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Results for “McCormick relaxations”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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18 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2016

Differentiable McCormick relaxations

McCormick's classical relaxation technique constructs closed-form convex and concave relaxations of compositions of simple intrinsic functions. These relaxations have several properties which make them useful for lower bounding problems in global optimization: they can be evaluated automatically, accurately, and computationally inexpensively, and they converge rapidly to the relaxed function as the underlying domain is reduced in size. They may also be adapted to yield relaxations of certain implicit functions and differential equation solutions. However, McCormick's relaxations may be nonsmooth, and this nonsmoothness can create theoretical and computational obstacles when relaxations are to be deployed. This article presents a continuously differentiable variant of McCormick's original relaxations in the multivariate McCormick framework of Tsoukalas and Mitsos. Gradients of the new differentiable relaxations may be computed efficiently using the standard forward or reverse modes of automatic differentiation. Furthermore, extensions to differentiable relaxations of implicit functions and solutions of parametric ordinary differential equations are discussed. A C++ implementation based on the library MC++ is described and applied to a case study in nonsmooth nonconvex optimization.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Tightening McCormick Relaxations Toward Global Solution of the ACOPF Problem

Here, we show that a strong upper bound on the objective of the alternating current optimal power flow (ACOPF) problem can significantly improve the effectiveness of optimization-based bounds tightening (OBBT) on a number of relaxations. We additionally compare the performance of relaxations of the ACOPF problem, including the rectangular form without reference bus constraints, the rectangular form with reference bus constraints, and the polar form. We find that relaxations of the rectangular form significantly strengthen existing relaxations if reference bus constraints are included. Overall, relaxations of the polar form perform the best. However, neither the rectangular nor the polar form dominates the other. Ultimately, with these strategies, we are able to reduce the optimality gap to less than 0.1% on all but 5 NESTA test cases with up to 300 buses by performing OBBT alone.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2021

Substation-Level Grid Topology Optimization Using Bus Splitting

Operations of substation circuit breakers are important for maintenance needs and topology reconfiguration in power systems. Bus splitting is one type of topology change where the two bus bars at a substation can become electrically disconnected under certain actions of circuit breakers. Because these events involve detailed substation modeling, they are typically not considered in routine power system operation and control. In this paper, an improved substation-level topology optimization framework is developed by expanding traditional line switching decisions by breaker-level bus splitting, which can further reduce grid congestion and generation costs. A tight McCormick relaxation is proposed to reformulate the bilinear terms in the resultant optimization problem to linear inequality constraints. Thus, a tractable mixed-integer linear program reformulation is attained that allows for efficient solutions in real-time operations. Numerical studies on the IEEE 14-bus and 118-bus systems demonstrate the computational performance and economic benefits of the proposed topology optimization approach.

bus split↗
DOE OSTI2021

Substation-Level Grid Topology Optimization Using Bus Splitting: Preprint

Operations of substation circuit breakers are of high significance for performing system maintenance and topology optimization. Bus splitting is one type of topology changes where the two bus-bars at a substation become electrically disconnected after certain actions of circuit breakers. As these events involve detailed substation modeling, they are not typically considered in power system routine operation and control. In this paper, an improved substation-level topology optimization is developed by expanding traditional line switching with breaker-level bus splitting, which can further reduce grid congestion and generation costs. A tight McCormick relaxation is proposed to reformulate the bi-linear terms in the resultant topology optimization model. Thus, a tractable mixed-integer linear program formulation is presented which can be efficiently solved for real-time control. Numerical studies on the IEEE 14-bus and 118-bus systems demonstrate the performance and economic benefits of the proposed topology optimization approach.

bus split↗
DOE OSTI2019

Preliminary analysis of market potential for a hydropneumatic ground-level integrated diverse energy storage system

With the increasing share of renewable energy in the power grid, energy storage is considered a viable and effective solution to mitigate intermittency and improve power system stability. To quantify and estimate market potential for a recently developed Ground-Level Integrated Diverse Energy Storage (GLIDES) technology, the first operation model is developed for this hydropneumatic system and the bilinear pressure-volume terms are then linearized using the binary McCormick Relaxation method to improve solving efficiency. Based on up-to-date 5-min price data in four different real-time markets, several sets of experiments have been designed with three objective scenarios: maximize arbitrage, maximize arbitrage + reserve, maximize arbitrage + reserve + regulation. The experimental results show that the monetary value of GLIDES could be boosted by participation in ancillary markets and that regulation service might contribute more than half of the total revenue, even as high as 90% in PJM and ISONE markets. In addition, a case study for 1-day operation is used to illustrate the volume control of the GLIDES system.

