Constrained optimization problems are prevalent in all areas of science and engineering, and many well-known numerical methods have been developed to solve these problems. Yet, when there exists random quantities in the model under consideration, most of the deterministic methods are no longer effective. In this dissertation, we address parameter uncertainty in the constraints by formulating the problem as a chance constrained problem, which requires that constraints are satisfied for a given probability. We discuss a new approximation to chance constraints that results in differentiable functions that can be integrated into any nonlinear optimization problem and solved with standard nonlinear programming techniques. For problems with joint chance constraints, our formulation can be used to develop a new trust-region sequential quadratic programming algorithm with provable convergence guarantees. Our formulation is based on sample average approximation and considers a set of random realizations of the constraints. Often, a mixed-integer programming formulation is used in this setting to select which of the resulting constraints should be enforced to achieve the desired probability of constraint satisfaction. However, this approach typically results in large computation times due to the implicit enumeration performed by the branch-and-bound algorithm. Instead, we propose a compact reformulation that combines all constraint scenarios in a single smooth constraint. This is achieved by means of a smoothed cumulative distribution function for the random constraint values. We provide theoretical statistical guarantees to the approximation. Finally, we show how our algorithm can be used to solve optimal power flow (OPF) problems with uncertainty in power injections. Managing variability in the power grid has become a major concern for power system operators due to the increasing levels of fluctuating renewable energy connected to the grid. This work addresses this uncertainty via a joint chance-constrained formulation of the OPF problem. The most common approaches that have been designed to solve joint-chance constrained OPF problems are typically either computationally intractable for large-scale problems or give overly conservative solutions, which result in very costly operations. By making two modifications to our general solver for chance constraints, we obtain a sample-based method that avoids making strong assumptions on the distribution of the uncertainty, scales favorably to large problems, and obtains less conservative results. We show first how this method can be applied to the DC-OPF chance-constrained problem, and extend it to work in the AC-OPF setting.

Creator

• Pena-Ordieres, Alejandra

DOI

• https://doi.org/10.21985/n2-84re-9w94

Subject

• Industrial engineering

Language

• en

Alternate Identifier

• http://dissertations.umi.com/northwestern:15173
• etdadmin_upload_752377

Keyword

• chance constraints
• optimal power flow
• nonlinear programming
• stochastic programming

Date created

• 2020-01-01

Resource type

• Dissertation

Rights statement

• In Copyright