Results 1  10
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27
On the exactness of convex relaxation for optimal power flow in tree networks,” in
 Proc. IEEE Conf. Decision Control,
, 2012
"... AbstractThe optimal power flow (OPF) problem determines a network operating point that minimizes a certain objective such as generation cost or power loss. It is nonconvex. We prove that a global optimum of OPF can be obtained by solving a secondorder cone program, under a mild condition after sh ..."
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Cited by 14 (5 self)
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AbstractThe optimal power flow (OPF) problem determines a network operating point that minimizes a certain objective such as generation cost or power loss. It is nonconvex. We prove that a global optimum of OPF can be obtained by solving a secondorder cone program, under a mild condition after shrinking the OPF feasible set slightly, for radial power networks. The condition can be checked a priori, and holds for the IEEE 13, 34, 37, 123bus networks and two realworld networks. Index TermsOptimal power flow (OPF).
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 13 (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.
Global Optimization of Optimal Power Flow Using a Branch & Bound Algorithm
, 2012
"... We propose two algorithms for the solution of the Optimal Power Flow (OPF) problem to global optimality. The algorithms are based on the spatial branch and bound framework with lower bounds on the optimal objective function value calculated by solving either the Lagrangian dual or the semidefinite p ..."
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Cited by 12 (1 self)
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We propose two algorithms for the solution of the Optimal Power Flow (OPF) problem to global optimality. The algorithms are based on the spatial branch and bound framework with lower bounds on the optimal objective function value calculated by solving either the Lagrangian dual or the semidefinite programming (SDP) relaxation. We show that this approach can solve to global optimality the general form of the OPF problem including: generation power bounds, apparent and real power line limits, voltage limits and thermal loss limits. The approach makes no assumption on the topology or resistive connectivity of the network. This work also removes some of the restrictive assumptions of the SDP approaches [1], [2], [3], [4], [5]. We present the performance of the algorithms on a number of standard IEEE systems, which are known to have a zero duality gap. We also make parameter perturbations to the test cases that result in solutions that fail to satisfy the SDP rank condition and have a nonzero duality gap. The proposed branch and bound algorithms are able to solve these cases to global optimality.
Convex Relaxation of Optimal Power Flow Part II: Exactness
, 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 10 (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.
Branch Flow Model: Relaxations and Convexification
, 2012
"... 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) problems that consists of two relaxation steps. The first step eliminates the voltage and current angles and the second step app ..."
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Cited by 6 (3 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) problems 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 and we characterize when the angle relaxation may fail. We propose a simple method to convexify a mesh network using phase shifters so that both relaxation steps are always exact and OPF for the convexified network can always be solved efficiently for a globally optimal solution.
Application of the momentsos approach to global optimization of the opf problem,”
 IEEE Trans. on Power Syst.,
, 2014
"... Finding a global solution to the optimal power flow (OPF) problem is difficult due to its nonconvexity. A convex relaxation in the form of semidefinite programming (SDP) has attracted much attention lately as it yields a global solution in several practical cases. However, it does not in all cases, ..."
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Cited by 5 (0 self)
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Finding a global solution to the optimal power flow (OPF) problem is difficult due to its nonconvexity. A convex relaxation in the form of semidefinite programming (SDP) has attracted much attention lately as it yields a global solution in several practical cases. However, it does not in all cases, and such cases have been documented in recent publications. This paper presents another SDP method known as the momentsos (sum of squares) approach, which generates a sequence that converges towards a global solution to the OPF problem at the cost of higher runtime. Our finding is that in the small examples where the previously studied SDP method fails, this approach finds the global solution. The higher cost in runtime is due to an increase in the matrix size of the SDP problem, which can vary from one instance to another. Numerical experiment shows that the size is very often a quadratic function of the number of buses in the network, whereas it is a linear function of the number of buses in the case of the previously studied SDP method.
A rank minimization algorithm to enhance semidefinite relaxations of optimal power flow
 In 51st Annu. Allerton Conf. Commun., Control, and Comput
, 2013
"... Abstract — The Optimal Power Flow (OPF) problem is nonconvex and, for generic network structures, is NPhard. A recent flurry of work has explored the use of semidefinite relaxations to solve the OPF problem. For general network structures, however, this approach may fail to yield solutions that ar ..."