25 ENERGY STORAGE↗
DOE OSTI2022

Sizing battery energy storage and PV system in an extreme fast charging station considering uncertainties and battery degradation

In this paper, we present mixed integer linear programming (MILP) formulations to obtain optimal sizing for a battery energy storage system (BESS) and solar generation system in an extreme fast charging station (XFCS) to reduce the annualized total cost. The proposed model characterizes a typical year with eight representative scenarios and obtains the optimal energy management for the station and BESS operation to exploit the energy arbitrage for each scenario. Contrasting extant literature, this paper proposes a constant power constant voltage (CPCV) based improved probabilistic approach to model the XFCS charging demand for weekdays and weekends. This paper also accounts for the monthly and annual demand charges based on realistic utility tariffs. Furthermore, BESS life degradation is considered in the model to ensure no replacement is needed during the considered planning horizon. Different from the literature, this paper offers pragmatic MILP formulations to tally BESS charge/discharge cycles using the cumulative charge/discharge energy concept. McCormick relaxations and the Big-M method are utilized to relax the bi-linear terms in the BESS operational constraints. Finally, a robust optimization-based MILP model is proposed and leveraged to account for uncertainties in electricity price, solar generation, and XFCS demand. Case studies were performed to signify the efficacy of the proposed formulations.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2025

McCormick envelopes in mixed-integer PDE-constrained optimization

McCormick envelopes are a standard tool for deriving convex relaxations of optimization problems that involve polynomial terms. Such McCormick relaxations provide lower bounds, for example, in branch-and-bound procedures for mixed-integer nonlinear programs but have not gained much attention in PDE-constrained optimization so far. This lack of attention may be due to the distributed nature of such problems, which on the one hand leads to infinitely many linear constraints (generally state constraints that may be difficult to handle) in addition to the state equation for a pointwise formulation of the McCormick envelopes and renders bound-tightening procedures that successively improve the resulting convex relaxations computationally intractable. We analyze McCormick envelopes for a model problem class that is governed by a semilinear PDE involving a bilinearity and integrality constraints. We approximate the nonlinearity and in turn the McCormick envelopes by averaging the involved terms over the cells of a partition of the computational domain on which the PDE is defined. This yields convex relaxations that underestimate the original problem up to an a priori error estimate that depends on the mesh size of the discretization. These approximate McCormick relaxations can be improved by means of an optimization-based bound-tightening procedure. We show that their minimizers converge to minimizers to a limit problem with a pointwise formulation of the McCormick envelopes when driving the mesh size to zero. We provide a computational example, for which we certify all of our imposed assumptions. The results point to both the potential of the methodology and the gaps in the research that need to be closed. Our methodology provides a framework first for obtaining pointwise underestimators for nonconvexities and second for approximating them with finitely many linear inequalities in an infinite-dimensional setting.

Approximations and Expansions↗
DOE OSTI2016

Bounding the solutions of parametric weakly coupled second-order semilinear parabolic partial differential equations

Here in this paper, two novel techniques for bounding the solutions of parametric weakly coupled second-order semilinear parabolic partial differential equations are developed. The first provides a theorem to construct interval bounds, while the second provides a theorem to construct lower bounds convex and upper bounds concave in the parameter. The convex/concave bounds can be significantly tighter than the interval bounds because of the wrapping effect suffered by interval analysis in dynamical systems. Both types of bounds are computationally cheap to construct, requiring solving auxiliary systems twice and four times larger than the original system, respectively. An illustrative numerical example of bound construction and use for deterministic global optimization within a simple serial branch-and-bound algorithm, implemented numerically using interval arithmetic and a generalization of McCormick's relaxation technique, is presented. Finally, problems within the important class of reaction-diffusion systems may be optimized with these tools.

