Tractable Approximations to Robust Conic Optimization Problems

Bertsimas, Dimitris; Sim, Melvyn

doi:10.1007/s10107-005-0677-1

Cited by 263 publications

(244 citation statements)

References 13 publications

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“…The interested reader is referred to for example [4], [13], [15]. In [4] the authors suggest to assume that the uncertainty is bounded by a certain value, which effectively approximates the chance constraint with a hard constraint. The authors describe how the bounding value has to be chosen such that they can give a performance guarantee on each chance constraint.…”

Section: B Chance Constraintsmentioning

confidence: 99%

A tractable approximation of chance constrained stochastic MPC based on affine disturbance feedback

Oldewurtel

Jones

Morari

2008

2008 47th IEEE Conference on Decision and Control

141

104

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Abstract-This paper deals with model predictive control of uncertain linear discrete-time systems with polytopic constraints on the input and chance constraints on the states. Recently, it has been shown that when having merely polytopic constraints and bounded disturbances, the conservatism of a robust solution can be reduced by applying a closed-loop prediction formulation. We show that in the presence of chance constraints and stochastic disturbances, this closed-loop formulation can be used together with a tractable approximation of the chance constraints to further increase the performance while giving probabilistic guarantees on the constraints.

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Section: B Chance Constraintsmentioning

confidence: 99%

A tractable approximation of chance constrained stochastic MPC based on affine disturbance feedback

Oldewurtel

Jones

Morari

2008

2008 47th IEEE Conference on Decision and Control

141

104

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“…Helping to spark this recent growth was the work of Nemirovski (1999, 2000), who show that even small perturbations of uncertain quantities can result in highly infeasible solutions. More recent work has focused on the properties of the solutions and the tractability of various robust formulations, as well as extending robustness to more general conic problems (e.g., Bertsimas et al 2004, Bertsimas and Sim 2006, El Ghaoui and Lebret 1997, El Ghaoui et al 1998.…”

Section: Introductionmentioning

confidence: 99%

Constructing Uncertainty Sets for Robust Linear Optimization

Bertsimas

Brown

2009

Operations Research

Self Cite

290

167

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In this paper, we propose a methodology for constructing uncertainty sets within the framework of robust optimization for linear optimization problems with uncertain parameters. Our approach relies on decision maker risk preferences. Specifically, we utilize the theory of coherent risk measures initiated by Artzner et al. (1999) [Artzner, P., F. Delbaen, J. Eber, D. Heath. 1999. Coherent measures of risk. Math. Finance 9 203-228.], and show that such risk measures, in conjunction with the support of the uncertain parameters, are equivalent to explicit uncertainty sets for robust optimization. We explore the structure of these sets in detail. In particular, we study a class of coherent risk measures, called distortion risk measures, which give rise to polyhedral uncertainty sets of a special structure that is tractable in the context of robust optimization. In the case of discrete distributions with rational probabilities, which is useful in practical settings when we are sampling from data, we show that the class of all distortion risk measures (and their corresponding polyhedral sets) are generated by a finite number of conditional value-at-risk (CVaR) measures. A subclass of the distortion risk measures corresponds to polyhedral uncertainty sets symmetric through the sample mean. We show that this subclass is also finitely generated and can be used to find inner approximations to arbitrary, polyhedral uncertainty sets.

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“…In this case, enumeration of the vertices should be avoided to solve the problem more efficiently. This can be achieved by applying techniques proposed in [15].…”

Section: A Chance Constraint Reformulationmentioning

confidence: 99%

Risk-constrained optimal power flow with probabilistic guarantees

Roald¹,

Vrakopoulou²,

Oldewurtel³

et al. 2014

2014 Power Systems Computation Conference

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Abstract-Higher penetration of renewable energy and market liberalization increase both the need for transmission capacity and the uncertainty in power system operation. New methods for power system operational planning are needed to allow for efficient use of the grid, while maintaining security and robustness against disturbances. In this paper, we propose a risk model for risk related to outages, accounting for available remedial measures and the impact of cascading events. The new risk model is used to formulate risk-based constraints for the postcontingency line flows, which are included in an optimal power flow (OPF) formulation. Forecast uncertainty is accounted for by formulating the relevant constraints as a chance constraint, and the problem is solved using a sampling based technique. In a case study for the IEEE 30 bus system, we demonstrate how the proposed risk-based, probabilistic OPF allows us to control the risk level, even in presence of uncertainty. We investigate the trade-off between generation cost and risk level in the system, and show how accounting for uncertainty leads to a more expensive, but more secure dispatch.Keywords-risk-based optimal power flow, chance constrained optimal power flow, security, wind power integration

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Tractable Approximations to Robust Conic Optimization Problems

Cited by 263 publications

References 13 publications

A tractable approximation of chance constrained stochastic MPC based on affine disturbance feedback

A tractable approximation of chance constrained stochastic MPC based on affine disturbance feedback

Constructing Uncertainty Sets for Robust Linear Optimization

Risk-constrained optimal power flow with probabilistic guarantees

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