 i y k i ) +p k 2i (u k i z k i ) +p k 3i (w k i z k i ) + 2 (x k+1 i y k i ) 2 + (u k i z k i ) 2 + (w k i z k i ) 2 where p k 1i , p k 2i and p k 3i are the dual variables corresponding respectively to the three constraints in (6.15). We divide the primal variables into two blocks as follows. Block 1: fq k v ;z k+1 i jv2V;i2E;k = 0; 2; 4;:::g Block 2: fq k+1 v ;z k i jv2V;i2E;k = 0; 2; 4;:::g In the primal updates, each variables block is updated alternatively, given the value of the other block variables and the dual variables. Each primal update involves solving an optimization problem over one of the two block variables. With the above choice of auxiliary variables and variable blocks, the variables on the dierent sides of the equalities in (6.15) are from dierent blocks and hence the optimization problem for each block is decomposable and could be solved in a distributed way over the network. The details of the iterations are shown in Table 6.1. Notice that the update equations for z k i are in closed-form in these iterations. 193 Variables Iterations Primal update for Block 1 For all v2V and k = 0; 2; 4;::: q k v (l + 1) = argmin q k v 2D k v X i2E + v x i (y k i ) +p k1 1i (l)x k i p k 1i (l)y k i +p k 3i (l)w k i + X i2E + v 2 (x k i y k1 i (l)) 2 + (x k+1 i (l)y k i ) 2 + (w k i z k i (l)) 2 + X i2E v z i (u k i ) +p k 2i (l)u k i + 2 (u k i z k i (l)) 2 (Note for k = 0, the terms p k1 1i (l)x i and (x k i y k1 i (l)) 2 need to be removed from the above iteration.) For all i2E and k = 1; 3; 5;::: z k i (l + 1) = argmin z k i 2 (u k i (l)z k i ) 2 + (w k i (l)z k i ) 2 (p k 2i (l) +p 3i (l) k )z k i = 1 2 u k i (l) +w k i (l) + p k 2i (l) +p k 3i (l) Primal update for Block 2 For all v2V and k = 1; 3; 5;::: q k v (l + 1) = argmin q k v 2D k v X i2E + v x i (y k i ) +p k1 1i (l)x k i p k 1i (l)y k i +p k 3i (l)w k i + X i2E + v 2 (x k i y k1 i (l + 1)) 2 + (x k+1 i (l + 1)y k i ) 2 + (w k i z k i (l + 1)) 2 + X i2E v z i (u k i ) +p k 2i (l)u k i + 2 (u k i z k i (l + 1)) 2 For all i2E and k = 0; 2; 4;::: z k i (l + 1) = argmin z k i 2 (u k i (l + 1)z k i ) 2 + (w k i (l + 1)z k i ) 2 (p k 2i (l) +p 3i (l) k )z k i = 1 2 u k i (l + 1) +w k i (l + 1) + p k 2i (l) +p k 3i (l) Dual update For all i2E and k = 0; 1; 2;::: p k 1i (l + 1) =p k 1i (l) +(x k+1 i (l + 1)y k i (l + 1)) p k 2i (l + 1) =p k 2i (l) +(u k i (l + 1)z k i (l + 1)) p k 3i (l + 1) =p k 3i (l) +(w k i (l + 1)z k i (l + 1)) Table 6.1: The ADMM iterations. Remark 72. One naive choice of the blocks of variables would befq k v ;z k i g for odd and even time steps, respectively. Due to the presence of the quadratic terms in the augmented Lagrangian, such a choice would make the variable z coupled with variables u and w over the network (see Fig. 6.1). As a result, the optimization problems for the primal iteration are not decomposable and hence the iterations are not distributed. 194 6.4 Convergence Analysis We show that the iterations in Table 6.1 converge to the optimal solution under reasonable conditions on the network topology and the value of initial density x 0 . The condition on the network topology is as follows. Assumption 6. For every cell i, there exists at least one directed path from an on-ramp j2R to i and one from i to an o-ramp k2S . The routing matrix R, density vector x and ow vector z can be partitioned into blocks corresponding to setR,EnfR[Sg andS, as shown in (6.16). Notice by denition, the rst column and last row of R are 0. 2 6 6 6 6 6 6 4 0 R 1 0 0 ^ R R 2 0 0 0 3 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4 z R ^ z z S 3 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4 x R ^ x x S 3 7 7 7 7 7 7 5 (6.16) where all the submatrices and subvectors are of appropriate sizes, e.g., ^ R2R (jEjjRjjSj)(jEjjRjjSj) . We have the following result for matrix ^ R. Lemma 28. The spectral radius of matrix ^ R is strictly less than 1. Proof. Since R satises (6.1) and (6.2), ^ R is a strictly sub-stochastic matrix. Due to the As- sumption 6, there exists at least one path from any link to an o-ramp link. ^ R can be seen as a probability transition matrix between the transient states of an appropriate Markov chain with hidden absorbing states. By completing the absorbing states, the probability transition matrix of the absorbing Markov chain is as follows. 2 6 6 4 ^ R 0 1 3 7 7 5 195 Therefore, lim k!1 ^ R k = 0 and the spectral radius of matrix ^ R is strictly less than 1. The convergence result for the ADMM iterations in Table 6.1 is stated next, where we recall the standing Assumption 4. Proposition 24. Under Assumption 6, and if x 0 2 (0;x jam ), then the ADMM iterations in Table 6.1 converge to an optimal solution to (6.15). Moreover, if the objective functions x i (x i ) and z i (z i ) are strictly convex for all i2E, then the ADMM iterations converge to the unique optimal solution. Proof. We rst verify standard conditions [21, Section 3.2] for convergence of the primal residual to zero under ADMM iterations. In order to cast the formulation in [21] in our setting, we need to include all the inequality constraints into the objectives. It is easy to see that this objective function is closed, proper and convex, which ensures that the subproblems arising in the primal update are solvable. We now establish the existence of a saddle point for the unaugmented Lagrangian. It is sucient to show that Slater's condition is satised [6], i.e., there exists interior point ( x; y; z; u; w). It is equivalent to nd ( x; z) 2 Int(D) = f(x k+1 ;z k ) N1 k=0 > 0jR T z k < s(x k );R T z k <s(x k+1 );z k <d(x k );z k <d(x k+1 );8 0 k N 1g where x =f x 1 ; x 2 ;:::; x