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DOE OSTI2025

Benders Decomposition Using Graph Modeling and Multi-Parametric Programming

Benders decomposition is a widely used method for solving large and structured optimization problems, but its performance is affected by the repeated solution of subproblems. We propose a flexible and modular algorithmic framework for accelerating Benders decomposition. Specifically, we express the problem structure by using a graph-theoretic modeling abstraction in which nodes represent optimization subproblems and edges represent connectivity between subproblems. A key innovation of our approach is that we embed multiparametric programming (mp) surrogates for node subproblems, which maps the exact analytical map of the subproblem solution space. The use of mp surrogates allows us to replace subproblem solves with fast look-ups and function evaluations for primal and dual variables during the iterative Benders process. We formally show the equivalence between classical Benders cuts and those derived from the mp solution. We implement our framework in the open-source PlasmoBenders.jl software package. To demonstrate the capabilities of the proposed framework, we apply it to a two-stage stochastic programming problem, which aims to make optimal capacity expansion decisions under market uncertainty. We evaluate both single-cut and multicut variants of Benders decomposition and show that the use of mp surrogates achieves substantial speedups in subproblem solve time, while preserving the convergence guarantees of Benders decomposition. We highlight advantages in solution analysis and interpretability that is enabled by mp critical region tracking; specifically, we show that these reveal how decisions evolve geometrically across the Benders search. Our results aim to demonstrate that combining surrogate modeling with graph modeling offers a promising and extensible foundation for structure-exploiting decomposition. In addition, by decomposing the problem into more tractable subproblems, the proposed approach also aims to overcome scalability issues of mp. Finally, the use of mp surrogates provides a unifying and modular optimization framework that enables the representation of heterogeneous node subproblems as modeling objects with a homogeneous structure.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2019

A Multi-Parametric optimization approach for bilevel mixed-integer linear and quadratic programming problems

Optimization problems involving two decision makers at two different decision levels are referred to as bi-level programming problems. Herein, we present novel algorithms for the exact and global solution of two classes of bi-level programming problems, namely (i) bi-level mixed-integer linear programming problems (B-MILP) and (ii) bi-level mixed-integer convex quadratic programming problems (B-MIQP) containing both integer and bounded continuous variables at both optimization levels. Based on multi-parametric programming theory, the main idea is to recast the lower level problem as a multi-parametric programming problem, in which the optimization variables of the upper level problem are considered as bounded parameters for the lower level. The resulting exact multi-parametric mixed-integer linear or quadratic solutions are then substituted into the upper level problem, which can be solved as a set of single-level, independent, deterministic mixed-integer optimization problems. Extensions to problems including right-hand-side uncertainty on both lower and upper levels are also discussed. Finally, computational implementation and studies are presented through test problems.

42 ENGINEERING↗

DOE OSTI2024

Multi-parametric analysis for mixed integer linear programming: An application to transmission upgrade and congestion management

Upgrading the capacity of existing transmission lines is essential for meeting the growing energy demands, facilitating the integration of renewable energy, and ensuring the security of the transmission system. This study focuses on the selection of lines whose capacities and by how much should be expanded from the perspective of the Independent System Operators (ISOs) to minimize the total system cost. We employ advanced multi-parametric programming and an enhanced branch-and-bound algorithm to address complex mixed-integer linear programming (MILP) problems, considering multi-period time constraints and physical limitations of generators and transmission lines. To characterize the various decisions in transmission expansion, we model the increased capacity of existing lines as parameters within a specified range. This study first relaxes the binary variables to continuous variables and applies the Lagrange method and Karush-Kuhn-Tucker (KKT) conditions to obtain optimal solutions and identify critical regions associated with active and inactive constraints. Moreover, we extend the traditional branch-and-bound (B&B) method by determining the problem’s upper and lower bounds at each node of the B&B decision tree, helping to manage computational challenges in large-scale MILP problems. Here, we compare the difference between the upper and lower bounds to obtain an approximate optimal solution within the decision-makers’ tolerable error range. In addition, the first derivative of the objective function on the parameters of each line is used to inform the selection of lines for easing congestion and maximizing social welfare. Finally, the capacity upgrades are selected by weighing the reductions in system costs against the expense of upgrading line capacities. The findings are supported by numerical simulations and provide transmission-line planners with decision-making guidance.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2019

