arXiv subjects

Michael Chertkov

Publications and source records attributed to Michael Chertkov.

At least 109 records · Page 6Linked to original sources

arXiv2016

Tractable Structure Learning in Radial Physical Flow Networks

Physical Flow Networks are different infrastructure networks that allow the flow of physical commodities through edges between its constituent nodes. These include power grid, natural gas transmission network, water pipelines etc. In such networks, the flow on each edge is characterized by a function of the nodal potentials on either side of the edge. Further the net flow in and out of each node is conserved. Learning the structure and state of physical networks is necessary for optimal control as well as to quantify its privacy needs. We consider radial flow networks and study the problem of learning the operational network from a loopy graph of candidate edges using statistics of nodal potentials. Based on the monotonic properties of the flow functions, the key result in this paper shows that if variance of the difference of nodal potentials is used to weight candidate edges, the operational edges form the minimum spanning tree in the loopy graph. Under realistic conditions on the statistics of nodal injection (consumption or production), we provide a greedy structure learning algorithm with quasilinear computational complexity in the number of candidate edges in the network. Our learning framework is very general due to two significant attributes. First it is independent of the specific marginal distributions of nodal potentials and only uses order properties in their second moments. Second, the learning algorithm is agnostic to exact flow functions that relate edge flows to corresponding potential differences and is applicable for a broad class of networks with monotonic flow functions. We demonstrate the efficacy of our work through realistic simulations on diverse physical flow networks and discuss possible extensions of our work to other regimes.

math.OC↗

arXiv2016

Monotone Order Properties for Control of Nonlinear Parabolic PDE on Graphs

We derive conditions for the propagation of monotone ordering properties for a class of nonlinear parabolic partial differential equation (PDE) systems on metric graphs. For such systems, PDE equations with a general nonlinear dissipation term define evolution on each edge, and balance laws create Kirchhoff-Neumann boundary conditions at the vertices. Initial conditions, as well as time-varying parameters in the coupling conditions at vertices, provide an initial value problem (IVP). We first prove that ordering properties of the solution to the IVP are preserved when the initial conditions and time-varying coupling law parameters at vertices are appropriately ordered. In addition, we prove that when monotone ordering is not preserved, the first crossing of solutions occurs at a graph vertex. We consider the implications for robust optimal control formulations and real-time monitoring involving uncertain dynamic flows on networks, and discuss application to subsonic compressible fluid flow with energy dissipation on physical networks.

math.OC↗

arXiv2016

Graphical Models for Optimal Power Flow

Optimal power flow (OPF) is the central optimization problem in electric power grids. Although solved routinely in the course of power grid operations, it is known to be strongly NP-hard in general, and weakly NP-hard over tree networks. In this paper, we formulate the optimal power flow problem over tree networks as an inference problem over a tree-structured graphical model where the nodal variables are low-dimensional vectors. We adapt the standard dynamic programming algorithm for inference over a tree-structured graphical model to the OPF problem. Combining this with an interval discretization of the nodal variables, we develop an approximation algorithm for the OPF problem. Further, we use techniques from constraint programming (CP) to perform interval computations and adaptive bound propagation to obtain practically efficient algorithms. Compared to previous algorithms that solve OPF with optimality guarantees using convex relaxations, our approach is able to work for arbitrary distribution networks and handle mixed-integer optimization problems. Further, it can be implemented in a distributed message-passing fashion that is scalable and is suitable for "smart grid" applications like control of distributed energy resources. We evaluate our technique numerically on several benchmark networks and show that practical OPF problems can be solved effectively using this approach.

eess.SY↗

arXiv2016

Monotonicity of Actuated Flows on Dissipative Transport Networks

We derive a monotonicity property for general, transient flows of a commodity transferred throughout a network, where the flow is characterized by density and mass flux dynamics on the edges with density continuity and mass balance conditions at the nodes. The dynamics on each edge are represented by a general system of partial differential equations that approximates subsonic compressible fluid flow with energy dissipation. The transferred commodity may be injected or withdrawn at any of the nodes, and is propelled throughout the network by nodally located compressors. These compressors are controllable actuators that provide a means to manipulate flows through the network, which we therefore consider as a control system. A canonical problem requires compressor control protocols to be chosen such that time-varying nodal commodity withdrawal profiles are delivered and the density remains within strict limits while an economic or operational cost objective is optimized. In this manuscript, we consider the situation where each nodal commodity withdrawal profile is uncertain, but is bounded within known maximum and minimum time-dependent limits. We introduce the monotone parameterized control system property, and prove that general dynamic dissipative network flows possess this characteristic under certain conditions. This property facilitates very efficient formulation of optimal control problems for such systems in which the solutions must be robust with respect to commodity withdrawal uncertainty. We discuss several applications in which such control problems arise and where monotonicity enables simplified characterization of system behavior.