N g and z =f z 0 ; z 1 ;:::; z N1 g. At the same time, ( x; z) needs to satises the dynamic equations (6.12). First, notice the dynamic equations (6.12) can be rewritten as follows using the submatrices and subvectors in Eq. (6.16). x k+1 R =x k R +4( k z k R ) ^ x k+1 = ^ x k +4(R T 1 z k R + ^ R T ^ z k ^ z k ) x k+1 S =x k S +4(R T 2 ^ z k z k S ) 196 We begin to construct an interior point ( x; z) by considering a constant control action specied by (6.17) for all time steps. z> 0 z R =z 0 1 ^ z = ^ R T ^ z +z 0 R T 1 1 z S =R T 2 ^ z (6.17) where z 0 > 0 is some scalar. We rst show that such a z 0 always exists that (6.17) is satised. Since z R = z 0 1 > 0 is always satised andz S =R T 2 ^ z> 0 if ^ z> 0, it is sucient to show that positive ^ z exists to satisfy ^ z = ^ R T ^ z +cR T 1 1, i.e., (I ^ R T )^ z =z 0 R T 1 1 As shown in Lemma 28, the spectral radius of ^ R is strictly less than 1. Hence, (I ^ R T ) is invertible and ^ z can be solved as follows. ^ z =z 0 1 X k=0 ( ^ R T ) k R T 1 1 =z 0 R T 1 1 + ^ R T R T 1 1 + ( ^ R T ) 2 R T 1 1 +::: With z 0 > 0 and both R 1 and ^ R are nonnegative matrices in mind, we show ^ z > 0. Notice that R T 1 1 has positive values on links that are immediately connected to the on-ramps. ^ R T R T 1 1 has positive values on link that are one hop away from the on-ramps. In general, ( ^ R T ) k R T 1 1 has positive values on link that are k hop away from the on-ramps. The connectivity guarantee provided by Assumption 6 implies ^ z> 0. As the network contains only nite number of links, for brevity, we are able to set z i =c i z 0 8i2E 197 for 0 < c i <1. Notice that z i is proportional with z 0 , hence we can scale z i by adjusting z 0 without violating (6.17). Under the control specied by (6.17), the dynamic equations (6.12) are as follows. x k+1 i =x k i +4( k i z 0 ) 8i2R x k+1 i =x k i 8i2EnR It is easy to solvex k i from the above equations. For on-ramp links i2R,x k i =x 0 i +4 P k j=1 j i kz 0 4x 0 i z 0 T for all 1kN. Leta i > 0 be such thata i =d i (x 0 i a i T ). And then for any i2R, d i (x k i )d i (x 0 i z 0 T )>z 0 for all z 0 <a i . Also s i (x k i ) =1>z k i is trivial. To have the demand and supply constraints strictly satised for i2EnR, x k i =x 0 i , c i z 0 < minfs i (x 0 i );d i (x 0 i )g 8i2E=R Such z 0 could be easily found so that z 0 < min i minfs i (x 0 i );d i (x 0 i )g c i In conclusion, an interior point ( x; z)2Int(D) is obtained by setting z 0 as follows z 0 = 0:9 min min i2R a i ; min i2EnR minfs i (x 0 i );d i (x 0 i )g c i Until now, we've proven that Slater's condition is satised and a saddle point exists for the unaugmented Lagrangian. Therefore, the primal residual converges to 0. On top of this, we now show that the primal variables converge to an optimal solution. 198 The equality constraints 6.13 are as follows after introducing the auxiliary variables. y k =x k + k +4R T u k 4w k ; y k =x k+1 ; z k =u k ; z k =w k where k = 0; 1;:::;N 1. Write the equality constraints in matrix form AX 1 +BX 2 =b where X 1 = [y 0 ;u 0 ;w 0 ;z 1 ;x 2 ;y 2 ;u 2 ;w 2 ;z 3 :::] T X 2 = [z 0 ;x 1 ;y 1 ;u 1 ;w 1 ;z 2 ;x 3 ;y 3 ;u 3 ;w 3 ;:::] T and A = diagfM 1 ;M 0 ;:::;M 0 ;M 2 g2 R (8N1)E(5N1)E and B = diagfM 0 ;:::;M 0 ;M 3 g2 R (8N1)E5NE . M 0 = 0 B B B B B B B B B B B B B B B B B B B B B B B B B B @ 0 0 I I I I I 4R T 4I I I I 1 C C C C C C C C C C C C C C C C C C C C C C C C C C A 2 R 8E5E 199 M 1 = 0 B B B B B B B B B B @ I 4R T 4I I I I 1 C C C C C C C C C C A 2 R 4E3E M 2 = [0 I I] T 2 R 3EE M 3 = 0 B B B B B B B B B B B B B B B B B B B B B B @ 0 0 I I I I I 4R T 4I I I 1 C C C C C C C C C C C C C C C C C C C C C C A 2 R 7E3E It is easy to see that M 0 , M 1 , M 2 and M 3 have independent columns. Thus, both A and B have full column ranks. Standard ADMM convergence results, e.g., [21, Section 3.2.1], then imply that both AX 1 (k) and BX 2 (k) converge. Since A and B contain independent columns, this implies convergence of X 1 (k) andX 2 (k), and hence also of (x(k);z(k)). Since the cost function converges to the optimal value ( [21, Section 3.2.1]), this implies that (x(k);z(k)) converges to an optimal solution. If the cost function is also strictly convex, the optimal solution is unique and hence (x(k);z(k)) converges to the unique optimal solution. 200 6.5 Conclusions and Future Work In this chapter, we adapted the alternating direction method of multipliers to design distributed algorithms to compute optimal solution to the problem of nite horizon optimal control for trac ow over networks. The distributed implementation of the ADMM iterations and the convergence guarantees for the primal variables requires careful choice of auxiliary variables and the splitting of the primal variables into two blocks. Such a choice of variable selection could be of interest to other dynamical network ow control settings. In our previous work on optimal trac equilibrium selection [6], we also adapted the Accelerated Dual Descent (ADD) method, and illustrated its superior convergence rate in comparison to the ADMM-based method via simulations. Extension of the ADD method to the present case, along with convergence guarantees is of interest. 