42 ENGINEERING↗
DOE OSTI2025

A Fast Dynamic Internal Predictive Power Scheduling Approach for Power Management in Microgrids: Preprint

This paper presents a Dynamic Internal Predictive Power Scheduling (DIPPS) approach for optimizing power management in microgrids, particularly focusing on external power exchanges among diverse prosumers. DIPPS utilizes a dynamic objective function with a time-varying binary parameter to control the timing of power transfers to the external grid, facilitated by efficient usage of energy storage for surplus renewable power. The microgrid power scheduling problem is modeled as a mixed-integer nonlinear programming (MINLP-PS) and subsequently transformed into a mixed-integer linear programming (MILPPS) optimization through McCormick's relaxation to reduce computational complexity. A predictive window window with 6 data points is solved at an average of 0.92s, a 97.6% improvement over the 38.27s required for the MINLP-PS formulation, implying the numerical feasibility of the DIPPS approach for real-time implementation. Finally, the approach is validated against a static objective using real-world load data across three case studies with different time-varying parameters, demonstrating the ability of DIPPS to optimize power exchanges and efficiently utilize distributed resources while shifting the external power transfers to specified time durations.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2019

An adaptive, multivariate partitioning algorithm for global optimization of nonconvex programs

We develop an adaptive, multivariate partitioning algorithm for solving nonconvex, Mixed-Integer Nonlinear Programs (MINLPs) with polynomial functions to global optimality. In particular, we present an iterative algorithm that exploits piecewise, convex relaxation approaches via disjunctive formulations to solve MINLPs that is different than conventional spatial branch-and-bound approaches. The algorithm partitions the domains of variables in an adaptive and non-uniform manner at every iteration to focus on productive areas of the search space. Furthermore, domain reduction techniques based on sequential, optimization-based bound-tightening and piecewise relaxation techniques, as a part of a presolve step, are integrated into the main algorithm. Finally, we demonstrate the effectiveness of the algorithm on well-known benchmark problems (including Pooling and Blending instances) from MINLPLib and compare our algorithm with state-of-the-art global optimization solvers. With our novel approach, we solve several large-scale instances, some of which are not solvable by state-of-the-art solvers. We also succeed in reducing the best known optimality gap for a hard, generalized pooling problem instance.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1996

Multigrid Method for Modeling Multi-Dimensional Combustion with Detailed Chemistry

A highly accurate and efficient numerical method is developed for modeling 3-D reacting flows with detailed chemistry. A contravariant velocity-based governing system is developed for general curvilinear coordinates to maintain simplicity of the continuity equation and compactness of the discretization stencil. A fully-implicit backward Euler technique and a third-order monotone upwind-biased scheme on a staggered grid are used for the respective temporal and spatial terms. An efficient semi-coarsening multigrid method based on line-distributive relaxation is used as the flow solver. The species equations are solved in a fully coupled way and the chemical reaction source terms are treated implicitly. Example results are shown for a 3-D gas turbine combustor with strong swirling inflows.

Zheng, Xiaoqing↗
NASA NTRS1989

Radiative relaxation in the polar stratosphere - The role of ozone

Radiative relaxation processes of the thermal disturbances associated with planetary waves in the Antarctic stratospheric polar region have been studied by using satellite ozone measurements of the Stratospheric Aerosol and Gas Experiment (SAGE II) and the associated temperature data from the National Meteorological Center (NMC).

Wang, Pi-Huan↗
DOE OSTI2025

Host Cavities Enhance the Photocatalytic Conversion of α‐Terpinene to Ascaridole Under Visible‐Light Irradiation

BODIPY-functionalized host molecules have been used as effective visible-light photosensitizers in the conversion of α-terpinene to ascaridole in the presence of molecular oxygen. Host-guest interactions enhance the effective local concentration of the substrate and singlet oxygen generated by the photosensitizing host. This results in up to a 28-fold increase in the rate of conversion depending on the host employed. A tetramethyl BODIPY analogue, in which guest access to the host cavity is blocked, was also employed to confirm that the cavities do indeed enhance substrate turnover. Whilst the latter photocatalyst is more efficient due to the suppression of nonradiative relaxation pathways associated with rotation of the meso-phenyl substituent, the rate enhancement induced by host-guest binding is not present. Combined, this provides insight for the future design of enhanced systems.

