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Register a new userDistributionally robust stochastic programs with side information based on trimmings -- Extended version
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We consider stochastic programs conditional on some covariate information, where the only knowledge of the possible relationship between the uncertain parameters and the covariates is reduced to a finite data sample of their joint distribution. By exploiting the close link between the notion of trimmings of a probability measure and the partial mass transportation problem, we construct a data-driven Distributionally Robust Optimization (DRO) framework to hedge the decision against the intrinsic error in the process of inferring conditional information from limited joint data. We show that our approach is computationally as tractable as the standard (without side information) Wasserstein-metric-based DRO and enjoys performance guarantees. Furthermore, our DRO framework can be conveniently used to address data-driven decision-making problems under contaminated samples and naturally produces distributionally robu
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In this paper, we develop a distributionally robust chance-constrained formulation of the Optimal Power Flow problem (OPF) whereby the system operator can leverage contextual information. For this purpose, we exploit an ambiguity set based on probability trimmings and optimal transport through which the dispatch solution is protected against the incomplete knowledge of the relationship between the OPF uncertainties and the context that is conveyed by a sample of their joint probability distribution. We provide an exact reformulation of the proposed distributionally robust chance-constrained OPF problem under the popular conditional-value-at-risk approximation. By way of numerical experiments run on a modified IEEE-118 bus network with wind uncertainty, we show how the power system can substantially benefit from taking into account the well-known statistical dependence between the point forecast of wind power outputs and its associated prediction error. Furthermore, the experiments conducted also reveal that the distributional robustness conferred on the OPF solution by our probability-trimmings-based approach is superior to that bestowed by alternative approaches in terms of expected cost and system reliability.
Aggregation of heating, ventilation, and air conditioning (HVAC) loads can provide reserves to absorb volatile renewable energy, especially solar photo-voltaic (PV) generation. However, the time-varying PV generation is not perfectly known when the system operator decides the HVAC control schedules. To consider the unknown uncertain PV generation, in this paper, we formulate a distributionally robust chance-constrained (DRCC) building load control problem under two typical ambiguity sets: the moment-based and Wasserstein ambiguity sets. We derive mixed-integer linear programming (MILP) reformulations for DRCC problems under both sets. Especially for the DRCC problem under the Wasserstein ambiguity set, we utilize the right-hand side (RHS) uncertainty to derive a more compact MILP reformulation than the commonly known MILP reformulations with big-M constants. All the results also apply to general individual chance constraints with RHS uncertainty. Furthermore, we propose an adjustable chance-constrained variant to achieve a trade-off between the operational risk and costs. We derive MILP reformulations under the Wasserstein ambiguity set and second-order conic programming (SOCP) reformulations under the moment-based set. Using real-world data, we conduct computational studies to demonstrate the efficiency of the solution approaches and the effectiveness of the solutions.
Probabilistic methods have attracted much interest in fault detection design, but its need for complete distributional knowledge is seldomly fulfilled. This has spurred endeavors in distributionally robust fault detection (DRFD) design, which secures robustness against inexact distributions by using moment-based ambiguity sets as a prime modelling tool. However, with the worst-case distribution being implausibly discrete, the resulting design suffers from over-pessimisim and can mask the true fault. This paper aims at developing a new DRFD design scheme with reduced conservatism, by assuming unimodality of the true distribution, a property commonly encountered in real-life practice. To tackle the chance constraint on false alarms, we first attain a new generalized Gauss bound on the probability outside an ellipsoid, which is less conservative than known Chebyshev bounds. As a result, analytical solutions to DRFD design problems are obtained, which are less conservative than known ones disregarding unimodality. We further encode bounded support information into ambiguity sets, derive a tightened multivariate Gauss bound, and develop approximate reformulations of design problems as convex programs. Moreover, the derived generalized Gauss bounds are broadly applicable to versatile change detection tasks for setting alarm thresholds. Results on a laborotary system shown that, the incorporation of unimodality information helps reducing conservatism of distributionally robust design and leads to a better tradeoff between robustness and sensitivity.
Adaptive robust optimization problems are usually solved approximately by restricting the adaptive decisions to simple parametric decision rules. However, the corresponding approximation error can be substantial. In this paper we show that two-stage robust and distributionally robust linear programs can often be reformulated exactly as conic programs that scale polynomially with the problem dimensions. Specifically, when the ambiguity set constitutes a 2-Wasserstein ball centered at a discrete distribution, then the distributionally robust linear program is equivalent to a copositive program (if the problem has complete recourse) or can be approximated arbitrarily closely by a sequence of copositive programs (if the problem has sufficiently expensive recourse). These results directly extend to the classical robust setting and motivate strong tractable approximations of two-stage problems based on semidefinite approximations of the copositive cone. We also demonstrate that the two-stage distributionally robust optimization problem is equivalent to a tractable linear program when the ambiguity set constitutes a 1-Wasserstein ball centered at a discrete distribution and there are no support constraints.
To ensure a successful bid while maximizing of profits, generation companies (GENCOs) need a self-scheduling strategy that can cope with a variety of scenarios. So distributionally robust opti-mization (DRO) is a good choice because that it can provide an adjustable self-scheduling strategy for GENCOs in the uncertain environment, which can well balance robustness and economics compared to strategies derived from robust optimization (RO) and stochastic programming (SO). In this paper, a novel mo-ment-based DRO model with conditional value-at-risk (CVaR) is proposed to solve the self-scheduling problem under electricity price uncertainty. Such DRO models are usually translated into semi-definite programming (SDP) for solution, however, solving large-scale SDP needs a lot of computational time and resources. For this shortcoming, two effective approximate models are pro-posed: one approximate model based on vector splitting and an-other based on alternate direction multiplier method (ADMM), both can greatly reduce the calculation time and resources, and the second approximate model only needs the information of the current area in each step of the solution and thus information private is guaranteed. Simulations of three IEEE test systems are conducted to demonstrate the correctness and effectiveness of the proposed DRO model and two approximate models.
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