Tight and Compact Sample Average Approximation for Joint Chance-Constrained Problems with Applications to Optimal Power Flow.

In this paper, we tackle the resolution of chance-constrained problems reformulated via sample average approximation. The resulting data-driven deterministic reformulation takes the form of a large-scale mixed-integer program (MIP) cursed with Big-Ms. We introduce an exact resolution method for the MIP that combines the addition of a set of valid inequalities to tighten the linear relaxation bound with coefficient strengthening and constraint screening algorithms to improve its Big-Ms and considerably reduce its size. The proposed valid inequalities are based on the notion of k-envelopes and can be computed off-line using polynomial-time algorithms and added to the MIP program all at once. Furthermore, they are equally useful to boost the strengthening of the Big-Ms and the screening rate of superfluous constraints. We apply our procedures to a probabilistically constrained version of the DC optimal power flow problem with uncertain demand. The chance constraint requires that the probability of violating any of the power system’s constraints be lower than some positive threshold. In a series of numerical experiments that involve five power systems of different size, we show the efficiency of the proposed methodology and compare it with some of the best performing convex inner approximations currently available in the literature.This work was supported in part by the European Research Council under the EU Horizon 2020 research and innovation program [Grant 755705], in part by the Spanish Ministry of Science and Innovation [Grant AEI/10.13039/501100011033] through project PID2020-115460GB-I00, and in part by the Junta de Andalucía and the European Regional Development Fund through the research project P20_00153. Á. Porras is also financially supported by the Spanish Ministry of Science, Innovation and Universities through the University Teacher Training Program with fellowship number FPU19/03053

On the Convexity of Level-sets of Probability Functions

In decision-making problems under uncertainty, probabilistic constraints are a valuable tool to express safety of decisions. They result from taking the probability measure of a given set of random inequalities depending on the decision vector. Even if the original set of inequalities is convex, this favourable property is not immediately transferred to the probabilistically constrained feasible set and may in particular depend on the chosen safety level. In this paper, we provide results guaranteeing the convexity of feasible sets to probabilistic constraints when the safety level is greater than a computable threshold. Our results extend all the existing ones and also cover the case where decision vectors belong to Banach spaces. The key idea in our approach is to reveal the level of underlying convexity in the nominal problem data (e.g., concavity of the probability function) by auxiliary transforming functions. We provide several examples illustrating our theoretical developments

Power System Operations with Probabilistic Guarantees

This study is motivated by the fact that uncertainties from deepening penetration of renewable energy resources have posed critical challenges to the secure and reliable operations of future electrical grids. Among various tools for decision making in uncertain environments, this study focuses on chance-constrained optimization, which provides explicit probabilistic guarantees on the feasibility of optimal solutions. Although quite a few methods have been proposed to solve chance-constrained optimization problems, there is a lack of comprehensive review and comparative analysis of the proposed methods. In this work, we provide a detailed tutorial on existing algorithms and a survey of major theoretical results of chance-constrained optimization theory. Data-driven methods, which are not constrained by any specific distributions of the underlying uncertainties, are of particular interest. Built upon chance-constrained optimization, we propose a three-stage power system operation framework with probabilistic guarantees: (1) the optimal unit commitment in the operational planning stage; (2) the optimal reactive power dispatch to address the voltage security issue in the hours-ahead adjustment period; and (3) the secure and reliable power system operation under uncertainties in real time. In the day-ahead operational planning stage, we propose a chance-constrained SCUC (c-SCUC) framework, which ensures that the risk of violating constraints is within an acceptable threshold. Using the scenario approach, c-SCUC is reformulated to the scenario-based SCUC (s-SCUC) problem. By choosing an appropriate number of scenarios, we provide theoretical guarantees on the posterior risk level of the solution to s-SCUC. Inspired by the latest progress of the scenario approach on non-convex problems, we demonstrate the structural properties of general scenario problems and analyze the specific characteristics of s-SCUC. Those characteristics were exploited to benefit the scalability and computational performance of s-SCUC. In the adjustment period, this work first investigates the benefits of look-ahead coordination of both continuous-state and discrete-state reactive power support devices across multiple control areas. The conventional static optimal reactive power dispatch is extended to a “moving-horizon” type formulation for the consideration of spatial and temporal variations. The optimal reactive power dispatch problem is further enhanced with chance constraints by considering the uncertainties from both renewables and contingencies. This chance-constrained optimal reactive power dispatch (c-ORPD) formulation offers system operators an effective tool to schedule voltage support devices such that the system voltage security can be ensured with quantifiable level of risk. Security-constrained Economic Dispatch (SCED) lies at the center of real-time operation of power systems and modern electricity markets. It determines the most cost-efficient output levels of generators while keeping the real-time balance between supply and demand. In this study, we formulate and solve chance-constrained SCED (c-SCED), which ensures system security under uncertainties from renewables. The c-SCED problem also serves as a benchmark problem for a critical comparison of existing algorithms to solve chance-constrained optimization problems