math.OC↗

arXiv2016

Learning Topology of the Power Distribution Grid with and without Missing Data

Distribution grids refer to the part of the power grid that delivers electricity from substations to the loads. Structurally a distribution grid is operated in one of several radial/tree-like topologies that are derived from an original loopy grid graph by opening switches on some lines. Due to limited presence of real-time switch monitoring devices, the operating structure needs to be estimated indirectly. This paper presents a new learning algorithm that uses only nodal voltage measurements to determine the operational radial structure. The algorithm is based on the key result stating that the correct operating structure is the optimal solution of the minimum-weight spanning tree problem over the original loopy graph where weights on all permissible edges/lines (open or closed) is the variance of nodal voltage difference at the edge ends. Compared to existing work, this spanning tree based approach has significantly lower complexity as it does not require information on line parameters. Further, a modified learning algorithm is developed for cases when the input voltage measurements are limited to only a subset of the total grid nodes. Performance of the algorithms (with and without missing data) is demonstrated by experiments on test cases.

math.OC↗

arXiv2016

Estimating Distribution Grid Topologies: A Graphical Learning based Approach

Distribution grids represent the final tier in electric networks consisting of medium and low voltage lines that connect the distribution substations to the end-users. Traditionally, distribution networks have been operated in a radial topology that may be changed from time to time. Due to absence of a significant number of real-time line monitoring devices in the distribution grid, estimation of the topology is a problem critical for its observability and control. This paper develops a novel graphical learning based approach to estimate the radial operational grid structure using voltage measurements collected from the grid loads. The learning algorithm is based on conditional independence tests for continuous variables over chordal graphs and has wide applicability. It is proven that the scheme can be used for several power flow laws (DC or AC approximations) and more importantly is independent of the specific probability distribution controlling individual bus power usage. The complexity of the algorithm is discussed and its performance is demonstrated by simulations on distribution test cases.

math.OC↗

arXiv2016

Linear PDEs and eigenvalue problems corresponding to ergodic stochastic optimization problems on compact manifolds

We consider long term average or `ergodic' optimal control poblems with a special structure: Control is exerted in all directions and the control costs are proportional to the square of the norm of the control field with respect to the metric induced by the noise. The long term stochastic dynamics on the manifold will be completely characterized by the long term density $ρ$ and the long term current density $J$. As such, control problems may be reformulated as variational problems over $ρ$ and $J$. We discuss several optimization problems: the problem in which both $ρ$ and $J$ are varied freely, the problem in which $ρ$ is fixed and the one in which $J$ is fixed. These problems lead to different kinds of operator problems: linear PDEs in the first two cases and a nonlinear PDE in the latter case. These results are obtained through through variational principle using infinite dimensional Lagrange multipliers. In the case where the initial dynamics are reversible we obtain the result that the optimally controlled diffusion is also symmetrizable. The particular case of constraining the dynamics to be reversible of the optimally controlled process leads to a linear eigenvalue problem for the square root of the density process.

math.OC↗

arXiv2016

Chance Constrained Optimal Power Flow with Curtailment and Reserves from Wind Power Plants

Over the past years, the share of electricity production from wind power plants has increased to significant levels in several power systems across Europe and the United States. In order to cope with the fluctuating and partially unpredictable nature of renewable energy sources, transmission system operators (TSOs) have responded by increasing their reserve capacity requirements and by requiring wind power plants to be capable of providing reserves or following active power set-point signals. This paper addresses the issue of efficiently incorporating these new types of wind power control in the day-ahead operational planning. We review the technical requirements the wind power plants must fulfill, and propose a mathematical framework for modeling wind power control. The framework is based on an optimal power flow formulation with weighted chance constraints, which accounts for the uncertainty of wind power forecasts and allows us to limit the risk of constraint violations. In a case study based on the IEEE 118 bus system, we use the developed method to assess the effectiveness of different types of wind power control in terms of operational cost, system security and wind power curtailment.

math.OC↗

arXiv2015

A differential analysis of the power flow equations

The AC power flow equations are fundamental in all aspects of power systems planning and operations. They are routinely solved using Newton-Raphson like methods. However, there is little theoretical understanding of when these algorithms are guaranteed to find a solution of the power flow equations or how long they may take to converge. Further, it is known that in general these equations have multiple solutions and can exhibit chaotic behavior. In this paper, we show that the power flow equations can be solved efficiently provided that the solution lies in a certain set. We introduce a family of convex domains, characterized by Linear Matrix Inequalities, in the space of voltages such that there is at most one power flow solution in each of these domains. Further, if a solution exists in one of these domains, it can be found efficiently, and if one does not exist, a certificate of non-existence can also be obtained efficiently. The approach is based on the theory of monotone operators and related algorithms for solving variational inequalities involving monotone operators. We validate our approach on IEEE test networks and show that practical power flow solutions lie within an appropriately chosen convex domain.