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Cited by 5 (0 self)
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Abstract — The Optimal Power Flow (OPF) problem is nonconvex and, for generic network structures, is NPhard. A recent flurry of work has explored the use of semidefinite relaxations to solve the OPF problem. For general network structures, however, this approach may fail to yield solutions that are physically meaningful, in the sense that they are high rank – precluding their efficient mapping back to the original feasible set. In certain cases, however, there may exist a hidden rankone optimal solution. In this paper, an iterative linearizationminimization algorithm is proposed to uncover rankone solutions for the relaxation. The iterates are shown to converge to a stationary point. A simple bisection method is also proposed to address problems for which the linearizationminimization procedure fails to yield a rankone optimal solution. The algorithms are tested on representative power system examples. In many cases, the linearizationminimization procedure obtains a rankone optimal solution where the naive semidefinite relaxation fails. Furthermore, a 14bus example is provided for which the linearizationminimization algorithm achieves a rankone solution with a cost strictly lower than that obtained by a conventional solver. We close by discussing some rank monotonicity properties of the proposed methodology.
Resistive network optimal power flow: uniqueness and algorithms
 IEEE Trans. on Power Systems
, 2014
"... Abstract—The optimal power flow (OPF) problem minimizes the power loss in an electrical network by optimizing the voltage and power delivered at the network buses, and is a nonconvex problem that is generally hard to solve. By leveraging a recent development on the zero duality gap of OPF, we propos ..."
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Cited by 5 (3 self)
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Abstract—The optimal power flow (OPF) problem minimizes the power loss in an electrical network by optimizing the voltage and power delivered at the network buses, and is a nonconvex problem that is generally hard to solve. By leveraging a recent development on the zero duality gap of OPF, we propose a secondorder cone programming convex relaxation of the resistive network OPF, and study the uniqueness of the optimal solution using differential topology especially the Poincare–Hopf Index Theorem. We characterize the global uniqueness for different network topologies, e.g., line, radial and mesh networks. This serves as a starting point to design distributed local algorithms with global behaviors that have low complexity, computationally fast and can run under synchronous and asynchronous settings in practical power grids. I.
Equivalent relaxations of optimal power flow
 IEEE Trans. Automatic Control
, 2014
"... Abstract—Several convex relaxations of the optimal power flow (OPF) problem have recently been developed using both bus injection models and branch flow models. In this paper, we prove relations among three convex relaxations: a semidefinite relaxation that computes a full matrix, a chordal relaxati ..."
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Abstract—Several convex relaxations of the optimal power flow (OPF) problem have recently been developed using both bus injection models and branch flow models. In this paper, we prove relations among three convex relaxations: a semidefinite relaxation that computes a full matrix, a chordal relaxation based on a chordal extension of the network graph, and a secondorder cone relaxation that computes the smallest partial matrix. We prove a bijection between the feasible sets of the OPF in the bus injection model and the branch flow model, establishing the equivalence of these two models and their secondorder cone relaxations. Our results imply that, for radial networks, all these relaxations are equivalent and one should always solve the secondorder cone relaxation. For mesh networks, the semidefinite relaxation is tighter than the secondorder cone relaxation but requires a heavier computational effort, and the chordal relaxation strikes a good balance. Simulations are used to illustrate these results.
Optimization decomposition of resistive power networks with energy storage
 IEEE Journal on Selected Areas in Comms
, 2014
"... Abstract—A fundamental challenge of a smart grid is: to what extent can moving energy through space and time be optimized to benefit the power network with largescale energy storage integration? With energy storage, there is a possibility to generate more energy when the demand is low and store it ..."
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Cited by 3 (3 self)
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Abstract—A fundamental challenge of a smart grid is: to what extent can moving energy through space and time be optimized to benefit the power network with largescale energy storage integration? With energy storage, there is a possibility to generate more energy when the demand is low and store it for later use. In this paper, we study a dynamic optimal power flow problem with energy storage dynamics in purely resistive power networks. By exploiting a recentlydiscovered zero duality gap property in the optimal power flow (OPF) problem, we apply optimization decomposition techniques to decouple the coupling energy storage constraints and obtain the global optimal solution using distributed message passing algorithms. The decomposition methods offer new interesting insights on the equilibrium load profile smoothing feature over space and time through the relationship between the optimal dual solution in the OPF and the energy storage dynamics. We evaluate the performance of the distributed algorithms in several IEEE benchmark systems and show that they converge fast to the global optimal solution by numerical simulations. Index Terms — Optimal power flow, energy storage, decomposition method, distributed optimization, smart grid, message passing algorithm. I.