Matt Kraning

Bio: Matt Kraning is an academic researcher from Stanford University. The author has contributed to research in topics: Optimization problem & Constraint satisfaction problem. The author has an hindex of 8, co-authored 8 publications receiving 573 citations.

Dynamic Network Energy Management Via Proximal Message Passing

Abstract: We consider a network of devices, such as generators, fixed loads, deferrable loads, and storage devices, each with its owndynamic constraints and objective, connected by AC and DC lines. The problem is to minimize the total network objective subject tothe device and line constraints, over a given time horizon. This is a large optimization problem, with variables for consumptionor generation for each device, power flow for each line, and voltage phase angles at AC buses, in each time period. In this paperwe develop a decentralized method for solving this problem called proximal message passing. 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 itsown objective function, augmented by a term determined by the messages it has received. We show that this message passing methodconverges to a solution when the device objective and constraints are convex. The method is completely decentralized, and needs noglobal coordination other than synchronizing iterations; the problems to be solved by each device can typically be solved extremelyefficiently and in parallel. The method is fast enough that even a serial implementation can solve substantial problems inreasonable time frames. We report results for several numerical experiments, demonstrating the method's speed and scaling,including the solution of a problem instance with over 10 million variables in under 50 minutes for a serial implementation;with decentralized computing, the solve time would be less than one second.

Message Passing for Dynamic Network Energy Management

Abstract: 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 in reasonable time frames. We report results for several numerical experiments, demonstrating the method’s speed and scaling, including the solution of a problem instance with over 30 million variables in 52 minutes for a serial implementation; with decentralized computing, the solve time would be less than one second.

Security Constrained Optimal Power Flow via proximal message passing

Abstract: In this paper, we propose a distributed algorithm to solve the Security Constrained Optimal Power Flow (SC-OPF) Problem. We consider a network of devices, each with its own dynamic constraints and objective, subject to reliability constraints across multiple scenarios. Each scenario corresponds to the failure or degradation of a set of devices and has an associated probability of occurrence. The network objective is to minimize the cost of operation of all devices, over a given time horizon, across all scenarios subject to the constraints of transmission limit, upper and lower generating limits, generation-load balance etc. This is a large optimization problem, with variables for consumption and generation for each device, in each scenario. In this paper, we extend the proximal message passing framework to handle reliability constraints across scenarios. The resulting algorithm is extremely scalable with respect to both network size and the number of scenarios.

The set of solutions of random XORSAT formulae

Abstract: The XOR-satisfiability (XORSAT) problem requires finding an assignment of $n$ Boolean variables that satisfy $m$ exclusive OR (XOR) clauses, whereby each clause constrains a subset of the variables. We consider random XORSAT instances, drawn uniformly at random from the ensemble of formulae containing $n$ variables and $m$ clauses of size $k$. This model presents several structural similarities to other ensembles of constraint satisfaction problems, such as $k$-satisfiability ($k$-SAT), hypergraph bicoloring and graph coloring. For many of these ensembles, as the number of constraints per variable grows, the set of solutions shatters into an exponential number of well-separated components. This phenomenon appears to be related to the difficulty of solving random instances of such problems. We prove a complete characterization of this clustering phase transition for random $k$-XORSAT. In particular, we prove that the clustering threshold is sharp and determine its exact location. We prove that the set of solutions has large conductance below this threshold and that each of the clusters has large conductance above the same threshold. Our proof constructs a very sparse basis for the set of solutions (or the subset within a cluster). This construction is intimately tied to the construction of specific subgraphs of the hypergraph associated with an instance of $k$-XORSAT. In order to study such subgraphs, we establish novel local weak convergence results for them.

Random graphs

Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Proximal Algorithms

Abstract: This monograph is about a class of optimization algorithms called proximal algorithms. Much like Newton's method is a standard tool for solving unconstrained smooth optimization problems of modest size, proximal algorithms can be viewed as an analogous tool for nonsmooth, constrained, large-scale, or distributed versions of these problems. They are very generally applicable, but are especially well-suited to problems of substantial recent interest involving large or high-dimensional datasets. Proximal methods sit at a higher level of abstraction than classical algorithms like Newton's method: the base operation is evaluating the proximal operator of a function, which itself involves solving a small convex optimization problem. These subproblems, which generalize the problem of projecting a point onto a convex set, often admit closed-form solutions or can be solved very quickly with standard or simple specialized methods. Here, we discuss the many different interpretations of proximal operators and algorithms, describe their connections to many other topics in optimization and applied mathematics, survey some popular algorithms, and provide a large number of examples of proximal operators that commonly arise in practice.

A Survey on Demand Response Programs in Smart Grids: Pricing Methods and Optimization Algorithms

Abstract: The smart grid concept continues to evolve and various methods have been developed to enhance the energy efficiency of the electricity infrastructure. Demand Response (DR) is considered as the most cost-effective and reliable solution for the smoothing of the demand curve, when the system is under stress. DR refers to a procedure that is applied to motivate changes in the customers' power consumption habits, in response to incentives regarding the electricity prices. In this paper, we provide a comprehensive review of various DR schemes and programs, based on the motivations offered to the consumers to participate in the program. We classify the proposed DR schemes according to their control mechanism, to the motivations offered to reduce the power consumption and to the DR decision variable. We also present various optimization models for the optimal control of the DR strategies that have been proposed so far. These models are also categorized, based on the target of the optimization procedure. The key aspects that should be considered in the optimization problem are the system's constraints and the computational complexity of the applied optimization algorithm.

A Survey of Distributed Optimization and Control Algorithms for Electric Power Systems

Abstract: Historically, centrally computed algorithms have been the primary means of power system optimization and control. With increasing penetrations of distributed energy resources requiring optimization and control of power systems with many controllable devices, distributed algorithms have been the subject of significant research interest. This paper surveys the literature of distributed algorithms with applications to optimization and control of power systems. In particular, this paper reviews distributed algorithms for offline solution of optimal power flow (OPF) problems as well as online algorithms for real-time solution of OPF, optimal frequency control, optimal voltage control, and optimal wide-area control problems.