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Optimization problems under uncertain conditions abound in many real-life applications. While solution approaches for probabilistic constraints are often developed in case the uncertainties can be assumed to follow a certain probability distribution, robust approaches are usually applied in case solutions are sought that are feasible for all realizations of uncertainties within some predefined uncertainty set. As many applications contain different types of uncertainties that require robust as well as probabilistic treatments, we deal with a class of joint probabilistic/robust constraints. Focusing on complex uncertain gas network optimization problems, we show the relevance of this class of problems for the task of maximizing free booked capacities in an algebraic model for a stationary gas network. We furthermore present approaches for finding their solution. Finally, we study the problem of controlling a transient system that is governed by the wave equation. The task consists in determining controls such that a certain robustness measure remains below some given upper bound with high probability.

The paper investigates analytical properties of dynamic probabilistic constraints (chance constraints). The underlying random distribution is supposed to be continuous. In the first part, a general multistage model with decision rules depending on past observations of the random process is analyzed. Basic properties like (weak sequential) (semi-) continuity of the probability function or existence of solutions are studied. It turns out that the results differ significantly according to whether decision rules are embedded into Lebesgue or Sobolev spaces. In the second part, the simplest meaningful two-stage model with decision rules from $$L^2$$ L 2 is investigated. More specific properties like Lipschitz continuity and differentiability of the probability function are considered. Explicitly verifiable conditions for these properties are provided along with explicit gradient formulae in the Gaussian case. The application of such formulae in the context of necessary optimality conditions is discussed and a concrete identification of solutions presented.

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Renewable-powered “undergrid mini-grids” (UMGs) are instrumental for electrification in developing countries. An UMG can be installed under a—possibly unreliable— main grid to improve the local reliability or the main grid may “arrive” and connect to a previously isolated mini-grid. Minimising costs is key to reducing risks associated with UMG development. This article presents a novel market-logic strategy for the optimal operation of UMGs that can incorporate multiple types of controllable loads, customer smart curtailment based on reliability requirements, storage management, and exports to and imports from a main grid, which is subject to failure. The formulation results in a mixed-integer linear programming model (MILP) and assumes accurate predictions of the following uncertain parameters: grid spot prices, outages of the main grid, solar availability and demand profiles. An AC hybrid solar-battery-diesel UMG configuration from Nigeria is used as a case example, and numerical simulations are presented. The load-following (LF) and cycle-charging (CC) strategies are compared with our predictive strategy and HOMER Pro’s Predictive dispatch. Results prove the generality and adequacy of the market-logic dispatch model and help assess the relevance of outages of the main grid and of spot prices above the other uncertain input factors. Comparison results show that the proposed market-logic operation approach performs better in terms of cost minimisation, higher renewable fraction and lower diesel use with respect to the conventional LF and CC operating strategies.