Increasing penetration levels of renewables have transformed how power systems are operated. High levels of uncertainty in production make it increasingly difficulty to guarantee operational feasibility; instead, constraints may only be satisfied with high probability. We present a chance-constrained economic dispatch model that efficiently integrates energy storage and high renewable penetration to satisfy renewable portfolio requirements. Specifically, we require that wind energy contribute at least a prespecified proportion of the total demand and that the scheduled wind energy is deliverable with high probability. We develop an approximate partial sample average approximation (PSAA) framework to enable efficient solution of large-scale chance-constrained economic dispatch problems. Computational experiments on the IEEE-24 bus system show that the proposed PSAA approach is more accurate, closer to the prescribed satisfaction tolerance, and approximately 100 times faster than standard sample average approximation. Finally, the improved efficiency of our PSAA approach enables solution of a larger WECC-240 test system in minutes.
One of the major challenges in deciding where to build new transmission lines is that there is uncertainty regarding future loads, renewal generation output and equipment failures. We propose a robust optimization model whose transmission expansion solutions ensure that demand can be met over a wide range of conditions. Specifically, we require feasible operation for all loads and renewable generation levels within given ranges, and for all single transmission line failures. Furthermore, we consider transmission switching as an allowable recovery action. This relatively inexpensive method of redirecting power flows improves resiliency, but introduces computational challenges. We present a novel algorithm to solve this model. Computational results are discussed.
Stochastic economic dispatch models address uncertainties in forecasts of renewable generation output by considering a finite number of realizations drawn from a stochastic process model, typically via Monte Carlo sampling. Accurate evaluations of expectations or higher order moments for quantities of interest, e.g., generating cost, can require a prohibitively large number of samples. We propose an alternative to Monte Carlo sampling based on polynomial chaos expansions. These representations enable efficient and accurate propagation of uncertainties in model parameters, using sparse quadrature methods. We also use Karhunen-Loève expansions for efficient representation of uncertain renewable energy generation that follows geographical and temporal correlations derived from historical data at each wind farm. Considering expected production cost, we demonstrate that the proposed approach can yield several orders of magnitude reduction in computational cost for solving stochastic economic dispatch relative to Monte Carlo sampling, for a given target error threshold.
We describe new capabilities for modeling bilevel programs within the Pyomo modeling software. These capabilities include new modeling components that represent subproblems, modeling transformations for re-expressing models with bilevel structure in other forms, and optimize bilevel programs with meta-solvers that apply transformations and then perform op- timization on the resulting model. We illustrate the breadth of Pyomo's modeling capabilities for bilevel programs, and we describe how Pyomo's meta-solvers can perform local and global optimization of bilevel programs.
This report summarizes the work performed under the project project Next-Generation Algo- rithms for Assessing Infrastructure Vulnerability and Optimizing System Resilience. The goal of the project was to improve mathematical programming-based optimization technology for in- frastructure protection. In general, the owner of a network wishes to design a network a network that can perform well when certain transportation channels are inhibited (e.g. destroyed) by an adversary. These are typically bi-level problems where the owner designs a system, an adversary optimally attacks it, and then the owner can recover by optimally using the remaining network. This project funded three years of Deon Burchett's graduate research. Deon's graduate advisor, Professor Jean-Philippe Richard, and his Sandia advisors, Richard Chen and Cynthia Phillips, supported Deon on other funds or volunteer time. This report is, therefore. essentially a replication of the Ph.D. dissertation it funded [12] in a format required for project documentation. The thesis had some general polyhedral research. This is the study of the structure of the feasi- ble region of mathematical programs, such as integer programs. For example, an integer program optimizes a linear objective function subject to linear constraints, and (nonlinear) integrality con- straints on the variables. The feasible region without the integrality constraints is a convex polygon. Careful study of additional valid constraints can significantly improve computational performance. Here is the abstract from the dissertation: We perform a polyhedral study of a multi-commodity generalization of variable upper bound flow models. In particular, we establish some relations between facets of single- and multi- commodity models. We then introduce a new family of inequalities, which generalizes traditional flow cover inequalities to the multi-commodity context. We present encouraging numerical results. We also consider the directed edge-failure resilient network design problem (DRNDP). This problem entails the design of a directed multi-commodity flow network that is capable of fulfilling a specified percentage of demands in the event that any G arcs are destroyed, where G is a constant parameter. We present a formulation of DRNDP and solve it in a branch-column-cut framework. We present computational results.
Stochastic unit commitment models typically handle uncertainties in forecast demand by considering a finite number of realizations from a stochastic process model for loads. Accurate evaluations of expectations or higher moments for the quantities of interest require a prohibitively large number of model evaluations. In this paper we propose an alternative approach based on using surrogate models valid over the range of the forecast uncertainty. We consider surrogate models based on Polynomial Chaos expansions, constructed using sparse quadrature methods. Considering expected generation cost, we demonstrate that the approach can lead to several orders of magnitude reduction in computational cost relative to using Monte Carlo sampling on the original model, for a given target error threshold.