201 Chapter 7 Conclusions and Future Work This dissertation aims at developing a unied framework of control and optimization for infras- tructure networks. Several tools, involving modeling, non-convex optimization and distributed optimization, are developed in order to achieve this goal. As for modeling, we propose a static diusion network model and a dynamic cascading failure model, both having the capability to model various infrastructure systems. The diusion network model is an extension of ow network model and includes DC network model as a special case. As for non-convex optimization, we pro- vide two approaches that enable ecient computation by exploiting network sparsity. The rst approach is the multilevel programming approach used to solve the robust weight control problem for reducible networks. It reduces satisfying subnetworks to single links. The second approach is the decomposition approach used to compute optimal control of cascading failure for type II tree reducible networks. It can be similarly treated as an approach to reduce star networks to single nodes. These two approaches demonstrate the potential of topological reduction in solving challenging problems for DC networks. As for distributed optimization, ADMM is shown to be an eective tool to generate distributed algorithms with convergence guarantee in the study of optimal dynamic assignment of trac ow. Furthermore, this dissertation builds unprecedented connections of DC network problems with two well-studied topics. The rst one is the maximal ow problem. While it gives an upper 202 bound on the maximal DC ow problem, it can also be solved indirectly via a gradient related algorithm for the maximal DC ow problem. For this purpose, we present useful results and interpretations on the pseudo-inverse of Laplacian matrix and the oblique projection matrix. The second topic is computational geometry. Geometric ideas, especially in the domain of arrangement of hyperplanes, show their power in characterizing the complexity of optimal control of cascading failure in DC networks. In future, we plan to extend the current tool set for robust operation of infrastructure networks in several dierent directions. It is of interest to develop a, possible distributed, descent algorithm with provable convergence guarantee for the maximal DC ow problem, and study its relationship with the algorithm used in [26] to solve the maximal ow problem. It is also of interest to develop a unied theory for network reduction that considers both reduction to links and reduction to nodes. As for distributed methods, we plan to explore the use of ADMM for solving non-convex problems in DC networks, by leveraging advancements from the active development of ADMM, e.g., see [87], and the applicability of second order distributed methods such as Newton's method [6,91]. Finally, we plan to extend our analysis on DC networks to realistic power grid dynamics, e.g., see [67], and stochastic failure rules, e.g., see [19]. 203 Reference List [1] G Andersson, P Donalek, R Farmer, N Hatziargyriou, I Kamwa, P Kundur, N Martins, J Paserba, P Pourbeik, J Sanchez-Gasca, et al. Causes of the 2003 major grid blackouts in north america and europe, and recommended means to improve system dynamic performance. Power Systems, IEEE Transactions on, 20(4):1922{1928, 2005. [2] Sanjeev Arora, Elad Hazan, and Satyen Kale. The multiplicative weights update method: a meta-algorithm and applications. Theory of Computing, 8(1):121{164, 2012. [3] Q. Ba and K. Savla. A dynamic programming approach to optimal load shedding control of cascading failure in DC power networks. In IEEE Conference on Decision and Control, Las Vegas, NV, 2016. [4] Q. Ba and K. Savla. On decentralized robust weight control for DC power networks. In American Control Conference, pages 5933{5938, Boston, MA, 2016. [5] Q. Ba and K. Savla. Robustness of DC networks under controllable link weights. IEEE Transactions on Control of Network Systems, 2017. In Press. Available at http://arxiv.org/abs/1609.02179. [6] Q. Ba, K. Savla, and G. Como. Distributed optimal equilibrium selection for trac ow over networks. In IEEE Conference on Decision and Control, pages 6942{6947, Osaka, Japan, 2015. [7] Qin Ba and Ketan Savla. Computing optimal control of cascading failure in dc networks. arXiv preprint arXiv:1712.06064, 2017. [8] Fr ed eric Babonneau, Yurii Nesterov, and Jean-Philippe Vial. Design and operations of gas transmission networks. Operations research, 60(1):34{47, 2012. [9] AS Bakshi, A Velayutham, SC Srivastava, KK Agrawal, RN Nayak, SK Soonee, and B Singh. Report of the enquiry committee on grid disturbance in northern region on 30th july 2012 and in northern, eastern & north-eastern region on 31st july 2012. New Delhi, India, 2012. [10] Albert-L aszl o Barab asi. Linked: The new science of networks, 2003. [11] A. Barrat, M. Barthelemy, and A. Vespignani. Dynamical Processes on Complex Networks. Cambridge University Press, 2008. [12] D Behmardi and ED Nayeri. Introduction of Fr echet and G^ ateaux derivative. Applied Mathematical Sciences, 2(20):975{980, 2008. [13] Andrey Bernstein, Daniel Bienstock, David Hay, Meric Uzunoglu, and Gil Zussman. Power grid vulnerability to geographically correlated failures-analysis and control implications. arXiv preprint arXiv:1206.1099, 2012. 