Nonlinear modeling and optimization is a valuable tool for aiding decisions by engineering practitioners, but programming an optimization problem based on a complex electrical, mechanical, or chemical process is a time-consuming and error-prone activity. Therefore, there is a need for model analysis and debugging tools that can detect and diagnose modeling errors. One such tool is the Dulmage–Mendelsohn decomposition, which identifies structurally under- and over-determined subsets in systems of equations and variables by partitioning the bipartite graph of the system. This work provides the necessary background to understand the Dulmage–Mendelsohn decomposition and its application to the analysis of nonlinear optimization problems, demonstrates its use in diagnosing a variety of modeling errors, and introduces software implementations for analyzing nonlinear optimization problems in the Pyomo and JuMP algebraic modeling languages.
Stinchfield, Georgia; Biegler, Lorenz T.; Eslick, John C.; Jacobson, Clas; Miller, David C.; Siirola, John D.; Zamarripa; Zhang, Chen; Zhang, Qi; Laird, Carl D.
Gao, Xian; Knueven, Bernard; Siirola, John D.; Miller, David C.; Dowling, Alexander W.
Accelerating the deep decarbonization of the world's electric grids requires the coordination of complex energy systems and infrastructures across timescales from seconds to decades. In this paper, we present a new multiscale simulation framework that integrates process- and grid-centric modeling paradigms to better design, operate, and control integrated energy systems (IESs), which combine multiple technologies, in wholesale energy markets. Traditionally, IESs are analyzed with a process-centric paradigm such as levelized cost of electricity (LCOE) or annualized net revenue, ignoring important interactions with electricity markets. This framework explicitly models the complex interactions between an IES's bidding, scheduling, and control decisions and the energy market's clearing and settlement processes, while incorporating operational uncertainties. Through two case studies, we show the importance of understanding and quantifying complex resource-grid interactions. In case study 1, we demonstrate that optimized bidding from one resource shifts the profit distribution for all energy systems in the market. This result suggests new and more flexible IES technologies can disrupt the economics of all market participants, possibly leading to accelerated retirements of less flexible resources. Interestingly, the optimized bidding has little impact on grid-level aggregate statistics, such as total generation costs and renewable penetration rate. While aggregate modeling strategies may remain valid under some IES adoption scenarios for analysis focused on regional outcomes, direct comparisons of IES technologies at specific locations without considering these interactions may lead to misleading or incorrect conclusions. In case study 2, we consider the design and flexible operation of IESs that hybridize conventional generators with energy storage. Through a sensitivity analysis, we find that as the size of the storage system increases, the total number of start-ups for coal- and natural gas-based IESs reduced by 25% and 33.6%, and the total thermal generator ramping (i.e., mileage) reduced by 86.5% and 62.5%, respectively. This shows the primary benefit of storage may not be reduced operational costs (which do not change significantly) but fewer start-ups and less ramping, which may greatly simplify the design, operation, and control of carbon capture systems. The new modeling and optimization capabilities from this work enable the coupling of rigorous, dynamic process models with grid-level production cost models to quantitatively identify the nuanced interdependencies across these vast timescales that must be addressed to realize clean, safe, and secure energy production. Moreover, the proposed general multiscale simulation framework is applicable to all IES technologies and can be easily extended to consider other energy carriers (e.g., hydrogen, ammonia) and energy infrastructures (e.g., natural gas pipelines).
Chen, Qi; Johnson, Emma S.; Bernal, David E.; Valentin, Romeo; Kale, Sunjeev; Bates, Johnny; Siirola, John D.; Grossmann, Ignacio E.
We present three core principles for engineering-oriented integrated modeling and optimization tool sets—intuitive modeling contexts, systematic computer-aided reformulations, and flexible solution strategies—and describe how new developments in Pyomo.GDP for Generalized Disjunctive Programming (GDP) advance this vision. We describe a new logical expression system implementation for Pyomo.GDP allowing for a more intuitive description of logical propositions. The logical expression system supports automated reformulation of these logical constraints to linear constraints. We also describe two new logic-based global optimization solver implementations built on Pyomo.GDP that exploit logical structure to avoid “zero-flow” numerical difficulties that arise in nonlinear network design problems when nodes or streams disappear. These new solvers also demonstrate the capability to link to external libraries for expanded functionality within an integrated implementation. We present these new solvers in the context of a flexible array of solution paths available to GDP models. Finally, we present results on a new library of GDP models demonstrating the value of multiple solution approaches.
This manuscript presents the recent advances in Mixed-Integer Nonlinear Programming (MINLP) and Generalized Disjunctive Programming (GDP) with a particular scope for superstructure optimization within Process Systems Engineering (PSE). We present an environment of open-source software packages written in Python and based on the algebraic modeling language Pyomo. These packages include MindtPy, a solver for MINLP that implements decomposition algorithms for such problems, CORAMIN, a toolset for MINLP algorithms providing relaxation generators for nonlinear constraints, Pyomo.GDP, a modeling extension for Generalized Disjunctive Programming that allows users to represent their problem as a GDP natively, and GDPOpt, a collection of algorithms explicitly tailored for GDP problems. Combining these tools has allowed us to solve several problems relevant to PSE, which we have gathered in an easily installable and accessible library, GDPLib. We show two examples of these models and how the flexibility of modeling given by Pyomo.GDP allows for efficient solutions to these complex optimization problems. Finally, we show an example of integrating these tools with the framework IDAES PSE, leading to optimal process synthesis and conceptual design with advanced multi-scale PSE modeling systems.
Zhang, Chen; Jacobson, Clas; Zhang, Qi; Biegler, Lorenz T.; Eslick, John C.; Zamarripa, Miguel A.; Stinchfeld, Georgia; Siirola, John D.; Laird, Carl D.
For many industries addressing varied customer needs means producing a family of products that satisfy a range of design requirements. Manufacturers seek to design this family of products while exploiting opportunities for shared components to reduce manufacturing cost and complexity. We present a mixed-integer programming formulation that determines the optimal design for each product, the number and design of shared components, and the allocation of those shared components across the products in the family. This formulation and workflow for product family design has created significant business impact on the industrial design of product families for large-scale commercial HVAC chillers in Carrier Global Corporation. We demonstrate the approach on an open case study based on a transcritical CO2 refrigeration cycle. This case study and our industrial experience show that the formulation is computationally tractable and can significantly reduce engineering time by replacing the manual design process with an automated approach.
This report summarizes the guidance provided to Sustainable Engineering to help them learn about equation-oriented optimization and the Sandia-developed software packages Pyomo and IDAESPSE. This was a short 10-week project (October 2021 – December 2021) and the goal was to help the company learn about the IDAES framework and how it could be used for their future projects. The company submitted an SBIR proposal related to developing a green ammonia process model with IDAES and if that proposal is successful this NMSBA project could lead to future collaboration opportunities.
This work focuses on estimation of unknown states and parameters in a discrete-time, stochastic, SEIR model using reported case counts and mortality data. An SEIR model is based on classifying individuals with respect to their status in regards to the progression of the disease, where S is the number individuals who remain susceptible to the disease, E is the number of individuals who have been exposed to the disease but not yet infectious, I is the number of individuals who are currently infectious, and R is the number of recovered individuals. For convenience, we include in our notation the number of infections or transmissions, T, that represents the number of individuals transitioning from compartment S to compartment E over a particular interval. Similarly, we use C to represent the number of reported cases.