S.H. Low

California Institute of Technology, Pasadena, California, United States

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Publications (251)119.21 Total impact

  • Qiuyu Peng, Steven H. Low
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    ABSTRACT: The optimal power flow (OPF) problem is fundamental in power system operations and planning. Large-scale renewable penetration calls for real-time feedback control, and hence the need for fast and distributed solutions for OPF. This is difficult because OPF is nonconvex and Kirchhoff's laws are global. In this paper we propose a solution for radial networks. It exploits recent results that suggest solving for a globally optimal solution of OPF over a radial network through the second-order cone program (SOCP) relaxation. Our distributed algorithm is based on alternating direction method of multiplier (ADMM), but unlike standard ADMM algorithms that often require iteratively solving optimization subproblems in each ADMM iteration, our decomposition allows us to derive closed form solutions for these subproblems, greatly speeding up each ADMM iteration. We present simulations on a real-world 2,065-bus distribution network to illustrate the scalability and optimality of the proposed algorithm.
    04/2014;
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    ABSTRACT: We study the role of a market maker (or market operator) in a transmission constrained electricity market. We model the market as a one-shot networked Cournot competition where generators supply quantity bids and load serving entities provide downward sloping inverse demand functions. This mimics the operation of a spot market in a deregulated market structure. In this paper, we focus on possible mechanisms employed by the market maker to balance demand and supply. In particular, we consider three candidate objective functions that the market maker optimizes - social welfare, residual social welfare, and consumer surplus. We characterize the existence of Generalized Nash Equilibrium (GNE) in this setting and demonstrate that market outcomes at equilibrium can be very different under the candidate objective functions.
    03/2014;
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    ABSTRACT: Deferrable load control is essential for handling the uncertainties associated with the increasing penetration of renewable generation. Model predictive control has emerged as an effective approach for deferrable load control, and has received considerable attention. In particular, previous work has analyzed the average-case performance of model predictive deferrable load control. However, to this point, distributional analysis of model predictive deferrable load control has been elusive. In this paper, we prove strong concentration results on the distribution of the load variance obtained by model predictive deferrable load control. These concentration results highlight that the typical performance of model predictive deferrable load control is tightly concentrated around the average-case performance.
    03/2014;
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    Changhong Zhao, Steven Low
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    ABSTRACT: We augment existing generator-side primary frequency control with load-side control that are local, ubiquitous, and continuous. The mechanisms on both the generator and the load sides are decentralized in that their control decisions are functions of locally measurable frequency deviations. These local algorithms interact over the network through nonlinear power flows. We design the local frequency feedback control so that any equilibrium point of the closed-loop system is the solution to an optimization problem that minimizes the total generation cost and user disutility subject to power balance across entire network. With Lyapunov method we derive a sufficient condition ensuring an equilibrium point of the closed-loop system is asymptotically stable. Simulation demonstrates improvement in both the transient and steady-state performance over the traditional control only on the generators, even when the total control capacity remains the same.
    03/2014;
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    Lingwen Gan, Ufuk Topcu, Steven H. Low
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    ABSTRACT: Electric vehicles (EVs) should be charged with some given profile to protect its battery, making the optimal charging problem discrete. We propose a stochastic distributed algorithm to approximately solve the discrete optimization in an iterative procedure. In each iteration, a center node broadcasts the average electricity load per EV; this information is used by each EV to generate a probability distribution over its potential charging profiles; then EVs sample from the distributions to update their charging profiles. We prove that the algorithm converges {\em almost surely} to one of its {\em equilibrium} charging profiles in {\em finite} iterations, and each of its equilibrium charging profiles has a sub-optimality upper bounded that scales as $\hO(1/N)$, where $N$ is the number of EVs.
    01/2014;
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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 second-order 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 second-order cone relaxations. Our results imply that, for radial networks, all these relaxations are equivalent and one should always solve the second-order cone relaxation. For mesh networks, the semidefinite relaxation is tighter than the second-order cone relaxation but requires a heavier computational effort, and the chordal relaxation strikes a good balance. Simulations are used to illustrate these results.
    01/2014;
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    Lingwen Gan, Na Li, Ufuk Topcu, Steven H. Low
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    ABSTRACT: The optimal power flow (OPF) problem determines power generation/demand that minimize a certain objective such as generation cost or power loss. It is nonconvex. We prove that, for radial networks, after shrinking its feasible set slightly, the global optimum of OPF can be recovered via a second-order cone programming (SOCP) relaxation under a condition that can be checked a priori. The condition holds for the IEEE 13-, 34-, 37-, 123-bus networks and two real-world networks, and has a physical interpretation.
    11/2013;
  • Qiuyu Peng, Steven H. Low
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    ABSTRACT: The feeder reconfiguration problem chooses the on/off status of the switches in a distribution network in order to minimize a certain cost such as power loss. It is a mixed integer nonlinear program and hence hard to solve. A popular heuristic search consists of repeated application of branch exchange, where some loads are transferred from one feeder to another feeder while maintaining the radial structure of the network, until no load transfer can further reduce the cost. Optimizing each branch exchange step is itself a mixed integer nonlinear program. In this paper we propose an efficient algorithm for optimizing a branch exchange step. It uses an AC power flow model and is based on the recently developed convex relaxation of optimal power flow. We provide a bound on the gap between the optimal cost and that of our solution. We prove that our algorithm is optimal when the voltage magnitudes are the same at all buses. We illustrate the effectiveness of our algorithm through the simulation of real-world distribution feeders.
    09/2013;
  • Qiuyu Peng, Steven H. Low
    [show abstract] [hide abstract]
    ABSTRACT: The feeder reconfiguration problem chooses the on/off status of the switches in a distribution network in order to minimize a certain cost such as power loss. It is a mixed integer nonlinear program and hence hard to solve. A popular heuristic search consists of repeated application of branch exchange, where some loads are transferred from one feeder to another feeder while maintaining the radial structure of the network, until no load transfer can further reduce the cost. Optimizing each branch exchange step is itself a mixed integer nonlinear program. In this paper we propose an efficient algorithm for optimizing a branch exchange step. It uses an AC power flow model and is based on the recently developed convex relaxation of optimal power flow. We provide a bound on the gap between the optimal cost and that of our solution. We prove that our algorithm is optimal when the voltage magnitudes are the same at all buses. We illustrate the effectiveness of our algorithm through the simulation of real- world distribution feeders.