A Bi-Level Formulation And Solution Method For The Integration Of Process Design And Scheduling

Design and operation decisions of manufacturing plants can often be expressed within a hierarchical structure, where optimal decisions at lower operating level provide constraints for the decision raking at a higher decision level. In this work, we are addressing the case of a hierarchical design and scheduling optimization problem. The integration of design and scheduling decisions can play a big role in designing economically profitable plants and improving their operational performance. This problem can be expressed as a bi-level problem, where design related decisions occur at the upper level and operational scheduling decisions at the lower level. Since discrete decisions are involved in both optimization levels, the resulting formulation typically corresponds to bi-level mixed-integer programming problems (B-MIP). The solution of B-MIP problems is very challenging, and typically requires the use of global optimization techniques, with many algorithms not able to guarantee feasibility. To overcome the challenges in solving this class of problems we propose the use of a multi-parametric based algorithm for the solution of bi-level mixed-integer linear and quadratic programming problems, capable of providing the exact and global solution. The main idea of this approach is to treat the lower scheduling problem as a multi-parametric programming problem in which the design decisions are considered as parameters. Here, the resulting parametric solutions can then be substituted into the upper level design problem, which can be solved as a set of single-level deterministic programming problems. Through the developed formulation and algorithm we are able to supply the decision makers with the exact solution of the integrated design and scheduling problem.

42 ENGINEERING↗

DOE OSTI2022

A parametric approach to identify synergistic domains of process intensification for reactive separation

Process intensification aims to combine multiple tasks within multi-functional units to drastically improve economic, energy or sustainability metrics of a chemical process. Limited work exists to systematically identify the synergistic domains where intensification outperforms its nonintensified counterpart. In this work, we computationally derive the synergistic domains of a reactive separation system. Specifically, we first postulate general models for both intensified and nonintensified systems. We use these models to generate data to train a ReLU-type artifical neural network (ANN). Further, the trained ReLU-NN model is formulated as a multi-parametric mixed-integer linear program (mp-MILP), and the critical regions of this mp-MILP define the synergistic feasible domains of intensification. We have derived these synergistic domains of vapor–liquid equilibrium (VLE)-based reactive separation for several industrial applications. These synergistic domains enable quick screening of properties that favor intensification.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1980

Analytical investigation of efficiency and performance limits in klystron amplifiers using multidimensional computer programs; multi-stage depressed collectors; and thermionic cathode life studies

An extensive parametric investigation was performed of the extraction of energy in output gaps of klystron amplifiers, using 3-D computer programs. Due to complexity of the program which used a hydrodynamic, axially and radially deformable disk ring model and the resulting long computing time, the investigation was limited to the output gap, by far the most important and difficult part of the klystron interaction. Results show that, for a confined flow focused beam throughout the penultimate cavity, radial velocities remain very small and the beam is highly laminar. It was, therefore, concluded that possible errors resulting from treating only the output cavity in 3-D would remain small.

Kosmahl, H. G.↗

NASA NTRS2013

Development of FIAT-Based Parametric Thermal Protection System Mass Estimating Relationships for NASA's Multi-Mission Earth Entry Concept

Part of NASAs In-Space Propulsion Technology (ISPT) program is the development of the tradespace to support the design of a family of multi-mission Earth Entry Vehicles (MMEEV) to meet a wide range of mission requirements. An integrated tool called the Multi Mission System Analysis for Planetary Entry Descent and Landing or M-SAPE tool is being developed as part of Entry Vehicle Technology project under In-Space Technology program. The analysis and design of an Earth Entry Vehicle (EEV) is multidisciplinary in nature, requiring the application many disciplines. Part of M-SAPE's application required the development of parametric mass estimating relationships (MERs) to determine the vehicle's required Thermal Protection System (TPS) for safe Earth entry. For this analysis, the heat shield was assumed to be made of a constant thickness TPS. This resulting MERs will then e used to determine the pre-flight mass of the TPS. Two Mers have been developed for the vehicle forebaody. One MER was developed for PICA and the other consisting of Carbon Phenolic atop an Advanced Carbon-Carbon composition. For the the backshell, MERs have been developed for SIRCA, Acusil II, and LI-900. How these MERs were developed, the resulting equations, model limitations, and model accuracy are discussed in this poster.

