arXiv subjects

Marc Vuffray

Publications and source records attributed to Marc Vuffray.

54 records · Page 3Linked to original sources

arXiv2017

Fast and Robust Determination of Power System Emergency Control Actions

This paper outlines an optimization framework for choosing fast and reliable control actions in a transmission grid emergency situation. We consider contractual load shedding and generation re-dispatch as exemplary emergency actions. To achieve computational efficiency and scalability, this novel formulation of the robust corrective action optimization is stated in terms of voltages and currents, as opposed to standard formulation in terms of power flows and voltages. The current-voltage formulation is natural for expressing voltage and thermal transmission line constraints, and its major advantage is in the linearity of the power flow equations. The complexity of the current-voltage formulation, which is mainly related to the transformation of voltages and currents into active and reactive power injections, can be largely avoided by stating the cost function directly in terms of currents and using linearized or robust estimates for load and generation constraints. The paper considers five different optimization problem formulations, including the full current-voltage formulation, as well as two convex and two linear approximation to the problem. In a case study on an illustrative case study for the IEEE RTS96 system, we demonstrate pros and cons of the different formulations based on simulations for both normal and contingency operation.

math.OC↗

arXiv2017

Graphical Models and Belief Propagation-hierarchy for Optimal Physics-Constrained Network Flows

In this manuscript we review new ideas and first results on application of the Graphical Models approach, originated from Statistical Physics, Information Theory, Computer Science and Machine Learning, to optimization problems of network flow type with additional constraints related to the physics of the flow. We illustrate the general concepts on a number of enabling examples from power system and natural gas transmission (continental scale) and distribution (district scale) systems.

eess.SY↗

arXiv2016

Interaction Screening: Efficient and Sample-Optimal Learning of Ising Models

We consider the problem of learning the underlying graph of an unknown Ising model on p spins from a collection of i.i.d. samples generated from the model. We suggest a new estimator that is computationally efficient and requires a number of samples that is near-optimal with respect to previously established information-theoretic lower-bound. Our statistical estimator has a physical interpretation in terms of "interaction screening". The estimator is consistent and is efficiently implemented using convex optimization. We prove that with appropriate regularization, the estimator recovers the underlying graph using a number of samples that is logarithmic in the system size p and exponential in the maximum coupling-intensity and maximum node-degree.

cs.LG↗

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↗

arXiv2015

Concentration to Zero Bit-Error Probability for Regular LDPC Codes on the Binary Symmetric Channel: Proof by Loop Calculus

In this paper we consider regular low-density parity-check codes over a binary-symmetric channel in the decoding regime. We prove that up to a certain noise threshold the bit-error probability of the bit-sampling decoder converges in mean to zero over the code ensemble and the channel realizations. To arrive at this result we show that the bit-error probability of the sampling decoder is equal to the derivative of a Bethe free entropy. The method that we developed is new and is based on convexity of the free entropy and loop calculus. Convexity is needed to exchange limit and derivative and the loop series enables us to express the difference between the bit-error probability and the Bethe free entropy. We control the loop series using combinatorial techniques and a first moment method. We stress that our method is versatile and we believe that it can be generalized for LDPC codes with general degree distributions and for asymmetric channels.

cs.IT↗

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

Approaching the Rate-Distortion Limit with Spatial Coupling, Belief propagation and Decimation

We investigate an encoding scheme for lossy compression of a binary symmetric source based on simple spatially coupled Low-Density Generator-Matrix codes. The degree of the check nodes is regular and the one of code-bits is Poisson distributed with an average depending on the compression rate. The performance of a low complexity Belief Propagation Guided Decimation algorithm is excellent. The algorithmic rate-distortion curve approaches the optimal curve of the ensemble as the width of the coupling window grows. Moreover, as the check degree grows both curves approach the ultimate Shannon rate-distortion limit. The Belief Propagation Guided Decimation encoder is based on the posterior measure of a binary symmetric test-channel. This measure can be interpreted as a random Gibbs measure at a "temperature" directly related to the "noise level of the test-channel". We investigate the links between the algorithmic performance of the Belief Propagation Guided Decimation encoder and the phase diagram of this Gibbs measure. The phase diagram is investigated thanks to the cavity method of spin glass theory which predicts a number of phase transition thresholds. In particular the dynamical and condensation "phase transition temperatures" (equivalently test-channel noise thresholds) are computed. We observe that: (i) the dynamical temperature of the spatially coupled construction saturates towards the condensation temperature; (ii) for large degrees the condensation temperature approaches the temperature (i.e. noise level) related to the information theoretic Shannon test-channel noise parameter of rate-distortion theory. This provides heuristic insight into the excellent performance of the Belief Propagation Guided Decimation algorithm. The paper contains an introduction to the cavity method.

cs.IT↗

arXiv2015

The Bethe Free Energy Allows to Compute the Conditional Entropy of Graphical Code Instances. A Proof from the Polymer Expansion