    09/2013;
  • Qiuyu Peng, Anwar Walid, Steven H. Low
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    ABSTRACT: Multi-path TCP (MP-TCP) has the potential to greatly improve application performance by using multiple paths transparently. We propose a fluid model for a large class of MP-TCP algorithms and identify design criteria that guarantee the existence, uniqueness, and stability of system equilibrium. We clarify how algorithm parameters impact TCP-friendliness, responsiveness, and window oscillation and demonstrate an inevitable tradeoff among these properties. We discuss the implications of these properties on the behavior of existing algorithms and motivate a new design that generalizes existing algorithms and strikes a good balance among TCP-friendliness, responsiveness, and window oscillation. We illustrate our analysis and the behavior of the new algorithm using ns2 simulations.
    08/2013;
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    ABSTRACT: We formulate an optimal load control (OLC) problem in power networks where the objective is to minimize the aggregate cost of tracking an operating point subject to power balance over the network. We prove that the swing dynamics and the branch power flows, coupled with frequency-based load control, serve as a distributed primal-dual algorithm to solve OLC. Even though the system has multiple equilibrium points, we prove that it nonetheless converges to an optimal point. This result implies that the local frequency deviations at each bus convey exactly the right information about the global power imbalance for the loads to make individual decisions that turn out to be globally optimal. It allows a completely decentralized solution without explicit communication among the buses. Simulations show that the proposed OLC mechanism can resynchronize bus frequencies with significantly improved transient performance.
    05/2013;
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    ABSTRACT: We propose a decentralized optimal load control scheme that provides contingency reserve in the presence of sudden generation drop. The scheme takes advantage of flexibility of frequency responsive loads and neighborhood area communication to solve an optimal load control problem that balances load and generation while minimizing end-use disutility of participating in load control. Local frequency measurements enable individual loads to estimate the total mismatch between load and generation. Neighborhood area communication helps mitigate effects of inconsistencies in the local estimates due to frequency measurement noise. Case studies show that the proposed scheme can balance load with generation and restore the frequency within seconds of time after a generation drop, even when the loads use a highly simplified power system model in their algorithms. We also investigate tradeoffs between the amount of communication and the performance of the proposed scheme through simulation-based experiments.
    IEEE Transactions on Power Systems 01/2013; 28(4):3576-3587. · 2.92 Impact Factor
  • [show abstract] [hide abstract]
    ABSTRACT: The electric power infrastructure is transforming into a smarter, increasingly flexible, and more efficient system with the potential to play an integral role in the changing energy landscape. This paper and the accompanying tutorial session at the 2013 American Control Conference provide an overview of this transformation. It then highlights three emerging optimization and controls techniques that can help facilitate the necessary changes to power grid design, operation and management.
    American Control Conference (ACC), 2013; 01/2013
  • M. Farivar, S.H. Low
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    ABSTRACT: 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.
    IEEE Transactions on Power Systems 01/2013; 28(3):2554-2564. · 2.92 Impact Factor
  • S. Low, D. Gayme, U. Topcu
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    ABSTRACT: The optimal power flow (OPF) problem is nonconvex and generally hard to solve, see e.g. [1], [2]. In this tutorial, we will provide an overview of two different solution approaches. The first uses the bus injection model, which is the standard model for power flow analysis and optimization. It focuses on nodal variables such as voltages, current and power injections and does not directly deal with power flows on individual branches. A key advantage is the simple linear relationship I = Y V between the nodal current injections I and the bus voltages V through the admittance matrix Y. Recently, it has been observed that this form of OPF can be reformulated as a nonconvex QCQP (quadratic constrained quadratic program), which leads to a standard convex relaxation through semidefinite programming [3]-[5]. For radial networks, different sufficient conditions have been derived under which the semidefinite relaxation turns out to be exact [6]-[8]. The second solution technique employs the branch flow model, which focuses on currents and powers on the branches rather than the nodal variables. The branch flow model has been historically used primarily for modeling distribution circuits, which tend to be radial. It has therefore received far less attention. A branch flow model has recently been proposed for the analysis and optimization of mesh as well as radial networks. The model leads to a new approach to solving 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, both relaxation steps are always exact, provided there are no upper bounds on loads [9]. For mesh networks, the conic relaxation is always exact and we provide a simple way to determine if a relaxed solution is globally optimal. We describe a simple method to convexify a mesh network using phase shifters so that both relaxation steps are alw- ys exact and OPF for the convexified network can always be solved efficiently for a globally optimal solution. We prove that convexification requires phase shifters only outside a spanning tree of the network graph and their placement depends only on network topology, not on power flows, generation, loads, or operating constraints [10]. The tutorial will describe precisely the bus injection model and the semidefinite relaxation of OPF as well as the branch flow model and its associated relaxations. We prove sufficient conditions for exact relaxations and verify our results on simulations of various IEEE test systems. Finally we explain the equivalence between the bus injection model and branch flow model.
    American Control Conference (ACC), 2013; 01/2013
  • M. Farivar, S.H. Low
    [show abstract] [hide abstract]
    ABSTRACT: 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.
    IEEE Transactions on Power Systems 01/2013; 28(3):2565-2572. · 2.92 Impact Factor
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    IEEE Transactions on Signal Processing 10/2012; 60(10):5384-5395. · 2.81 Impact Factor
  • Source
    Masoud Farivar, Steven H. Low
    [show abstract] [hide abstract]
    ABSTRACT: We propose a branch flow model for the anal- ysis 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.
    04/2012;
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    ABSTRACT: This paper proves that non-convex quadratically constrained quadratic programs can be solved in polynomial time when their underlying graph is acyclic, provided the constraints satisfy a certain technical condition. When this condition is not satisfied, we propose a heuristic to obtain a feasible point. We demonstrate this approach on optimal power flow problems over radial networks.
    03/2012;
  • [show abstract] [hide abstract]
    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 non-decreasing 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.
    Decision and Control (CDC), 2012 IEEE 51st Annual Conference on; 01/2012