204 [14] Andrey Bernstein, Daniel Bienstock, David Hay, Meric Uzunoglu, and Gil Zussman. Sensi- tivity analysis of the power grid vulnerability to large-scale cascading failures. ACM SIG- METRICS Performance Evaluation Review, 40(3):33{37, 2012. [15] Dimitri P Bertsekas. Network optimization: continuous and discrete models. Athena Scien- tic Belmont, 1998. [16] Dimitri P Bertsekas. Convex optimization algorithms. Athena Scientic Belmont, 2015. [17] Dimitris Bertsimas and John N Tsitsiklis. Introduction to linear optimization, volume 6. Athena Scientic Belmont, MA, 1997. [18] Daniel Bienstock. Optimal control of cascading power grid failures. In Decision and Control and European Control Conference (CDC-ECC), 2011 50th IEEE Conference on, pages 2166{ 2173. IEEE, 2011. [19] Daniel Bienstock. Electrical Transmission System Cascades and Vulnerability: An Operations Research Viewpoint, volume 22. SIAM, 2016. [20] Daniel Bienstock and Abhinav Verma. The n-k problem in power grids: New models, for- mulations, and numerical experiments. SIAM Journal on Optimization, 20(5):2352{2380, 2010. [21] Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, and Jonathan Eckstein. Distributed op- timization and statistical learning via the alternating direction method of multipliers. Foun- dations and Trends in Machine Learning, 3(1):1{122, 2011. [22] Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, 2004. [23] F. Bullo. Lectures on Network Systems. Version 0.96, 2018. With contributions by J. Cortes, F. Dor er, and S. Martinez. [24] Thomas Burger, Peter Gritzmann, and Victor Klee. Polytope projection and projection polytopes. The American mathematical monthly, 103(9):742{755, 1996. [25] Jie Chen, James S Thorp, and Ian Dobson. Cascading dynamics and mitigation assessment in power system disturbances via a hidden failure model. International Journal of Electrical Power and Energy Systems, 27(4):318{326, 2005. [26] Paul Christiano, Jonathan A Kelner, Aleksander Madry, Daniel A Spielman, and Shang- Hua Teng. Electrical ows, laplacian systems, and faster approximation of maximum ow in undirected graphs. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages 273{282. ACM, 2011. [27] Fan RK Chung. Spectral graph theory (CBMS regional conference series in mathematics, No. 92). American Mathematical Society, 1996. [28] Carleton Corin and Pascal Van Hentenryck. A linear-programming approximation of AC power ows. INFORMS Journal on Computing, 26(4):718{734, 2014. [29] Reuven Cohen, Keren Erez, Daniel Ben-Avraham, and Shlomo Havlin. Resilience of the internet to random breakdowns. Physical review letters, 85(21):4626, 2000. [30] G. Como, E. Lovisari, and K. Savla. Convexity and robustness of dynamic trac assignment for control of freeway networks. Transportation Research Part B: Methodological, 91:446{465, September 2016. 205 [31] G. Como, K. Savla, D. Acemoglu, M. A. Dahleh, and E. Frazzoli. Robust distributed routing in dynamical networks { part I: Locally responsive policies and weak resilience. IEEE Trans. on Automatic Control, 58(2):317{332, 2013. [32] G. Como, K. Savla, D. Acemoglu, M. A. Dahleh, and E. Frazzoli. Robust distributed routing in dynamical networks { part II: Strong resilience, equilibrium selection and cascaded failures. IEEE Trans. on Automatic Control, 58(2):333{348, 2013. [33] P. Crucitti, V. Latora, and M. Marchiori. Model for cascading failures in complex networks. Physical Review E, 69(4), 2004. [34] Carlos F Daganzo. The cell transmission model: A dynamic representation of highway trac consistent with the hydrodynamic theory. Transportation Research Part B: Methodological, 28(4):269{287, 1994. [35] Carlos F Daganzo. The cell transmission model, part ii: network trac. Transportation Research Part B: Methodological, 29(2):79{93, 1995. [36] Mark De Berg, Otfried Cheong, Marc Van Kreveld, and Mark Overmars. Computational Geometry: Introduction. Springer, 2008. [37] Reinhard Diestel. Graph theory, volume 173 of graduate texts in mathematics, 2005. [38] Herbert Edelsbrunner. Algorithms in combinatorial geometry, volume 10 of eatcs monographs on theoretical computer science, 1987. [39] Herbert Edelsbrunner, Joseph ORourke, and Raimund Seidel. Constructing arrangements of lines and hyperplanes with applications. SIAM Journal on Computing, 15(2):341{363, 1986. [40] Francisco Facchinei and Jong-Shi Pang. Finite-dimensional variational inequalities and com- plementarity problems. Springer Science & Business Media, 2007. [41] Lester R Ford and Delbert R Fulkerson. Maximal ow through a network. Canadian Journal of Mathematics, 8(3):399{404, 1956. [42] Terry L Friesz. Network science, nonlinear science and infrastructure systems, volume 102. Springer Science & Business Media, 2007. [43] St ephane Gerbex, Rachid Cherkaoui, and Alain J Germond. Optimal location of multi-type FACTS devices in a power system by means of genetic algorithms. Power Systems, IEEE Transactions on, 16(3):537{544, 2001. [44] Isidoro Gitler and Feli u Sagols. On terminal delta-wye reducibility of planar graphs. Net- works, 57(2):174{186, 2011. [45] J Duncan Glover, Mulukutla Sarma, and Thomas Overbye. Power System Analysis & Design, SI Version. Cengage Learning, 2011. [46] Christopher David Godsil and Gordon Royle. Algebraic graph theory, volume 8. Springer New York, 2001. [47] Gene H Golub and Victor Pereyra. The dierentiation of pseudo-inverses and nonlinear least squares problems whose variables separate. SIAM Journal on numerical analysis, 10(2):413{ 432, 1973. [48] G. Gomes and R. Horowitz. Optimal freeway ramp metering using the asymmetric cell transmission model. Transportation Research Part C, 14(4):244{268, 2006. 