eess.SY↗

arXiv2015

Minimum Weight Perfect Matching via Blossom Belief Propagation

Max-product Belief Propagation (BP) is a popular message-passing algorithm for computing a Maximum-A-Posteriori (MAP) assignment over a distribution represented by a Graphical Model (GM). It has been shown that BP can solve a number of combinatorial optimization problems including minimum weight matching, shortest path, network flow and vertex cover under the following common assumption: the respective Linear Programming (LP) relaxation is tight, i.e., no integrality gap is present. However, when LP shows an integrality gap, no model has been known which can be solved systematically via sequential applications of BP. In this paper, we develop the first such algorithm, coined Blossom-BP, for solving the minimum weight matching problem over arbitrary graphs. Each step of the sequential algorithm requires applying BP over a modified graph constructed by contractions and expansions of blossoms, i.e., odd sets of vertices. Our scheme guarantees termination in O(n^2) of BP runs, where n is the number of vertices in the original graph. In essence, the Blossom-BP offers a distributed version of the celebrated Edmonds' Blossom algorithm by jumping at once over many sub-steps with a single BP. Moreover, our result provides an interpretation of the Edmonds' algorithm as a sequence of LPs.

cs.DS↗

arXiv2015

Optimal Control of Transient Flow in Natural Gas Networks

We outline a new control system model for the distributed dynamics of compressible gas flow through large-scale pipeline networks with time-varying injections, withdrawals, and control actions of compressors and regulators. The gas dynamics PDE equations over the pipelines, together with boundary conditions at junctions, are reduced using lumped elements to a sparse nonlinear ODE system expressed in vector-matrix form using graph theoretic notation. This system, which we call the reduced network flow (RNF) model, is a consistent discretization of the PDE equations for gas flow. The RNF forms the dynamic constraints for optimal control problems for pipeline systems with known time-varying withdrawals and injections and gas pressure limits throughout the network. The objectives include economic transient compression (ETC) and minimum load shedding (MLS), which involve minimizing compression costs or, if that is infeasible, minimizing the unfulfilled deliveries, respectively. These continuous functional optimization problems are approximated using the Legendre-Gauss-Lobatto (LGL) pseudospectral collocation scheme to yield a family of nonlinear programs, whose solutions approach the optima with finer discretization. Simulation and optimization of time-varying scenarios on an example natural gas transmission network demonstrate the gains in security and efficiency over methods that assume steady-state behavior.

math.OC↗

arXiv2015

Uncertainty Sets For Wind Power Generation

As penetration of wind power generation increases, system operators must account for its stochastic nature in a reliable and cost-efficient manner. These conflicting objectives can be traded-off by accounting for the variability and uncertainty of wind power generation. This letter presents a new methodology to estimate uncertainty sets for parameters of probability distributions that capture wind generation uncertainty and variability.

math.OC↗

arXiv2015

Solving the power flow equations: a monotone operator approach

The AC power flow equations underlie all operational aspects of power systems. They are solved routinely in operational practice using the Newton-Raphson method and its variants. These methods work well given a good initial "guess" for the solution, which is always available in normal system operations. However, with the increase in levels of intermittent generation, the assumption of a good initial guess always being available is no longer valid. In this paper, we solve this problem using the theory of monotone operators. We show that it is possible to compute (using an offline optimization) a "monotonicity domain" in the space of voltage phasors. Given this domain, there is a simple efficient algorithm that will either find a solution in the domain, or provably certify that no solutions exist in it. We validate the approach on several IEEE test cases and demonstrate that the offline optimization can be performed tractably and the computed "monotonicity domain" includes all practically relevant power flow solutions.

eess.SY↗

arXiv2015

Natural Gas Flow Solutions with Guarantees: A Monotone Operator Theory Approach

We consider balanced flows in a natural gas transmission network and discuss computationally hard problems such as establishing if solution of the underlying nonlinear gas flow equations exists, if it is unique, and finding the solution. Particular topologies, e.g. trees, are known to be easy to solve based on a variational description of the gas flow equations, but these approaches do not generalize. In this paper, we show that the gas flow problem can be solved efficiently using the tools of monotone operator theory, provided that we look for solution within certain monotonicity domains. We characterize a family of monotonicity domains, described in terms of Linear Matrix Inequalities (LMI) in the state variables, each containing at most one solution. We also develop an efficient algorithm to choose a particular monotonicity domain, for which the LMI based condition simplifies to a bound on the flows. Performance of the technique is illustrated on exemplary gas networks.

eess.SY↗

arXiv2015

Maximum Throughput Problem in Dissipative Flow Networks with Application to Natural Gas Systems

