Predictability and Fairness in Load Aggregation with Deadband

Virtual power plants and load aggregation are becoming increasingly common. There, one regulates the aggregate power output of an ensemble of distributed energy resources (DERs). Marecek et al. [Automatica, Volume 147, January 2023, 110743, arXiv:2110.03001] recently suggested that long-term averages of prices or incentives offered should exist and be independent of the initial states of the operators of the DER, the aggregator, and the power grid. This can be seen as predictability, which underlies fairness. Unfortunately, the existence of such averages cannot be guaranteed with many traditional regulators, including the proportional-integral (PI) regulator with or without deadband. Here, we consider the effects of losses in the alternating current model and the deadband in the controller. This yields a non-linear dynamical system (due to the non-linear losses) exhibiting discontinuities (due to the deadband). We show that Filippov invariant measures enable reasoning about predictability and fairness while considering non-linearity of the alternating-current model and deadband.Comment: arXiv admin note: substantial text overlap with arXiv:2110.0300

Dynamics of Social Networks: Multi-agent Information Fusion, Anticipatory Decision Making and Polling

This paper surveys mathematical models, structural results and algorithms in controlled sensing with social learning in social networks. Part 1, namely Bayesian Social Learning with Controlled Sensing addresses the following questions: How does risk averse behavior in social learning affect quickest change detection? How can information fusion be priced? How is the convergence rate of state estimation affected by social learning? The aim is to develop and extend structural results in stochastic control and Bayesian estimation to answer these questions. Such structural results yield fundamental bounds on the optimal performance, give insight into what parameters affect the optimal policies, and yield computationally efficient algorithms. Part 2, namely, Multi-agent Information Fusion with Behavioral Economics Constraints generalizes Part 1. The agents exhibit sophisticated decision making in a behavioral economics sense; namely the agents make anticipatory decisions (thus the decision strategies are time inconsistent and interpreted as subgame Bayesian Nash equilibria). Part 3, namely {\em Interactive Sensing in Large Networks}, addresses the following questions: How to track the degree distribution of an infinite random graph with dynamics (via a stochastic approximation on a Hilbert space)? How can the infected degree distribution of a Markov modulated power law network and its mean field dynamics be tracked via Bayesian filtering given incomplete information obtained by sampling the network? We also briefly discuss how the glass ceiling effect emerges in social networks. Part 4, namely \emph{Efficient Network Polling} deals with polling in large scale social networks. In such networks, only a fraction of nodes can be polled to determine their decisions. Which nodes should be polled to achieve a statistically accurate estimates

A survey of random processes with reinforcement

The models surveyed include generalized P\'{o}lya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of some proofs. Applications are discussed in statistics, biology, economics and a number of other areas.Comment: Published at http://dx.doi.org/10.1214/07-PS094 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

First-order expansion of homogenized coefficients under Bernoulli perturbations

Divergence-form operators with stationary random coefficients homogenize over large scales. We investigate the effect of certain perturbations of the medium on the homogenized coefficients. The perturbations that we consider are rare at the local level, but when occurring, have an effect of the same order of magnitude as the initial medium itself. The main result of the paper is a first-order expansion of the homogenized coefficients, as a function of the perturbation parameter.Comment: 32 pages. V3: revised introduction, some corrections and clarification

Phase transition in a sequential assignment problem on graphs

We study the following game on a finite graph $G = (V, E)$. At the start, each edge is assigned an integer $n_e \ge 0$, $n = \sum_{e \in E} n_e$. In round $t$, $1 \le t \le n$, a uniformly random vertex $v \in V$ is chosen and one of the edges $f$ incident with $v$ is selected by the player. The value assigned to $f$ is then decreased by $1$. The player wins, if the configuration $(0, \dots, 0)$ is reached; in other words, the edge values never go negative. Our main result is that there is a phase transition: as $n \to \infty$, the probability that the player wins approaches a constant $c_G > 0$ when $(n_e/n : e \in E)$ converges to a point in the interior of a certain convex set $\mathcal{R}_G$, and goes to $0$ exponentially when $(n_e/n : e \in E)$ is bounded away from $\mathcal{R}_G$. We also obtain upper bounds in the near-critical region, that is when $(n_e/n : e \in E)$ lies close to $\partial \mathcal{R}_G$. We supply quantitative error bounds in our arguments.Comment: 28 pages, 2 eps figures. Some mistakes have been corrected, and the introduction has been re-written. Minor corrections throughou

Stochastic hybrid system : modelling and verification

Hybrid systems now form a classical computational paradigm unifying discrete and continuous system aspects. The modelling, analysis and verification of these systems are very difficult. One way to reduce the complexity of hybrid system models is to consider randomization. The need for stochastic models has actually multiple motivations. Usually, when building models complete information is not available and we have to consider stochastic versions. Moreover, non-determinism and uncertainty are inherent to complex systems. The stochastic approach can be thought of as a way of quantifying non-determinism (by assigning a probability to each possible execution branch) and managing uncertainty. This is built upon to the - now classical - approach in algorithmics that provides polynomial complexity algorithms via randomization. In this thesis we investigate the stochastic hybrid systems, focused on modelling and analysis. We propose a powerful unifying paradigm that combines analytical and formal methods. Its applications vary from air traffic control to communication networks and healthcare systems. The stochastic hybrid system paradigm has an explosive development. This is because of its very powerful expressivity and the great variety of possible applications. Each hybrid system model can be randomized in different ways, giving rise to many classes of stochastic hybrid systems. Moreover, randomization can change profoundly the mathematical properties of discrete and continuous aspects and also can influence their interaction. Beyond the profound foundational and semantics issues, there is the possibility to combine and cross-fertilize techniques from analytic mathematics (like optimization, control, adaptivity, stability, existence and uniqueness of trajectories, sensitivity analysis) and formal methods (like bisimulation, specification, reachability analysis, model checking). These constitute the major motivations of our research. We investigate new models of stochastic hybrid systems and their associated problems. The main difference from the existing approaches is that we do not follow one way (based only on continuous or discrete mathematics), but their cross-fertilization. For stochastic hybrid systems we introduce concepts that have been defined only for discrete transition systems. Then, techniques that have been used in discrete automata now come in a new analytical fashion. This is partly explained by the fact that popular verification methods (like theorem proving) can hardly work even on probabilistic extensions of discrete systems. When the continuous dimension is added, the idea to use continuous mathematics methods for verification purposes comes in a natural way. The concrete contribution of this thesis has four major milestones: 1. A new and a very general model for stochastic hybrid systems; 2. Stochastic reachability for stochastic hybrid systems is introduced together with an approximating method to compute reach set probabilities; 3. Bisimulation for stochastic hybrid systems is introduced and relationship with reachability analysis is investigated. 4. Considering the communication issue, we extend the modelling paradigm