206 [49] Branko Gr unbaum, Victor Klee, Micha A Perles, and Georey Colin Shephard. Convex polytopes, volume 16. Springer, 1967. [50] Dan Halperin and Micha Sharir. Arrangements and their applications in robotics: recent developments. In Proceedings of the workshop on Algorithmic foundations of robotics, pages 495{511. AK Peters, Ltd., 1995. [51] Kory W Hedman, Shmuel S Oren, and Richard P O'Neill. A review of transmission switching and network topology optimization. In 2011 IEEE power and energy society general meeting, pages 1{7. IEEE, 2011. [52] Harold V Henderson and Shayle R Searle. On deriving the inverse of a sum of matrices. Siam Review, 23(1):53{60, 1981. [53] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, 1990. [54] Reiner Horst and Panos M Pardalos. Handbook of global optimization, volume 2. Springer Science & Business Media, 2013. [55] Gabriela Hug-Glanzmann and G oran Andersson. Decentralized optimal power ow control for overlapping areas in power systems. Power Systems, IEEE Transactions on, 24(1):327{ 336, 2009. [56] Matthew O Jackson. Social and economic networks. Princeton University Press, 2010. [57] S. Jafarpour and F. Bullo. Synchronization of Kuramoto oscillators via cutset projections. IEEE Trans. on Automatic Control, Nov. 2017. Submitted. see also arXiv:1711.03711v1. [58] Douglas J Klein and Milan Randi c. Resistance distance. Journal of Mathematical Chemistry, 12(1):81{95, 1993. [59] Jon Kleinberg and Eva Tardos. Algorithm design. Pearson Education India, 2006. [60] Raymond P Klump and Thomas J Overbye. Assessment of transmission system loadability. Power Systems, IEEE Transactions on, 12(1):416{423, 1997. [61] B. Korte and J. Vygen. Combinatorial Optimization: Theory and Algorithms. Springer, 2002. [62] Ioannis Koutis, Gary L Miller, and Richard Peng. Approaching optimality for solving sdd linear systems. In Foundations of Computer Science (FOCS), 2010 51st Annual IEEE Sym- posium on, pages 235{244. IEEE, 2010. [63] C. Lai and Steven H. Low. The redistribution of power ow in cascading failures. In 51st Annual Allerton Conference on Communication, Control, and Computing, pages 1037{1044, 2013. [64] Yen-Yu Lee and Ross Baldick. A frequency-constrained stochastic economic dispatch model. IEEE Transactions on Power Systems, 28(3):2301{2312, 2013. [65] B Liscouski and W Elliot. Final report on the august 14, 2003 blackout in the united states and canada: Causes and recommendations. A report to US Department of Energy, 40(4), 2004. [66] Steven H. Low. Convex relaxation of optimal power ow. part I: Formulations and equiva- lence. IEEE Transactions on Control of Network Systems, 1(1):15 { 27, 2014. 207 [67] Enrique Mallada, Changhong Zhao, and Steven Low. Optimal load-side control for frequency regulation in smart grids. In Communication, Control, and Computing (Allerton), 2014 52nd Annual Allerton Conference on, pages 731{738. IEEE, 2014. [68] D.K. Merchant and G.L. Nemhauser. A model and an algorithm for the dynamic trac assignment problem. Transportation Science, 12:183{199, 1978. [69] D.K. Merchant and G.L. Nemhauser. Optimality conditions for a dynamic trac assignment model. Transportation Science, 12:200{207, 1978. [70] Carl D Meyer, Jr. Generalized inversion of modied matrices. SIAM Journal on Applied Mathematics, 24(3):315{323, 1973. [71] Adilson Motter. Cascade-based attacks on complex networks. Phys. Rev. E; Physical Review E, 66(6), 2002. [72] A. Muralidharan and R. Horowitz. Optimal control of freeway networks based on the link node cell transmission model. In Proceedings of the American Control Conference (ACC), pages 5769{5774, June 2012. [73] S. Peeta and A. K. Ziliaskopoulos. Foundations of dynamic trac assignment: The past, the present and the future. Networks and Spatial Economics, 1(3-4):233{265, 2001. [74] J. Reilly and A. M. Bayen. Distributed optimization for shared state systems: Applications to decentralized freeway control via subnetwork splitting. IEEE Transactions on Intelligent Transportation Systems, 16(6):3465 { 3472, 2015. [75] R. T. Rockafellar. Convex Analysis. Princeton University Press, 1970. [76] M. Rungger and G. Reissig. Arbitrarily precise abstractions for optimal controller synthesis. In IEEE Conf. on Decision and Control, Melbourne, Australia, 2017. To appear. [77] Stuart Jonathan Russell and Peter Norvig. Articial intelligence: a modern approach (3rd edition), 2009. [78] Raman Sanyal and G unter M Ziegler. Construction and analysis of projected deformed products. Discrete & Computational Geometry, 43(2):412{435, 2010. [79] K. Savla, G. Como, and M. A. Dahleh. Robust network routing under cascading failures. IEEE Transactions on Network Science and Engineering, 1(1):53{66, 2014. [80] Wei Shao and Vijay Vittal. LP-based OPF for corrective FACTS control to relieve overloads and voltage violations. IEEE Transactions on Power Systems, 21(4):1832{1839, 2006. [81] PGDJL Snell and Peter Doyle. Random walks and electric networks. Free Software Founda- tion, 2000. [82] S. Soltan, A. Loh, and G. Zussman. Analyzing and quantifying the eect of k-line failures in power grids. IEEE Transactions on Control of Network Systems, 2017. In press. [83] Daniel A Spielman and Shang-Hua Teng. Nearly linear time algorithms for precondition- ing and solving symmetric, diagonally dominant linear systems. SIAM Journal on Matrix Analysis and Applications, 35(3):835{885, 2014. [84] Brian Stott, Jorge Jardim, and Ongun Alsa c. Dc power ow revisited. Power Systems, IEEE Transactions on, 24(3):1290{1300, 2009. 208 [85] Csaba D Toth, Joseph O'Rourke, and Jacob E Goodman. Handbook of discrete and compu- tational geometry. CRC press, 2004. [86] Ulrike Von Luxburg. A tutorial on spectral clustering. Statistics and computing, 17(4):395{ 416, 2007. [87] Yu Wang, Wotao Yin, and Jinshan Zeng. Global convergence of admm in nonconvex nons- mooth optimization. arXiv preprint arXiv:1511.06324, 2015. [88] D. J. Watts. A simple model of global cascades on random networks. PNAS, 99(9):5766{ 5771, 2002. [89] Bruce Wollenberg and Allen Wood. Power generation, operation and control. John Wiley & Sons, Inc, pages 264{327, 1996. [90] Haibo You, Vijay Vittal, and Xiaoming Wang. Slow coherency-based islanding. Power Systems, IEEE Transactions on, 19(1):483{491, 2004. [91] Michael Zargham, Alejandro Ribeiro, Asuman Ozdaglar, and Ali Jadbabaie. Accelerated dual descent for network ow optimization. IEEE Trans. Automatic Control, 59(4):905 { 920, 2014. [92] G unter M Ziegler. Lectures on polytopes, volume 152. Springer Science & Business Media, 2012. [93] Ray Daniel Zimmerman, Carlos Edmundo Murillo-S anchez, and Robert John Thomas. Mat- power: Steady-state operations, planning, and analysis tools for power systems research and education. Power Systems, IEEE Transactions on, 26(1):12{19, 2011. 