Sepka, Steven A.↗

NASA NTRS2005

Results of the JIMO Follow-on Destinations Parametric Studies

NASA's proposed Jupiter Icy Moon Orbiter (JIMO) mission currently in conceptual development is to be the first one of a series of highly capable Nuclear Electric Propulsion (NEP) science driven missions. To understand the implications of a multi-mission capability requirement on the JIMO vehicle and mission, the NASA Prometheus Program initiated a set of parametric high-level studies to be followed by a series of more in-depth studies. The JIMO potential follow-on destinations identified include a Saturn system tour, a Neptune system tour, a Kuiper Belt Objects rendezvous, an Interstellar Precursor mission, a Multiple Asteroid Sample Return and a Comet Sample Return. This paper shows that the baseline JIMO reactor and design envelop can satisfy five out of six of the follow-on destinations. Flight time to these destinations can significantly be reduced by increasing the launch energy or/and by inserting gravity assists to the heliocentric phase.

nuclear electric propulsion↗

DOE OSTI2019

Rapid-Cycling Synchrotron for Multi-Megawatt Proton Facility at Fermilab

The Fermilab accelerator complex delivers intense high-energy proton beams to a variety of fixed-target scientific programs, including a flagship long-baseline neutrino program. With the advent of the Deep Underground Neutrino Experiment (DUNE) and Long Baseline Neutrino Facility (LBNF) program there is strong motivation for a 2.4 MW beam power upgrade of the Fermilab proton facility. We show the Fermilab proton facility can achieve 2.4~MW with a new rapid-cycling synchrotron (RCS) to replace the Fermilab Booster and we provide a comprehensive technical analysis of the RCS-based facility design. Past design efforts and operational experience at the Fermilab Booster, J-PARC RCS, and Oak Ridge SNS are leveraged to provide strong empirical precedent for the design. We provide a parametric study of slip-stacking accumulation, RCS extraction energy, space-charge limits, beampipe aperture, eddy current heating, injection foil heating, and lattice optics. The 2.4 MW benchmark for the long baseline neutrino program is achieved independently of a previously proposed multi-GeV linac program, but we assess the impact the linac upgrade would have on RCS performance.

43 PARTICLE ACCELERATORS↗

DOE OSTI2019

Rapid-Cycling Synchrotron for Multi-Megawatt Proton Facility at Fermilab

The Fermilab accelerator complex delivers intense high-energy proton beams to a variety of fixed-target scientific programs, including a flagship long-baseline neutrino program. With the advent of the Deep Underground Neutrino Experiment (DUNE) and Long Baseline Neutrino Facility (LBNF) program there is strong motivation for a 2.4 MW beam power upgrade of the Fermilab proton facility. We show the Fermilab proton facility can achieve 2.4~MW with a new rapid-cycling synchrotron (RCS) to replace the Fermilab Booster and we provide a comprehensive technical analysis of the RCS-based facility design. Past design efforts and operational experience at the Fermilab Booster, J-PARC RCS, and Oak Ridge SNS are leveraged to provide strong empirical precedent for the design. We provide a parametric study of slip-stacking accumulation, RCS extraction energy, space-charge limits, beampipe aperture, eddy current heating, injection foil heating, and lattice optics. The 2.4 MW benchm ark for the long baseline neutrino program is achieved independently of a previously proposed multi-GeV linac program, but we assess the impact the linac upgrade would have on RCS performance.