Publication Stats

7k Citations
119.21 Total Impact Points

Institutions

  • 2002–2013
    • California Institute of Technology
      • • Division of Engineering and Applied Science
      • • Department of Computing & Mathematical Sciences
      • • Department of Electrical Engineering
      Pasadena, California, United States
  • 2011
    • University of California, Berkeley
      • Department of Electrical Engineering and Computer Sciences
      Berkeley, CA, United States
  • 2009–2010
    • Cornell University
      • Department of Electrical and Computer Engineering
      Ithaca, New York, United States
  • 2005–2009
    • Rensselaer Polytechnic Institute
      • Department of Electrical, Computer, and Systems Engineering
      Troy, NY, United States
  • 2008
    • KTH Royal Institute of Technology
      Tukholma, Stockholm, Sweden
  • 2006–2007
    • Pusan National University
      • Division of Electrical and Electronics Engineering
      Pusan, Busan, South Korea
    • Princeton University
      • Department of Electrical Engineering
      Princeton, NJ, United States
  • 1996–2007
    • University of Melbourne
      • Department of Electrical and Electronic Engineering
      Melbourne, Victoria, Australia
  • 2001–2005
    • University of California, Los Angeles
      • Department of Electrical Engineering
      Los Angeles, CA, United States
  • 2003
    • Los Alamos National Laboratory
      Los Alamos, California, United States
  • 1994–1997
    • AT&T Labs
      Austin, Texas, United States