209 Appendix A A.1 Minimizing A Quasi-concave Function over A Polytope A polytope is the convex hull of nitely many pointsfb 1 ;:::;b m g [75, p. 12]. Lemma 29. Let h : R n ! R be a quasi-concave function and S R n be a polytope whose ele- ments can be expressed as convex combinations ofb 1 :::;b m . Then, min x2S h(x) = min i2[m] h(b i ). Proof. We prove by contradiction. Suppose argmin x2S h(x)\fb 1 ;:::;b m g =;. Since S is the convex hull offb 1 :::;b m g, for any x 2 argmin x2S h(x), there exist k 0;k 2 [m], with P m k=1 k = 1 such that x = P m k=1 k b k . Since x = 2fb 1 ;:::;b m g by assumption, we have k < 1 for all k2 [m]. Quasi-concavity of h(x) then implies: h(x ) =h( m X k=1 k b k ) =h( 1 b 1 + (1 1 ) m X k=2 k 1 1 b k ) minfh(b 1 );h( m X k=2 k 1 1 b k )g (A.1) where, in the second equality, it is easy to see that, due to P m k=2 k = 1 1 , we have P m k=2 k 11 b k 2 S. Recursive application of (A.1) then implies h(x ) min k2[m] h(b k ) 1 , giving a contradic- tion. A.2 Derivative of Pseudoinverse of Laplacian Matrix The following is an adaptation of the result from [47] on the derivative of the pseudo-inverse of a matrix. Theorem 7 (Theorem 4.3 [47]). LetX R k be an open set, and P (x)2 R mn , x2X , be a Fr echet dierentiable matrix function with local constant rank inX . Then for any x2X , P y (x) is Fr echet dierentiable and the Fr echet derivative is as follows. dP y (x) dx =P y dP dx P y +P y P y TdP T dx (IPP y ) + (IP y P ) dP T dx P y T P y where P y is the pseudo-inverse of P and d dx denotes the Fr echet derivative of a mapping. Proof. The local constant rank condition ensures that P y (x) and P (x) are continuous and dif- ferentiable in the calculations to follow. Since P y is the pseudo-inverse of matrix P , we have PP y P =P and P y PP y =P y . Then dP dx = d(PP y P ) dx = d(PP y ) dx P +PP y dP dx 1 If k = 0 for some k2 [m], then we exclude h(b k ) for that k from the minimization. 210 Multiplying from the right by P y and re-arranging, we get d(PP y ) dx (PP y ) = dP dx P y PP y dP dx P y = (IPP y ) dP dx P y Since (PP y )(PP y ) =PP y and PP y is symmetric, d(PP y ) dx = d(PP y ) 2 dx = d(PP y ) dx (PP y ) + (PP y ) d(PP y ) dx = d(PP y ) dx (PP y ) + d(PP y ) dx (PP y ) T = (IPP y ) dP dx P y +P y TdP T dx (IPP y ) (A.2) Likewise, we can get d(P y P ) dx =P y dP dx (IP y P ) + (IP y P ) dP T dx P y T (A.3) Since P y =P y PP y , we have following identities. dP y dx = d(P y PP y ) dx = dP y dx PP y +P y d(PP y ) dx (A.4a) dP y dx = d(P y PP y ) dx = d(P y P ) dx P y +P y P dP y dx (A.4b) dP y dx = d(P y PP y ) dx = dP y dx PP y +P y dP dx P y +P y P P y dx (A.4c) Computing (A.4a) + (A.4b) (A.4c), and substituting the resulting expression in (A.2) and (A.3), gives the theorem. A.3 Inverse of Sums of Matrices The following results on the inverse and Moore Penrose generalized inverse of sums of matrices are well known and its generalized versions are reviewed in [52]. Theorem 8. For P 2 R nm and Q2 R mn , I +PQ2 R nn is nonsingular if and only if I +QP2 R mm is nonsingular, in which case, (I +PQ) 1 =I (IPQ) 1 PQ =IP (I +QP ) 1 Q (A.5) (I +PQ) 1 P =P (I +QP ) 1 (A.6) Theorem 9. P2 R nn and Q2 R nn are symmetric matrices. If PP y Q =Q and I +QP y is nonsingular, then (P +Q) y =P y (I +QP y ) 1 = (I +P y Q) 1 P y (A.7) A.4 Extension of Proposition 3 Proposition 25. Consider a connected DC networkG = (V;E) containing single supply demand pair and with weightw2 R E >0 and balanced supply-demand vectorp2 R E . LetP be a simple path from the supply node to the demand node,E 0 :=fi2Pjf i (w) = 0g andP 0 := ([ i2E0 M i )[P, 211 whereM i is as dened in Denition 7. Without loss of generality, assume that links inP 0 has the same direction asP. Then there exists a direction d2 R E such that sgn [J(w)d] i = sgnf i for all i2EnP 0 and [J(w)d] i 0 for all i2P 0 . Moreover, the inequality for the latter is strict if and only ifM i \P 0 does not form a cut separating the supply node from the demand node. Proof. The proof proceeds along the same line as the proof of Proposition 3. We need to nd a direction d 2 R E with d i = 0 wherever f i (w) = 0 such that the assertion on J(w)d is true. Let d 0 2 R E be such that d 0 i := w i =f i (w) for i2fj2Pjf j (w)6= 0g and d 0 i := 0 else- where. Let := L y (w)p be an associated phase angle with ow f(w). For all k 2 E 0 , let V k := v2Vnf(k)gj v (k) , then (j) > (j) and f j (w)6= 0 for all j2C V k nfM k g, where we recallC V k denotes the cutV k EnV k andM k denotes the parallel links, if not unique, connecting node (k) with node (k). It is then possible to dene d k 2 R E to be such that d k j :=w 2 j = f j (w) P i2M k w i for allj2C V k nfkg andd k j := 0 elsewhere. Letd :=d 0 + P k2E0 d k . By denition, it is clear that d i = 0 wherever f i = 0. Let k := W 1 diag(f(w))d k for all k2E 0 [f0g and := P k2E0[f0g k . As d k = 0 for all k2E 0 , k = 0 can be written as k =w k = P j2M k w j w k = P j2M k w j . . By combining the rst term into 0 and the second term into k , we obtain the following. = 0 + X k2E0 w k P j2M k w j 1 k ! + X k2E0 k w k P j2M k w j ! = ~ X k2E0 1 P j2M k w j W1 C V k where ~ i = 1 for all i2PnE 0 , ~ i = w