43 PARTICLE ACCELERATORS↗

NASA NTRS2014

Development Of FIAT-Based Parametric Thermal Protection System Mass Estimating Relationships For NASA's Multi-Mission Earth Entry Concept

An integrated tool called the Multi Mission System Analysis for Planetary Entry Descent and Landing (M-SAPE) is being developed as part of NASAs In-Space Propulsion Technology (ISPT) program. Part of M-SAPEs development requires the formulation of mass estimating relationships (MERs) to determine the vehicle's Thermal Protection System (TPS) material and required thickness for safe Earth entry. The objective of this study was to develop MERs using simple correlations that were non-ITAR and matched as accurately as possible NASAs high-fidelity ablation modeling tool, the Fully Implicit Ablation and Thermal Analysis Program (FIAT ). These MERs would be a first-estimate for feasibility studies; it is understood that higher-fidelity modeling like FIAT would be necessary once a proposed trajectory was down-selected. The trajectory space for these MERS consisted of 840 different trajectories, and a materials heating limit was the main constraint for an allowable trajectory. MERs for the vehicle fore body included the ablating materials Phenolic Impregnated Carbon Ablator (PICA ) and Carbon Phenolic atop Advanced Carbon-Carbon. For the backshell the materials were Silicone Impregnated Reusable Ceramic Ablator (SIRCA ), Acusil II, SLA-561V, and LI-900. The MERFIAT ratio indicates MERs are accurate to within 14 percent (at one standard deviation) of FIAT prediction, and the most any MER can under-predict TPS thickness is 18.7 percent of FIAT prediction. This poster focuses on the development of these MERs, the resulting equations, model limitations, and model accuracy.

thermal protection system↗

NASA NTRS1984

Extended parametric representation of compressor fans and turbines. Volume 1: CMGEN user's manual

A modeling technique for fans, boosters, and compressors has been developed which will enable the user to obtain consistent and rapid off-design performance from design point input. The fans and compressors are assumed to be multi-stage machines incorporating front variable stators. The boosters are assumed to be fixed geometry machines. The modeling technique has been incorporated into time sharing program to facilitate its use. Because this report contains a description of the input output data, values of typical inputs, and examples cases, it is suitable as a user's manual. This report is the first of a three volume set describing the parametric representation of compressors, fans, and turbines. The titles of the three volumes are as follows: (1) Volume 1 CMGEN USER's Manual (Parametric Compressor Generator); (2) Volume 2 PART USER's Manual (parametric Turbine); (3) Volume 3 MODFAN USER's Manual (Parametric Modulating Flow Fan).

Converse, G. L.↗

NASA NTRS2013

Parametric Thermal Soak Model for Earth Entry Vehicles

The analysis and design of an Earth Entry Vehicle (EEV) is multidisciplinary in nature, requiring the application many disciplines. An integrated tool called Multi Mission System Analysis for Planetary Entry Descent and Landing or M-SAPE is being developed as part of Entry Vehicle Technology project under In-Space Technology program. Integration of a multidisciplinary problem is a challenging task. Automation of the execution process and data transfer among disciplines can be accomplished to provide significant benefits. Thermal soak analysis and temperature predictions of various interior components of entry vehicle, including the impact foam and payload container are part of the solution that M-SAPE will offer to spacecraft designers. The present paper focuses on the thermal soak analysis of an entry vehicle design based on the Mars Sample Return entry vehicle geometry and discusses a technical approach to develop parametric models for thermal soak analysis that will be integrated into M-SAPE. One of the main objectives is to be able to identify the important parameters and to develop correlation coefficients so that, for a given trajectory, can estimate the peak payload temperature based on relevant trajectory parameters and vehicle geometry. The models are being developed for two primary thermal protection (TPS) materials: 1) carbon phenolic that was used for Galileo and Pioneer Venus probes and, 2) Phenolic Impregnated Carbon Ablator (PICA), TPS material for Mars Science Lab mission. Several representative trajectories were selected from a very large trade space to include in the thermal analysis in order to develop an effective parametric thermal soak model. The selected trajectories covered a wide range of heatload and heatflux combinations. Non-linear, fully transient, thermal finite element simulations were performed for the selected trajectories to generate the temperature histories at the interior of the vehicle. Figure 1 shows the finite element model that was used for the simulations. The results indicate that it takes several hours for the thermal energy to soak into the interior of the vehicle and achieve maximum payload temperatures. In addition, a strong correlation between the heatload and peak payload container temperature is observed that will help establishing the parametric thermal soak model.