Main, Mawgan U↗
DOE OSTI2023

Refinery Perspective on Decarbonizing with Marine Biofuels

This work seeks to understand what biofuel production pathways a refinery might prefer to produce very low sulfur fuel oil (VLSFO) for marine applications. A comprehensive refinery optimization model was modified to allow for (1) direct blending of soy biodiesel, renewable diesel, Fischer-Tropsch diesel, and several pyrolysis oils and (2) indirect blending of all pyrolysis oils via co-processing in a fluidized catalytic cracker (FCC) and diesel hydrotreater into the marine fuel pool. Results showed that preferred pathways to bio-VLSFO production included co-processing low-quality pyrolysis oil in a FCC to blend the resulting biogenic light cycle oil, directly blending soy biodiesel, and directly blending small quantities of pyrolysis oil. Bio-VLSFO production costs were compared to those of fossil VLSFO subject to different marine fuel demands, benchmark crude oil prices, and biogenic fractions in the finished product. Given benchmark crude oil prices over 60 $/bbl, bio-VLSFO production appeared to be significantly cheaper than fossil VLSFO. Corresponding marginal abatement costs of CO 2 mostly ranging from -300 to 350 $/ton of CO 2 were also determined using a simplified but novel approach to allow for a comparison to other decarbonization strategies. In conclusion, this work indicates that low-sulfur contents in biofuels, relatively relaxed specifications for marine fuels, and current difficulties in meeting VLSFO specifications with crude oils can combine to make bio-VLSFO production cost-effective. Moreover, marine fuels appear to be a good entry point for refiners to start decarbonizing with biofuel pathways that could eventually be extended to other product pools.

09 BIOMASS FUELS↗
NASA NTRS1993

Multigrid methods for numerical simulation of laminar diffusion flames

This paper documents the result of a computational study of multigrid methods for numerical simulation of 2D diffusion flames. The focus is on a simplified combustion model, which is assumed to be a single step, infinitely fast and irreversible chemical reaction with five species (C3H8, O2, N2, CO2 and H2O). A fully-implicit second-order hybrid scheme is developed on a staggered grid, which is stretched in the streamwise coordinate direction. A full approximation multigrid scheme (FAS) based on line distributive relaxation is developed as a fast solver for the algebraic equations arising at each time step. Convergence of the process for the simplified model problem is more than two-orders of magnitude faster than other iterative methods, and the computational results show good grid convergence, with second-order accuracy, as well as qualitatively agreement with the results of other researchers.

Liu, C.↗
NASA NTRS1993

Multigrid methods for flow transition in three-dimensional boundary layers with surface roughness

The efficient multilevel adaptive method has been successfully applied to perform direct numerical simulations (DNS) of flow transition in 3-D channels and 3-D boundary layers with 2-D and 3-D isolated and distributed roughness in a curvilinear coordinate system. A fourth-order finite difference technique on stretched and staggered grids, a fully-implicit time marching scheme, a semi-coarsening multigrid method associated with line distributive relaxation scheme, and an improved outflow boundary-condition treatment, which needs only a very short buffer domain to damp all order-one wave reflections, are developed. These approaches make the multigrid DNS code very accurate and efficient. This allows us not only to be able to do spatial DNS for the 3-D channel and flat plate at low computational costs, but also to do spatial DNS for transition in the 3-D boundary layer with 3-D single and multiple roughness elements, which would have extremely high computational costs with conventional methods. Numerical results show good agreement with the linear stability theory, the secondary instability theory, and a number of laboratory experiments. The contribution of isolated and distributed roughness to transition is analyzed.

Liu, Chaoqun↗
DOE OSTI2023

Reduced dynamic complexity allows structure elucidation of an excited state of KRAS G13D

Localized dynamics of RAS, including regions distal to the nucleotide-binding site, is of high interest for elucidating the mechanisms by which RAS proteins interact with effectors and regulators and for designing inhibitors. Among several oncogenic mutants, methyl relaxation dispersion experiments reveal highly synchronized conformational dynamics in the active (GMPPNP-bound) KRAS G13D , which suggests an exchange between two conformational states in solution. Methyl and 31 P NMR spectra of active KRAS G13D in solution confirm a two-state ensemble interconverting on the millisecond timescale, with a major Pγ atom peak corresponding to the dominant State 1 conformation and a secondary peak indicating an intermediate state different from the known State 2 conformation recognized by RAS effectors. High-resolution crystal structures of active KRAS G13D and KRAS G13D -RAF1 RBD complex provide snapshots of the State 1 and 2 conformations, respectively. We use residual dipolar couplings to solve and cross-validate the structure of the intermediate state of active KRAS G13D , showing a conformation distinct from those of States 1 and 2 outside the known flexible switch regions. The dynamic coupling between the conformational exchange in the effector lobe and the breathing motion in the allosteric lobe is further validated by a secondary mutation in the allosteric lobe, which affects the conformational population equilibrium.

59 BASIC BIOLOGICAL SCIENCES↗