k = P j2M k w j for all i2[ k2E0 M k and ~ i = 0 for all i2EnP 0 . Since (IK(w))W1 C V k = 0 for all k2E 0 (cf. Remark 14), J(w)d = (IK(w)) = (IK(w)) ~ = ~ f(w; 2p=kpk 1 ) where the third equality follows from the fact thatP is a simple path ofG from the supply node to the demand node. Therefore, for i2EnP 0 , sgn [J(w)d] i = sgnf i (w) follows in the same way as in proof of Proposition 3; for i2PnE 0 , [J(w)d] i = 1f(w; 2p=kpk 1 ) 0 and the inequality is strict if and only iffig does not form a cut separating the supply node from the demand node, as followed from Lemma 2; for i2[ k2E0 M k , the result follows in a similar way from the proof of Lemma 2. A.5 Derivatives of g(w eq ) In this section, we provide explicit expression for the derivatives dened in (4.37). The derivatives depend on active links, which for the left derivative ofg, are dened asK + (x) := i2Ej i (w l i )<x for x2 [g l ;g max ], andK (x) :=fi2Ej i (w u i )<xg for x2 [g u ;g max ]. The left derivative, for w eq 2 (w l eq ; ^ g + (g max )], is then given by: g 0 (w eq ) = 1 ^ g + 0 (x ) x=g(weq) = 0 @ X i2K + (g(weq)) @H(w)=@w i 0 i (w i ) 1 A 1 w=! + (g(weq)) 212 where ^ g + 0 (x ) and 0 i (w i ) denote left derivatives, similar to g 0 (w eq ); the rst equality is because ^ g + is inverse ofg, and the second equality follows from chain rule. Following along the same lines, all the left and right derivatives of g are gathered as: g 0 (w eq ) = 8 > > > > > > > > > < > > > > > > > > > : 0 @ X i2K + (g(weq)) @H(w)=@w i 0 i (w i ) 1 A 1 w=! + (g(weq)) w eq 2 w l eq ; ^ g + (g max ) 0 w eq 2 (^ g + (g max ); ^ g (g max )] 0 @ X i2 ~ K (g(weq)) @H(w)=@w i 0 i (w i ) 1 A 1 w=! (g(weq)) w eq 2 ^ g (g max );w u eq g 0 (w + eq ) = 8 > > > > > > > > > < > > > > > > > > > : 0 @ X i2 ~ K + (g(weq)) @H(w)=@w i 0 i (w + i ) 1 A 1 w=! + (g(weq)) w eq 2 w l eq ; ^ g + (g max ) 0 w eq 2 [^ g + (g max ); ^ g (g max )) 0 @ X i2K (g(weq)) @H(w)=@w i 0 i (w + i ) 1 A 1 w=! (g(weq)) w eq 2 ^ g (g max );w u eq (A.8) where ~ K + (x) := i2Ej i (w l i )x for x2 [g l ;g max ], and ~ K (x) :=fi2Ej i (w u i )xg for x2 [g u ;g max ]. A.6 Proof of Proposition 19 Proof. We rst show that the conditions are sucient. We start by showing that the output functions from f(g in j ;X j )g j2[n] and f(g in j ; convX j )g j2[n] have the same domain under the given conditions. Since 1 j 2X j for all j2 [n] and P n j=1 X j \ (1; 1 j ] is connected, P n j=1 X j \ (1; 1 j ] = [ P n j=1 minX j ; P n j=1 1 j ]. Similarly, P n j=1 X j \ [ 1 j ;1) = [ P n j=1 1 j ; P n j=1 maxX j ]. Therefore, P n j=1 X j = [ P n j=1 minX j ; P n j=1 maxX j ] = P n j=1 convX j . It is straightforward that f(g in j ;X j )g j2[n] f(g in j ; convX j )g j2[n] . Hence it is sucient to prove the other direction. By Lemma 22, f(g in j ; convX j )g j2[n] is a function with top point ~ := ( P n j=1 1 j ; P n j=1 2 j ). We need to show that f(g in j ;X j )g j2[n] (z) ~ (z) for all z2 [ P n j=1 minX j ; P n j=1 maxX j ]. We rst consider z2 [ P n j=1 minX j ; P n j=1 1 j ]. Let X j contain m j pieces of intervals. Plus that 1 j 2X j separates an interval of X j into two pieces, with possibly one of the two containing the single point j , there are m j + 1 pieces in total. Without loss of generality, label the m j + 1 intervalsX k j in increasing order, that is,X k j such that maxX k1 j minX k j for allk2 [m j +1]. In particular, let l j 2 [m j + 1] be such that 1 j = maxX lj j = minX lj +1 j for all j2 [n]. Furthermore, let ~ := n j=1 [m j + 1] and ~ 0 := n 2 ~ j 0 o for all 0 2 ~ . The notation ~ > 0 has similar meaning. With these notations, X j \ (1; 1 j ] =[ jlj X j j . Then[ 2 ~ l P n j=1 X j j = P n j=1 [ jlj X j j = P n j=1 X j \ (1; 1 j ] = [ P n j=1 minX j ; P n j=1 1 j ]. It is then sucient to prove that for all 2 ~ l , () = ~ over the domain P n j=1 X j j . Pick arbitrary 2 ~ l , without loss of generality, let [q l j ;q u j ] := X j j . Restricted in this domain, g in j is a linear function with slope 1. Lemma 21 implies that (q u j ;q u j 1 j + 2 j ) can be treated as the top point of g in j over domain [q l j ;q u j ]. Lemma 22 and Remark 61 then implies that () is a linear function with top point ( P n j=1 q u j ; P n j=1 q u j P n j=1 1 j + P n j=1 2 j ). It is 213 straightforward to check that this top point lies on ~ . As a result, () = ~ when evaluated in the domain P n j=1 X j j . Due to symmetry, the same result can be shown for the case z 2 [ P n j=1 1 j ; P n j=1 maxX j ] by considering 2 ~ >l . We now show that the conditions are necessary. Lemma 22 implies that the corresponding problem f(g in j ; convX j )g j2[n] (z) to z = P n j=1 1 j has unique solution x j = 1 j for all j2 [n]. In order for f(g in j ;X j )g j2[n] = ~ to be true, it must be that 1 j 2X j for all j2 [n]. Previous arguments imply that it is sucient to prove that for all2 ~ n ( ~ l [ ~ >l ), either ()< ~ or () = ( 0 ) for some 0 2 ~ l [ ~ >l . Pick arbitrary2 ~ n( ~ l [ ~ >l ), bothfj : j l j g and fj : j l j + 1g are nonempty. Similarly as that in previous paragraphs, let [q l j ;q u j ] := X j j , then Lemma 21 and Lemma 22 imply that () is a function with top point (^ 1 ; ^ 2 ) := P fj:jljg q u j + P fj:jlj +1g q l j ; P fj:jljg (q u j 1 j ) + P fj:jlj +1g ( 1 j q l j ) + P n j=1 2 j and do- main [ P n j=1 q l j ; P n j=1 q u j ]. It is simple algebra to check the following: ~ (^ 1 ) ^ 2 = min 8 < : X fj:jljg ( 1 j q u j ); X fj:jlj +1g (q l j 1 j ) 9 = ; 0 If bothfj : j l j 1g andfj : j l j + 2g are nonempty, then the above inequality is strict. (^ 1 ; ^ 2 ) is below ~ and hence () < ~ . Otherwise, if eitherfj : j l j g =fj : j = l j g orfj : j l j + 1g =fj : j = l j + 1g, the above inequality becomes equality and (^ 1 ; ^ 2 ) lies on ~ . We consider the rst case and show that ( ()) (z) < ~ (z) for z2 [ P n j=1 q l j ; ^ 1 ) and ( ()) (z) = ( 0 ) for some 0 2 ~ l [ ~ >l for z2 [^ 1 ; P n j=1 q u j ]. Similar result can be obtained for the second case from the same argument. In this case, q u k = 1 k for all k2fj : j = l j g and q u k 1 k for all k2fj : j l j + 1g. Then ^ 1 P n j=1 1 j = ~ 1 . Remark 61 implies that for z2 [ P n j=1 q l j ; ^ 1 ), () is a linear function with slope1 over and hence ( ()) (z) < ~ (z), and for z 2 [^ 1 ; P n j=1 q u j ], x k = 1 k for all k2fj : j = l j g. Let 0 2 ~ >l be such that 0 k = l j + 1 for all k2fj : j = l j g and 0 k = k for all k2fj : j l j + 1g. Since by denition x k = 1 k 2X l k +1 k for all k2fj : j =l j g and by construction x k 2X l k +1 k for all k2fj : j l j + 1g, we have () = ( 0 ). 214

Abstract (if available)

Abstract An infrastructure system is formally modeled as a network, whose topology is described by a directed or undirected graph, and whose nodes and links are associated with physical quantities such as node potential and link flow. Two canonical physical laws occurring in multiple infrastructure networks are considered to relate these physical quantities. The first one is flow conservation law. It requires zero net flow at every node. The second physical law, referred to as diffusion law, requires a strictly monotonic relationship between link flow and the potential difference between the incident nodes. The linear version of diffusion law plus flow conservation law defines the class of DC networks. The flow over a DC network is uniquely determined by node supply-demands and link weights. We are interested in finding a weight that produces a maximal feasible flow, i.e, flow satisfying capacity constraints, for given supply-demands over a DC network. The solution relies on minimizing a congestion function that is neither convex nor differentiable. Based on the derived directional derivative of the congestion function and properties proved for flow-weight Jacobian matrix, we show that every local minimum of the congestion function is a global minimum. ❧ The flexibility of choosing weight is also adopted as a control mechanism to maximize the robustness of a DC network toward disturbances to nominal supply-demand. The margin of robustness is defined as the radius of the largest L1 ball of disturbances for which there exists a weight within specified bounds such that the resulting flow is feasible. Computation of this margin is posed as a non-convex optimization problem. A multi-level programming approach is developed to solve the problem for reducible networks, i.e., a class of networks with certain sparsity properties. The approach is based on a novel notion of equivalent capacity function, which is proven to possess a strong quasi-concavity property for link reducible networks. This property further facilitates an easy solution to the multilevel programming formulation for a subclass of reducible networks, that we call type I tree reducible networks. ❧ Another major consideration in robust operation of infrastructure networks is to prevent cascading failure, i.e., a phenomenon under which the initial failure of a small number of links due to an exogenous event is followed by a sequence of link failures due to the interplay between physics, network topology, and controls. We consider DC networks and model cascading failure as a discrete-time dynamical system featuring a hybrid state consisting of a set variable for the active link and a continuous variable for supply-demand. While link weight is assumed fixed, load shedding, i.e., decreasing the magnitude of supply-demand, is adopted as the control mechanism to stop failure propagation. The control objective is to minimize the amount of load to be shed. We propose two approaches for computing an optimal control, and provide time complexity analysis for these approaches. The first approach, geared towards a certain class of networks that we call type II tree reducible networks, decomposes the global non-convex problem into a system of coupled local non-convex problems, which can be solved to optimality in two iterations. The second approach transforms the continuous reachable set equivalently into a finite set by leveraging and extending tools for arrangement of hyperplanes and convex polytopes. ❧ Finally, we adapt the Alternating Direction Method of Multipliers (ADMM) to develop a distributed algorithm for computing finite horizon optimal control for traffic flow over transportation networks and present its convergence properties.

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Creator Ba, Qin (author)

Core Title Elements of robustness and optimal control for infrastructure networks

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Civil Engineering

Publication Date 02/22/2018

Defense Date 01/19/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag cascading failure,distributed optimization,infrastructure networks,non-convex optimization,OAI-PMH Harvest,optimal control,power grids,robustness,transportation networks

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Savla, Ketan (committee chair), Ghanem, Roger (committee member), Nayyar, Ashutosh (committee member), Pang, Jong-Shi (committee member)

Creator Email baqin101@gmail.com,qba@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-475855

Unique identifier UC11265766

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Legacy Identifier etd-BaQin-6053.pdf

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