We consider a dissipative flow network that obeys the standard linear nodal flow conservation, and where flows on edges are driven by potential difference between adjacent nodes. We show that in the case when the flow is a monotonically increasing function of the potential difference, solution of the network flow equations is unique and can be equivalently recast as the solution of a strictly convex optimization problem. We also analyze the maximum throughput problem on such networks seeking to maximize the amount of flow that can be delivered to the loads while satisfying bounds on the node potentials. When the dissipation function is differentiable we develop a representation of the maximum throughput problem in the form of a twice differentiable biconvex optimization problem exploiting the variational representation of the network flow equations. In the process we prove a special case of a certain monotonicity property of dissipative flow networks. When the dissipation function follows a power law with exponent greater than one, we suggest a mixed integer convex relaxation of the maximum throughput problem. Finally, we illustrate application of these general results to balanced, i.e. steady, natural gas networks also validating the theory results through simulations on a test case.

math.OC↗

arXiv2015

Monotonicity of Dissipative Flow Networks Renders Robust Maximum Profit Problem Tractable: General Analysis and Application to Natural Gas Flows

We consider general, steady, balanced flows of a commodity over a network where an instance of the network flow is characterized by edge flows and nodal potentials. Edge flows in and out of a node are assumed to be conserved, thus representing standard network flow relations. The remaining freedom in the flow distribution over the network is constrained by potentials so that the difference of potentials at the head and the tail of an edge is expressed as a nonlinear function of the edge flow. We consider networks with nodes divided into three categories: sources that inject flows into the network for a certain cost, terminals which buy the flow at a fixed price and "internal" customers each withdrawing an uncertain amount of flow, which has a priority and thus it is not priced. Our aim is to operate the network such that the profit, i.e. amount of flow sold to terminals minus cost of injection, is maximized, while maintaining the potentials within prescribed bounds. We also require that the operating point is robust with respect to the uncertainty of customers' withdrawals. In this setting we prove that potentials are monotonic functions of the withdrawals. This observation enables us to replace in the maximum profit optimization infinitely many nodal constraints, each representing a particular value of withdrawal uncertainty, by only two constraints representing the cases where all nodes with uncertainty consume their minimum and maximum amounts respectively. We illustrate this general result on example of the natural gas transmission network. In this enabling example gas withdrawals by consumers are assumed uncertain, the potentials are gas pressures squared, the potential drop functions are bilinear in the flow and its intensity with an added tunable factor representing compression.

math.OC↗

arXiv2015

Optimal Power Flow with Weighted Chance Constraints and General Policies for Generation Control

Due to the increasing amount of electricity generated from renewable sources, uncertainty in power system operation will grow. This has implications for tools such as Optimal Power Flow (OPF), an optimization problem widely used in power system operations and planning, which should be adjusted to account for this uncertainty. One way to handle the uncertainty is to formulate a Chance Constrained OPF (CC-OPF) which limits the probability of constraint violation to a predefined value. However, existing CC-OPF formulations and solutions are not immune to drawbacks. On one hand, they only consider affine policies for generation control, which are not always realistic and may be sub-optimal. On the other hand, the standard CC-OPF formulations do not distinguish between large and small violations, although those might carry significantly different risk. In this paper, we introduce the Weighted CC-OPF (WCC-OPF) that can handle general control policies while preserving convexity and allowing for efficient computation. The weighted chance constraints account for the size of violations through a weighting function, which assigns a higher risk to a higher overloads. We prove that the problem remains convex for any convex weighting function, and for very general generation control policies. In a case study, we compare the performance of the new WCC-OPF and the standard CC-OPF and demonstrate that WCC-OPF effectively reduces the number of severe overloads. Furthermore, we compare an affine generation control policy with a more general policy, and show that the additional flexibility allow for a lower cost while maintaining the same level of risk.

math.OC↗

arXiv2015

Structure Learning and Statistical Estimation in Distribution Networks - Part I

Traditionally power distribution networks are either not observable or only partially observable. This complicates development and implementation of new smart grid technologies, such as those related to demand response, outage detection and management, and improved load-monitoring. In this two part paper, inspired by proliferation of metering technology, we discuss estimation problems in structurally loopy but operationally radial distribution grids from measurements, e.g. voltage data, which are either already available or can be made available with a relatively minor investment. In Part I, the objective is to learn the operational layout of the grid. Part II of this paper presents algorithms that estimate load statistics or line parameters in addition to learning the grid structure. Further, Part II discusses the problem of structure estimation for systems with incomplete measurement sets. Our newly suggested algorithms apply to a wide range of realistic scenarios. The algorithms are also computationally efficient -- polynomial in time -- which is proven theoretically and illustrated computationally on a number of test cases. The technique developed can be applied to detect line failures in real time as well as to understand the scope of possible adversarial attacks on the grid.

math.OC↗