We study discrete time linear constrained switching systems with additive disturbances, in the general setting where the switching acts on the system matrices, the disturbance sets, and the state constraint sets. Our primary goal is to extend the existing invariant set constructions when the switching signal is constrained by a given automation. We achieve it by working with a relaxation of invariance, namely the multi-set invariance. By exploiting recent results on computing the stability metrics for these systems, we establish explicit bounds on the number of iterations required for each construction. Last, as an application, we develop new maximal invariant set constructions for the case of linear systems in far fewer iterations compared to the state-of-the-art.

We study discrete time linear switching systems subject to additive disturbances. We consider two types of constraints, namely on the states and on the switching signal. A switching sequence is admissible if it is accepted by an automaton. Contrary to the arbitrary switching case, stability does not imply the existence of an invariant ¹ set. In this article, we propose a generalization of a bounded invariant set, namely, the notion of an invariant multi-set and show its significance in terms of dynamical systems. Under standard assumptions, we provide an iterative algorithm to approximate the minimal invariant multi-set with a guarantee of accuracy and an algorithm to compute the maximal invariant multi-set. Application of the established framework to switching systems with minimum dwell time reveals potential computational benefits and allows formulations of more refined notions.

The increasing renewable energy sources RES penetration in today’s energy islands and rural energy communities with weak grid connections is expected to incur severe distribution network stability problems (i.e., congestion, voltage issues). Tackling these problems is even more challenging since RES spillage minimization and energy cost minimization for the local energy community are set as major pre-requisites. In this paper, we consider a Microgrid Operator (MGO) that: (i) gradually decides the optimal mix of its RES and flexibility assets’ (FlexAsset) sizing, siting and operation, (ii) respects the physical distribution network constraints in high RES penetration contexts, and (iii) is able to bid strategically in the existing day-ahead energy market. We model this problem as a Stackelberg game, expressed as a Mathematical Problem with Equilibrium Constraints (MPEC), which is finally transformed into a tractable Mixed Integer Linear Program (MILP). The performance evaluation results show that the MGO can lower its costs when bidding strategically, while the coordinated planning and scheduling of its FlexAssets result in higher RES utilization, as well as distribution network aware and cost-effective RES and FlexAsset operation.