Aggregation and Control of Populations of Thermostatically Controlled Loads by Formal Abstractions

This work discusses a two-step procedure, based on formal abstractions, to generate a finite-space stochastic dynamical model as an aggregation of the continuous temperature dynamics of a homogeneous population of Thermostatically Controlled Loads (TCL). The temperature of a single TCL is described by a stochastic difference equation and the TCL status (ON, OFF) by a deterministic switching mechanism. The procedure is formal as it allows the exact quantification of the error introduced by the abstraction -- as such it builds and improves on a known, earlier approximation technique in the literature. Further, the contribution discusses the extension to the case of a heterogeneous population of TCL by means of two approaches resulting in the notion of approximate abstractions. It moreover investigates the problem of global (population-level) regulation and load balancing for the case of TCL that are dependent on a control input. The procedure is tested on a case study and benchmarked against the mentioned alternative approach in the literature.Comment: 40 pages, 21 figures; the paper generalizes the result of conference publication: S. Esmaeil Zadeh Soudjani and A. Abate, "Aggregation of Thermostatically Controlled Loads by Formal Abstractions," Proceedings of the European Control Conference 2013, pp. 4232-4237. version 2: added references for section

Quantitative Approximation of the Probability Distribution of a Markov Process by Formal Abstractions

The goal of this work is to formally abstract a Markov process evolving in discrete time over a general state space as a finite-state Markov chain, with the objective of precisely approximating its state probability distribution in time, which allows for its approximate, faster computation by that of the Markov chain. The approach is based on formal abstractions and employs an arbitrary finite partition of the state space of the Markov process, and the computation of average transition probabilities between partition sets. The abstraction technique is formal, in that it comes with guarantees on the introduced approximation that depend on the diameters of the partitions: as such, they can be tuned at will. Further in the case of Markov processes with unbounded state spaces, a procedure for precisely truncating the state space within a compact set is provided, together with an error bound that depends on the asymptotic properties of the transition kernel of the original process. The overall abstraction algorithm, which practically hinges on piecewise constant approximations of the density functions of the Markov process, is extended to higher-order function approximations: these can lead to improved error bounds and associated lower computational requirements. The approach is practically tested to compute probabilistic invariance of the Markov process under study, and is compared to a known alternative approach from the literature.Comment: 29 pages, Journal of Logical Methods in Computer Scienc

On Decidability of Time-Bounded Reachability in CTMDPs

We consider the time-bounded reachability problem for continuous-time Markov decision processes. We show that the problem is decidable subject to Schanuel's conjecture. Our decision procedure relies on the structure of optimal policies and the conditional decidability (under Schanuel's conjecture) of the theory of reals extended with exponential and trigonometric functions over bounded domains. We further show that any unconditional decidability result would imply unconditional decidability of the bounded continuous Skolem problem, or equivalently, the problem of checking if an exponential polynomial has a non-tangential zero in a bounded interval. We note that the latter problems are also decidable subject to Schanuel's conjecture but finding unconditional decision procedures remain longstanding open problems