Publications

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The optimization of sensor placements is a key aspect of the design of contaminant warning systems for automatically detecting contaminants in water distribution systems. Although researchers have generally assumed that all sensors are placed at the same time, in practice sensor networks will likely grow and evolve over time. For example, limitations for a water utility's budget may dictate an staged, incremental deployment of sensors over many years. We describe optimization formulations of multi-stage sensor placement problems. The objective of these formulations includes an explicit trade-off between the value of the initially deployed and final sensor networks. This trade-off motivates the deployment of sensors in initial stages of the deployment schedule, even though these choices typically lead to a solution that is suboptimal when compared to placing all sensors at once. These multi-stage sensor placement problems can be represented as mixed-integer programs, and we illustrate the impact of this trade-off using standard commercial solvers. We also describe a multi-stage formulation that models budget uncertainty, expressed as a tree of potential budget scenarios through time. Budget uncertainty is used to assess and hedge against risks due to a potentially incomplete deployment of a planned sensor network. This formulation is a multi-stage stochastic mixed-integer program, which are notoriously difficult to solve. We apply standard commercial solvers to small-scale test problems, enabling us to effectively analyze multi-stage sensor placement problems subject to budget uncertainties, and assess the impact of accounting for such uncertainty relative to a deterministic multi-stage model. © 2012 ASCE.

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This manuscript presents a unified software framework for modeling and optimizing large-scale engineered systems with uncertainty. We propose a Python-based " block- oriented" modeling approach for representing the discrete components within the system. Through the use of a modeling components library, the block-oriented approach facilitates a clean separation of system superstructure from the details of individual components. This approach also lends itself naturally to expressing design and operational decisions as disjunctive expressions over the component blocks. We then apply a Python- based risk and uncertainty analysis library that leverages the explicit representation of the mathematical program in Python to automatically expand the deterministic system model into a multi-stage stochastic program, which can then be solved either directly or via decomposition-based solution strategies. This manuscript demonstrates the application of this modeling approach for risk-aware analysis of an electric distribution system. © 2012 Elsevier B.V.

Decision makers increasingly rely on large-scale computational models to simulate and analyze complex man-made systems. For example, computational models of national infrastructures are being used to inform government policy, assess economic and national security risks, evaluate infrastructure interdependencies, and plan for the growth and evolution of infrastructure capabilities. A major challenge for decision makers is the analysis of national-scale models that are composed of interacting systems: effective integration of system models is difficult, there are many parameters to analyze in these systems, and fundamental modeling uncertainties complicate analysis. This project is developing optimization methods to effectively represent and analyze large-scale heterogeneous system of systems (HSoS) models, which have emerged as a promising approach for describing such complex man-made systems. These optimization methods enable decision makers to predict future system behavior, manage system risk, assess tradeoffs between system criteria, and identify critical modeling uncertainties.

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We consider the problem of placing a limited number of sensors in a municipal water distribution network to minimize the impact over a given suite of contamination incidents. In its simplest form, the sensor placement problem is a p-median problem that has structure extremely amenable to exact and heuristic solution methods. We describe the solution of real-world instances using integer programming or local search or a Lagrangian method. The Lagrangian method is necessary for solution of large problems on small PCs. We summarize a number of other heuristic methods for effectively addressing issues such as sensor failures, tuning sensors based on local water quality variability, and problem size/approximation quality tradeoffs. These algorithms are incorporated into the TEVA-SPOT toolkit, a software suite that the US Environmental Protection Agency has used and is using to design contamination warning systems for US municipal water systems.

A range of core operations and planning problems for the national electrical grid are naturally formulated and solved as stochastic programming problems, which minimize expected costs subject to a range of uncertain outcomes relating to, for example, uncertain demands or generator output. A critical decision issue relating to such stochastic programs is: How many scenarios are required to ensure a specific error bound on the solution cost? Scenarios are the key mechanism used to sample from the uncertainty space, and the number of scenarios drives computational difficultly. We explore this question in the context of a long-term grid generation expansion problem, using a bounding procedure introduced by Mak, Morton, and Wood. We discuss experimental results using problem formulations independently minimizing expected cost and down-side risk. Our results indicate that we can use a surprisingly small number of scenarios to yield tight error bounds in the case of expected cost minimization, which has key practical implications. In contrast, error bounds in the case of risk minimization are significantly larger, suggesting more research is required in this area in order to achieve rigorous solutions for decision makers.

The Python Optimization Modeling Objects (Pyomo) package [1] is an open source tool for modeling optimization applications within Python. Pyomo provides an objected-oriented approach to optimization modeling, and it can be used to define symbolic problems, create concrete problem instances, and solve these instances with standard solvers. While Pyomo provides a capability that is commonly associated with algebraic modeling languages such as AMPL, AIMMS, and GAMS, Pyomo's modeling objects are embedded within a full-featured high-level programming language with a rich set of supporting libraries. Pyomo leverages the capabilities of the Coopr software library [2], which integrates Python packages (including Pyomo) for defining optimizers, modeling optimization applications, and managing computational experiments. A central design principle within Pyomo is extensibility. Pyomo is built upon a flexible component architecture [3] that allows users and developers to readily extend the core Pyomo functionality. Through these interface points, extensions and applications can have direct access to an optimization model's expression objects. This facilitates the rapid development and implementation of new modeling constructs and as well as high-level solution strategies (e.g. using decomposition- and reformulation-based techniques). In this presentation, we will give an overview of the Pyomo modeling environment and model syntax, and present several extensions to the core Pyomo environment, including support for Generalized Disjunctive Programming (Coopr GDP), Stochastic Programming (PySP), a generic Progressive Hedging solver [4], and a tailored implementation of Bender's Decomposition.

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Although stochastic programming is a powerful tool for modeling decision-making under uncertainty, various impediments have historically prevented its widespread use. One key factor involves the ability of non-specialists to easily express stochastic programming problems as extensions of deterministic models, which are often formulated first. A second key factor relates to the difficulty of solving stochastic programming models, particularly the general mixed-integer, multi-stage case. Intricate, configurable, and parallel decomposition strategies are frequently required to achieve tractable run-times. We simultaneously address both of these factors in our PySP software package, which is part of the COIN-OR Coopr open-source Python project for optimization. To formulate a stochastic program in PySP, the user specifies both the deterministic base model and the scenario tree with associated uncertain parameters in the Pyomo open-source algebraic modeling language. Given these two models, PySP provides two paths for solution of the corresponding stochastic program. The first alternative involves writing the extensive form and invoking a standard deterministic (mixed-integer) solver. For more complex stochastic programs, we provide an implementation of Rockafellar and Wets Progressive Hedging algorithm. Our particular focus is on the use of Progressive Hedging as an effective heuristic for approximating general multi-stage, mixed-integer stochastic programs. By leveraging the combination of a high-level programming language (Python) and the embedding of the base deterministic model in that language (Pyomo), we are able to provide completely generic and highly configurable solver implementations. PySP has been used by a number of research groups, including our own, to rapidly prototype and solve difficult stochastic programming problems.

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