To accommodate the increasing electric vehicle (EV) penetration in distribution grid, coordinated EV charging has been extensively studied in the literature. However, most of the existing works ...optimistically consider the EV charging rate as a continuous variable and implicitly ignore the capacity limitation in distribution transformers, which both have great impact on the efficiency and stability of practical grid operation. Towards a more realistic setting, this paper formulates the EV coordinated discrete charging problem as two successive binary programs. The first one is designed to achieve a desired aggregate load profile (e.g., valley-filling profile) at the distribution grid level while taking into account the capacity constraints of distribution transformers. Leveraging the properties of separable convex function and total unimodularity, the problem is transformed into an equivalent linear program, which can be solved efficiently and optimally. The second problem aims to minimize the total number of on-off switchings of all the EVs' charging profiles while preserving the optimality of the former problem. We prove the second problem is NP-hard and propose a heuristic algorithm to approximately achieve our target in an iterative manner. Case studies confirm the validity of our proposed scheduling methods and indicate our algorithm's potential for real-time implementations.
This paper investigates two related optimal input selection problems for fixed (non-switched) and switched structured (or structural) systems. More precisely, we consider selecting the minimum cost ...of inputs from a prior set of inputs, and selecting the inputs of the smallest possible cost with a bound on their cardinality, all to ensure system structural controllability. Those problems have attracted much attention recently; unfortunately, they are NP-hard in general. In this paper, it is found that, if the input structure satisfies certain ’regularizations’, which are characterized by the proposed restricted total unimodularity notion, those problems can be solvable in polynomial time via linear programming (LP) relaxations. Particularly, the obtained characterizations depend only on the incidence matrix relating the inputs and the source strongly connected components (SCC) of the system structure, irrespective of how the inputs actuate states within the same SCC. They cover all the currently known polynomially solvable cases (such as the dedicated input case), and contain many new cases unexplored in the past, among which the source-SCC separated input (SSSI) constraint is highlighted. Further, these results are extended to switched systems, and a polynomially solvable condition, namely the joint SSSI constraint, is obtained that does not require each of the subsystems to satisfy the SSSI constraint. We achieve these by first formulating those problems as equivalent integer linear programmings (ILPs), and then proving the total unimodularity of the corresponding constraint matrices. We also study solutions obtained via LP-relaxation and LP-rounding in the general case, resulting in some lower and upper bounds. Several examples are given to illustrate the obtained theoretical results.
UNIMODULARITY UNIFIED GARCÍA, DARÍO; WAGNER, FRANK O.
The Journal of symbolic logic,
09/2017, Letnik:
82, Številka:
3
Journal Article
Recenzirano
Odprti dostop
Unimodularity is localized to a complete stationary type, and its properties are analysed. Some variants of unimodularity for definable and type-definable sets are introduced, and the relationship ...between these different notions is studied. In particular, it is shown that all notions coincide for non-multidimensional theories where the dimensions are associated to strongly minimal types.
We consider a single-server scheduling problem given a fixed sequence of appointment arrivals with random no-shows and service durations. The probability distribution of the uncertain parameters is ...assumed to be ambiguous, and only the support and first moments are known. We formulate a class of distributionally robust (DR) optimization models that incorporate the worst-case expectation/conditional value-at-risk penalty cost of appointment waiting, server idleness, and overtime into the objective or constraints. Our models flexibly adapt to different prior beliefs of no-show uncertainty. We obtain exact mixed-integer nonlinear programming reformulations and derive valid inequalities to strengthen the reformulations that are solved by decomposition algorithms. In particular, we derive convex hulls for special cases of no-show beliefs, yielding polynomial-sized linear programming models for the least and the most conservative supports of no-shows. We test various instances to demonstrate the computational efficacy of our approaches and to compare the results of various DR models given perfect or ambiguous distributional information.
The e-companion is available at
https://doi.org/10.1287/opre.2017.1656
.
A long-standing open question in integer programming is whether integer programs with constraint matrices with bounded subdeterminants are efficiently solvable. An important special case thereof are ...congruency-constrained integer programs Formula: see text with a totally unimodular constraint matrix T. Such problems are shown to be polynomial-time solvable for m = 2, which led to an efficient algorithm for integer programs with bimodular constraint matrices, that is, full-rank matrices whose n × n subdeterminants are bounded by two in absolute value. Whereas these advances heavily rely on existing results on well-known combinatorial problems with parity constraints, new approaches are needed beyond the bimodular case, that is, for m > 2. We make first progress in this direction through several new techniques. In particular, we show how to efficiently decide feasibility of congruency-constrained integer programs with a totally unimodular constraint matrix for m = 3 using a randomized algorithm. Furthermore, for general m, our techniques also allow for identifying flat directions of infeasible problems and deducing bounds on the proximity between solutions of the problem and its relaxation.
Funding: This project received funding from the Swiss National Science Foundation Grants 200021_184622 and P500PT_206742, the European Research Council under the European Union’s Horizon 2020 research and innovation program Grant 817750, and the Deutsche Forschungsgemeinschaft (German Research Foundation) under Germany’s Excellence Strategy–GZ 2047/1 Grant 390685813.
This article investigates several cost-sparsity-induced optimal input selection problems for structured systems. Given an autonomous system and a prescribed set of input links, where each input link ...has a nonnegative cost, the problems include selecting the minimum cost of input links and selecting the input links with the smallest possible cost while bounding their cardinality to achieve system structural controllability. Current studies show that in the dedicated input case, the former problem is polynomially solvable by some graph-theoretic algorithms, while the general nontrivial constrained case is largely unexplored. We show that these problems can be formulated as equivalent integer linear programming (ILP) problems. Subject to the "source strongly connected component grouped input constraint," which contains the dedicated input one as a special case, we demonstrate that the constraint matrices of these ILPs are totally unimodular . This property allows us to solve these ILPs efficiently via their linear programming (LP) relaxations, leading to a unifying algebraic method for these problems with polynomial time complexity. We further show that these problems can be solved in strongly polynomial time, independent of the size of the costs and cardinality bounds. Finally, we provide an example to illustrate the power of the proposed method.
We study the limiting behavior of interacting particle systems indexed by large sparse graphs, which evolve either according to a discrete time Markov chain or a diffusion, in which particles ...interact directly only with their nearest neighbors in the graph. To encode sparsity we work in the framework of local weak convergence of marked (random) graphs. We show that the joint law of the particle system varies continuously with respect to local weak convergence of the underlying graph marked with the initial conditions. In addition, we show that the global empirical measure converges to a nonrandom limit for a large class of graph sequences including sparse Erdős–Rényi graphs and configuration models, whereas the empirical measure of the connected component of a uniformly random vertex converges to a random limit. Along the way, we develop some related results on the time-propagation of ergodicity and empirical field convergence, as well as some general results on local weak convergence of Gibbs measures in the uniqueness regime which appear to be new. The results obtained here are also useful for obtaining autonomous descriptions of marginal dynamics of interacting diffusions and Markov chains on sparse graphs. While limits of interacting particle systems on dense graphs have been extensively studied, there are relatively few works that have studied the sparse regime in generality.
Random interlacement is a factor of i.i.d Borbényi, Márton; Ráth, Balázs; Rokob, Sándor
Electronic journal of probability,
1/2023, Letnik:
28, Številka:
none
Journal Article