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Optimal power flow over tree networks
 PROCEEDINGS OF THE FORTHNINTH ANNUAL ALLERTON CONFERENCE
, 2011
"... The optimal power flow (OPF) problem is critical to power system operation but it is generally nonconvex and therefore hard to solve. Recently, a sufficient condition has been found under which OPF has zero duality gap, which means that its solution can be computed efficiently by solving the conve ..."
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Cited by 27 (12 self)
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The optimal power flow (OPF) problem is critical to power system operation but it is generally nonconvex and therefore hard to solve. Recently, a sufficient condition has been found under which OPF has zero duality gap, which means that its solution can be computed efficiently by solving the convex dual problem. In this paper we simplify this sufficient condition through a reformulation of the problem and prove that the condition is always satisfied for a tree network provided we allow oversatisfaction of load. The proof, cast as a complex semidefinite program, makes use of the fact that if the underlying graph of an n n Hermitian positive semidefinite matrix is a tree, then the matrix has rank at least n  1.
Quadratically constrained quadratic programs on acyclic graphs with application to power flow
, 2013
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A LinearProgramming Approximation of AC Power Flows
, 2013
"... Linear activepoweronly DC power flow approximations are pervasive in the planning and control of power systems. However, these approximations fail to capture reactive power and voltage magnitudes, both of which are necessary in many applications to ensure voltage stability and AC power flow feasib ..."
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Cited by 17 (4 self)
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Linear activepoweronly DC power flow approximations are pervasive in the planning and control of power systems. However, these approximations fail to capture reactive power and voltage magnitudes, both of which are necessary in many applications to ensure voltage stability and AC power flow feasibility. This paper proposes linearprogramming models (the LPAC models) that incorporate reactive power and voltage magnitudes in a linear power flow approximation. The LPAC models are built on a convex approximation of the cosine terms in the AC equations, as well as Taylor approximations of the remaining nonlinear terms. Experimental comparisons with AC solutions on a variety of standard IEEE and MATPOWER benchmarks show that the LPAC models produce accurate values for active and reactive power, phase angles, and voltage magnitudes. The potential benefits of the LPAC models are illustrated on two “proofofconcept” studies in power restoration and capacitor placement.
Monitoring and Optimization for Power Grids: A Signal Processing Perspective
 IEEE SIGNAL PROCESSING MAGAZINE
, 2012
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Chance Constrained Optimal Power Flow: RiskAware Network Control under Uncertainty
, 2012
"... Abstract. When uncontrollable resources fluctuate, Optimum Power Flow (OPF), routinely used by the electric power industry to redispatch hourly controllable generation (coal, gas and hydro plants) over control areas of transmission networks, can result in grid instability, and, potentially, cascadi ..."
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Cited by 16 (4 self)
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Abstract. When uncontrollable resources fluctuate, Optimum Power Flow (OPF), routinely used by the electric power industry to redispatch hourly controllable generation (coal, gas and hydro plants) over control areas of transmission networks, can result in grid instability, and, potentially, cascading outages. This risk arises because OPF dispatch is computed without awareness of major uncertainty, in particular fluctuations in renewable output. As a result, grid operation under OPF with renewable variability can lead to frequent conditions where power line flow ratings are significantly exceeded. Such a condition, which is borne by simulations of real grids, would likely resulting in automatic line tripping to protect lines from thermal stress, a risky and undesirable outcome which compromises stability. Smart grid goals include a commitment to large penetration of highly fluctuating renewables, thus calling to reconsider current practices, in particular the use of standard OPF. Our Chance Constrained (CC) OPF corrects the problem and mitigates dangerous renewable fluctuations with minimal changes in the current operational procedure. Assuming availability of a reliable wind forecast parameterizing the distribution function of the uncertain generation, our CCOPF satisfies all the constraints with high probability while simultaneously minimizing the cost of economic redispatch. CCOPF allows efficient implementation, e.g. solving a typical instance over the 2746bus Polish network in 20s on a standard laptop.