The main objective of this paper is to explore the precise relationship between the Bethe free energy (or entropy) and the Shannon conditional entropy of graphical error correcting codes. The main result shows that the Bethe free energy associated with a low-density parity-check code used over a binary symmetric channel in a large noise regime is, with high probability, asymptotically exact as the block length grows. To arrive at this result we develop new techniques for rather general graphical models based on the loop sum as a starting point and the polymer expansion from statistical mechanics. The true free energy is computed as a series expansion containing the Bethe free energy as its zero-th order term plus a series of corrections. It is easily seen that convergence criteria for such expansions are satisfied for general high-temperature models. We apply these general results to ensembles of low-density generator-matrix and parity-check codes. While the application to generator-matrix codes follows standard "high temperature" methods, the case of parity-check codes requires non-trivial new ideas because the hard constraints correspond to a zero-temperature regime. Nevertheless one can combine the polymer expansion with expander and counting arguments to show that the difference between the true and Bethe free energies vanishes with high probability in the large block

cs.IT↗

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↗

arXiv2014

The Inviscid, Compressible and Rotational, 2D Isotropic Burgers and Pressureless Euler-Coriolis Fluids; Solvable models with illustrations

The coupling between dilatation and vorticity, two coexisting and fundamental processes in fluid dynamics is investigated here, in the simplest cases of inviscid 2D isotropic Burgers and pressureless Euler-Coriolis fluids respectively modeled by single vortices confined in compressible, local, inertial and global, rotating, environments. The field equations are established, inductively, starting from the equations of the characteristics solved with an initial Helmholtz decomposition of the velocity fields namely a vorticity free and a divergence free part and, deductively, by means of a canonical Hamiltonian Clebsch like formalism, implying two pairs of conjugate variables. Two vector valued fields are constants of the motion: the velocity field in the Burgers case and the momentum field per unit mass in the Euler-Coriolis one. Taking advantage of this property, a class of solutions for the mass densities of the fluids is given by the Jacobian of their sum with respect to the actual coordinates. Implementation of the isotropy hypothesis results in the cancellation of the dilatation-rotational cross terms in the Jacobian. A simple expression is obtained for all the radially symmetric Jacobians occurring in the theory. Representative examples of regular and singular solutions are shown and the competition between dilatation and vorticity is illustrated. Inspired by thermodynamical, mean field theoretical analogies, a genuine variational formula is proposed which yields unique measure solutions for the radially symmetric fluid densities investigated. We stress that this variational formula, unlike the Hopf-Lax formula, enables us to treat systems which are both compressible and rotational. Moreover in the one-dimensional case, we show for an interesting application that both variational formulas are equivalent.

physics.flu-dyn↗

arXiv2012

Lossy Source Coding via Spatially Coupled LDGM Ensembles

We study a new encoding scheme for lossy source compression based on spatially coupled low-density generator-matrix codes. We develop a belief-propagation guided-decimation algorithm, and show that this algorithm allows to approach the optimal distortion of spatially coupled ensembles. Moreover, using the survey propagation formalism, we also observe that the optimal distortions of the spatially coupled and individual code ensembles are the same. Since regular low-density generator-matrix codes are known to achieve the Shannon rate-distortion bound under optimal encoding as the degrees grow, our results suggest that spatial coupling can be used to reach the rate-distortion bound, under a {\it low complexity} belief-propagation guided-decimation algorithm. This problem is analogous to the MAX-XORSAT problem in computer science.

cs.IT↗

arXiv2012

Beyond the Bethe Free Energy of LDPC Codes via Polymer Expansions

The loop series provides a formal way to write down corrections to the Bethe entropy (and/or free energy) of graphical models. We provide methods to rigorously control such expansions for low-density parity-check codes used over a highly noisy binary symmetric channel. We prove that in the asymptotic limit of large size, with high probability, the Bethe expression gives an exact formula for the entropy (per bit) of the input word conditioned on the output of the channel. Our methods also apply to more general models.

cs.IT↗

arXiv2012

Polymer Expansions for Cycle LDPC Codes

We prove that the Bethe expression for the conditional input-output entropy of cycle LDPC codes on binary symmetric channels above the MAP threshold is exact in the large block length limit. The analysis relies on methods from statistical physics. The finite size corrections to the Bethe expression are expressed through a polymer expansion which is controlled thanks to expander and counting arguments.

cs.IT↗

arXiv2008

Bounds states of the Schrödinger-Newton model in low dimensions

We prove the existence of quasi-stationary symmetric solutions with exactly n>=0 zeros and uniqueness for n=0 for the Schrödinger-Newton model in one dimension and in two dimensions along with an angular momentum m>=0. Our result is based on an analysis of the corresponding system of second-order differential equations.

math-ph↗

arXiv2008

Stationary solutions of the Schrödinger-Newton model - An ODE approach

We prove the existence and uniqueness of stationary spherically symmetric positive solutions for the Schrödinger-Newton model in any space dimension $d$. Our result is based on an analysis of the corresponding system of second order differential equations. It turns out that $d=6$ is critical for the existence of finite energy solutions and the equations for positive spherically symmetric solutions reduce to a Lane-Emden equation for all $d\geq 6$. Our result implies in particular the existence of stationary solutions for two-dimensional self-gravitating particles and closes the gap between the variational proofs in $d=1$ and $d=3$.

math-ph↗