Agrawal, Parul↗

NASA NTRS1988

Advancements in frequency-domain methods for rotorcraft system identification

A new method for frequency-domain identification of rotorcraft dynamics is presented. Nonparametric frequency-response identification and parametric tranfer-function modeling methods are extended to allow the extraction of state-space (stability and control derivative) representations. An interactive computer program DERIVID is described for the iterative solution of the multi-input/multi-output frequency-response matching approach used in the identification. Theoretical accuracy methods are used to determine the appropriate model structure and degree-of-confidence in the identified parameters. The method is applied to XV-15 tilt-rotor aircraft data in hover. Bare-airframe stability and control derivatives for the lateral/directional dynamics are shown to compare favorably with models previously obtained using time-domain identification methods and the XV-15 simulation program.

Tischler, Mark B.↗

NASA NTRS1989

Advancements in frequency-domain methods for rotorcraft system identification

A new method for frequency-domain identification of rotorcraft dynamics is presented. Nonparametric frequency-response identification and parametric transfer-function modeling methods are extended to allow the extraction of state-space (stability and control derivative) representations. An interactive computer program DERIVID is described for the iterative solution of the multi-input/multi-output frequency-response matching approach used in the identification. Theoretical accuracy methods are used to determine the appropriate model structure and degree-of-confidence in the identified parameters. The method is applied to XV-15 tilt-rotor aircraft data in hover. Bare-airframe stability and control derivatives for the lateral/directional dynamics are shown to compare favorably with models previously obtained using time-domain identification methods and the XV-15 simulation program.

Tischler, Mark B.↗

NASA NTRSDate not supplied

Recent Updates to the Object Reentry Survival Analysis Tool (ORSAT) Version 7.1

The Object Reentry Survival Analysis Tool (ORSAT) code is maintained and used by the NASA Orbital Debris Program Office (ODPO) and has been under continuous development and improvement since the mid-1990s. ORSAT is an object-oriented reentry simulation tool; it models a satellite as a collection of discrete components that follow independent trajectories upon the breakup of the parent object. Version 7.1 of the tool incorporates five years of new thermal and aerodynamic model development, multi-processor parametric study capability, codebase upgrades, and numerous bug-fixes. The thermal demise model was completely rewritten using a forward-time/central-space numerical stencil and incorporating a new pyrolysis model for fiber-reinforced plastic (FRP) materials. New aerodynamic and aeroheating models for hollow cylinders and hollow square prisms were developed using a combination of flow simulations in the direct simulation Monte Carlo (DSMC) Analysis Code (DAC) and Data Parallel Line Relaxation (DPLR) code and free-flight tests in the University of Texas at San Antonio’s Hypersonic Wind Tunnel. The latest version also incorporates a mechanical, strength-based demise model for FRP materials. Minor improvements include an update to the Fortran 2018 codebase; improved integration and speed with the Python-based, multi-core, parametric study tool, AutoORSAT; and fixes for many minor bugs. This new version of ORSAT will enable more accurate reentry risk assessments for modern satellites. This paper presents an overview of these changes and a summary of the verification and validation performed on the final code.

Benton R. Greene↗

NASA NTRSDate not supplied

Recent Updates to the Object Reentry Survival Analysis Tool (ORSAT) Version 7.1

Benton R. Greene↗

NASA NTRS1972

Isotope Brayton electric power system for the 500 to 2500 watt range.

An extensive study was conducted at the Lewis Research Center to evaluate an isotope Brayton electric power system for use in the 500 to 2500 W power range. The study emphasized overall system simplicity in order to reduce parasitic power losses and improve system reliability. The study included detailed parametric cycle analysis, conceptual component designs, and evaluation of system packaging. The study has resulted in the selection of a single-loop system (gas) with six major components including one rotating unit. Calculated net system efficiency varies from 23 to 28% over the power range. The use of the Pu-238 heat source being developed for the Multi-Hundred-Watt Radioisotope Thermoelectric Generator program was assumed.

Macosko, R. P.↗