Message passing for dynamic network energy management
, 2012
"... We consider a network of devices, such as generators, fixed loads, deferrable loads, and storage devices, each with its own dynamic constraints and objective, connected by lossy capacitated lines. The problem is to minimize the total network objective subject to the device and line constraints, over ..."
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Cited by 15 (0 self)
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We consider a network of devices, such as generators, fixed loads, deferrable loads, and storage devices, each with its own dynamic constraints and objective, connected by lossy capacitated lines. The problem is to minimize the total network objective subject to the device and line constraints, over a given time horizon. This is a large optimization problem, with variables for consumption or generation in each time period for each device. In this paper we develop a decentralized method for solving this problem. The method is iterative: At each step, each device exchanges simple messages with its neighbors in the network and then solves its own optimization problem, minimizing its own objective function, augmented by a term determined by the messages it has received. We show that this message passing method converges to a solution when the device objective and constraints are convex. The method is completely decentralized, and needs no global coordination other than synchronizing iterations; the problems to be solved by each device can typically be solved extremely efficiently and in parallel. The method is fast enough that even a serial implementation can solve substantial problems
Branch Flow Model: Relaxations and Convexification (Part I)
, 2013
"... We propose a branch flow model for the analysis and optimization of mesh as well as radial networks. The model leads to a new approach to solving optimal power flow (OPF) that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step approximates ..."
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Cited by 15 (7 self)
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We propose a branch flow model for the analysis and optimization of mesh as well as radial networks. The model leads to a new approach to solving optimal power flow (OPF) that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step approximates the resulting problem by a conic program that can be solved efficiently. For radial networks, we prove that both relaxation steps are always exact, provided there are no upper bounds on loads. For mesh networks, the conic relaxation is always exact but the angle relaxation may not be exact, and we provide a simple way to determine if a relaxed solution is globally optimal. We propose convexification of mesh networks using phase shifters so that OPF for the convexified network can always be solved efficiently for an optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints. Part I introduces our branch flow model, explains the two relaxation steps, and proves the conditions for exact relaxation. Part II describes convexification of mesh networks, and presents simulation results.
Power grid vulnerability to geographically correlated failures  analysis and control implications
, 2011
"... We consider power line outages in the transmission system of the power grid, and specifically those caused by a natural disaster or a large scale physical attack. In the transmission system, an outage of a line may lead to overload on other lines, thereby eventually leading to their outage. While s ..."
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Cited by 14 (5 self)
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We consider power line outages in the transmission system of the power grid, and specifically those caused by a natural disaster or a large scale physical attack. In the transmission system, an outage of a line may lead to overload on other lines, thereby eventually leading to their outage. While such cascading failures have been studied before, our focus is on cascading failures that follow an outage of several lines in the same geographical area. We provide an analytical model of such failures, investigate the model’s properties, and show that it differs from other models used to analyze cascades in the power grid (e.g., epidemic/percolationbased models). We then show how to identify the most vulnerable locations in the grid and perform extensive numerical experiments with real grid data to investigate the various effects of geographically correlated outages and the resulting cascades. These results allow us to
Convex Relaxation of Optimal Power Flow  Part I: Formulations and Equivalence
, 2014
"... This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. ..."
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Cited by 12 (0 self)
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This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. Part II presents sufficient conditions under which the convex relaxations are exact.
Exact convex relaxation for optimal power flow in tree networks
, 2012
"... Abstract — The optimal power flow problem is nonconvex, and a convex relaxation has been proposed to solve it. We prove that the relaxation is exact, if there are no upper bounds on the voltage, and any one of some conditions holds. One of these conditions requires that there is no reverse real powe ..."
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Cited by 11 (4 self)
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Abstract — The optimal power flow problem is nonconvex, and a convex relaxation has been proposed to solve it. We prove that the relaxation is exact, if there are no upper bounds on the voltage, and any one of some conditions holds. One of these conditions requires that there is no reverse real power flow, and that the resistance to reactance ratio is nondecreasing as transmission lines spread out from the substation to the branch buses. This condition is likely to hold if there are no distributed generators. Besides, avoiding reverse real power flow can be used as rule of thumb for placing distributed generators. I.