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DOE OSTI2019

On the Efficacy and High-Performance Implementation of Quaternion Matrix Multiplication

Quaternion symmetry is ubiquitous in the physical sciences. As such, much work has been afforded over the years to the development of efficient schemes to exploit this symmetry using real and complex linear algebra. Recent years have also seen many advances in the formal theoretical development of explicitly quaternion linear algebra with promising applications in image processing and machine learning. Despite these advances, there do not currently exist optimized software implementations of quaternion linear algebra. The leverage of optimized linear algebra software is crucial in the achievement of high levels of performance on modern computing architectures, and thus provides a central tool in the development of high-performance scientific software. In this work, a case will be made for the efficacy of high-performance quaternion linear algebra software for appropriate problems. In this pursuit, an optimized software implementation of quaternion matrix multiplication will be presented and will be shown to outperform a vendor tuned implementation for the analogous complex matrix operation. The results of this work pave the path for further development of high-performance quaternion linear algebra software which will improve the performance of the next generation of applicable scientific applications.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2012

Estimating the Maximum Potential Revenue for Grid Connected Electricity Storage: Arbitrage and Regulation

The valuation of an electricity storage device is based on the expected future cash flow generated by the device. Two potential sources of income for an electricity storage system are energy arbitrage and participation in the frequency regulation market. Energy arbitrage refers to purchasing (storing) energy when electricity prices are low, and selling (discharging) energy when electricity prices are high. Frequency regulation is an ancillary service geared towards maintaining system frequency, and is typically procured by the independent system operator in some type of market. This paper outlines the calculations required to estimate the maximum potential revenue from participating in these two activities. First, a mathematical model is presented for the state of charge as a function of the storage device parameters and the quantities of electricity purchased/sold as well as the quantities offered into the regulation market. Using this mathematical model, we present a linear programming optimization approach to calculating the maximum potential revenue from an electricity storage device. The calculation of the maximum potential revenue is critical in developing an upper bound on the value of storage, as a benchmark for evaluating potential trading strategies, and a tool for capital finance risk assessment. Then, we use historical California Independent System Operator (CAISO) data from 2010-2011 to evaluate the maximum potential revenue from the Tehachapi wind energy storage project, an American Recovery and Reinvestment Act of 2009 (ARRA) energy storage demonstration project. We investigate the maximum potential revenue from two different scenarios: arbitrage only and arbitrage combined with the regulation market. Our analysis shows that participation in the regulation market produces four times the revenue compared to arbitrage in the CAISO market using 2010 and 2011 data. Then we evaluate several trading strategies to illustrate how they compare to the maximum potential revenue benchmark. We conclude with a sensitivity analysis with respect to key parameters.

electricity storage↗

DOE OSTI2020

Algebraic multigrid preconditioning of the Hessian in optimization constrained by a partial differential equation

Summary We construct an algebraic multigrid (AMG) based preconditioner for the reduced Hessian of a linear‐quadratic optimization problem constrained by an elliptic partial differential equation. While the preconditioner generalizes a geometric multigrid preconditioner introduced in earlier works, its construction relies entirely on a standard AMG infrastructure built for solving the forward elliptic equation, thus allowing for it to be implemented using a variety of AMG methods and standard packages. Our analysis establishes a clear connection between the quality of the preconditioner and the AMG method used. The proposed strategy has a broad and robust applicability to problems with unstructured grids, complex geometry, and varying coefficients. The method is implemented using the Hypre package and several numerical examples are presented.

Barker, Andrew T.↗

DOE OSTI2018

Cascade energy optimization for waste heat recovery in distributed energy systems

The efficiency of distributed energy systems can be significantly increased through waste heat recovery from industry or power generation. The technologies used for this process are typically dependent on the quality and temperature grades of waste heat. To maximize the efficiency of cascade heat utilization, it is important to optimize the choice of waste heat recovery technologies and their operation. Here, a detailed mixed integer linear programming optimization model is proposed for waste heat recovery in a district-scale microgrid. The model can distinguish waste heat quality for planning and operation optimization of distributed energy systems. Heat utilization technologies are formulated in this developed model and categorized in different temperature grades. The developed model is validated using four typical cases under different settings of system operation and business models. It is found that the optimization model, by distinguishing waste heat temperature, can increase energy cost savings by around 5%, compared to models that do not consider waste heat temperature grades. Additionally, the results indicate that the developed model can provide more realistic configuration and technologies dispatch.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2024

Depot Charging Schedule Optimization for Medium- and Heavy-Duty Battery-Electric Trucks

Charge management, which lowers charging costs for fleets and prevents straining the electrical grid, is critical to the successful deployment of medium- and heavy-duty battery-electric trucks (MHD BETs). This study introduces an energy demand and cost management framework that optimizes depot charging for MHD BETs by combining an energy consumption machine learning model and a linear program optimization model. The framework considers key factors impacting real-world MHD BET operations, including vehicle and charger configurations, duty cycles, use cases, geographic and climate conditions, operation schedules, and utilities’ time-of-use (TOU) rates and demand charges. The framework was applied to a hypothetical fleet of 100 MHD BETs in California under three different utilities for 365 days, with results compared to unmanaged charging. The optimized charging solution avoided more than 90% of on-peak charging, reduced fleet charging peak load by 64–75%, and lowered fleet energy variable costs by 54–64%. This study concluded that the proposed charge management framework significantly reduces energy costs and peak loads for MHD BET fleets while making recommendations for fleet electrification infrastructure planning and the design of utility TOU rates and demand charges.

Song, Shuhan↗

DOE OSTI2013

Continuous-Time Mean-Variance Portfolio Selection with Random Horizon

This paper examines the continuous-time mean-variance optimal portfolio selection problem with random market parameters and random time horizon. Treating this problem as a linearly constrained stochastic linear-quadratic optimal control problem, I explicitly derive the efficient portfolios and efficient frontier in closed forms based on the solutions of two backward stochastic differential equations. Some related issues such as a minimum variance portfolio and a mutual fund theorem are also addressed. All the results are markedly different from those in the problem with deterministic exit time. A key part of my analysis involves proving the global solvability of a stochastic Riccati equation, which is interesting in its own right.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2024

Direct microstability optimization of stellarator devices

Turbulent transport is regarded as one of the key issues in magnetic confinement nuclear fusion, both for tokamaks and stellarators. Here, in this work, we show that a significant decrease in a microstability-based proxy, as opposed to a geometric one, for the turbulent heat flux, namely the quasilinear heat flux, can be obtained in an efficient manner by coupling stellarator optimization with linear gyrokinetic simulations. This is accomplished by computing the quasilinear heat flux at each step of the optimization process, as well as the deviation from quasisymmetry, and minimizing their sum, leading to a balance between neoclassical and the turbulent transport proxy.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2012

Sensitivity analysis and optimization of the nuclear fuel cycle

A sensitivity study has been conducted to assess the robustness of the conclusions presented in the MIT Fuel Cycle Study. The Once Through Cycle (OTC) is considered as the base-line case, while advanced technologies with fuel recycling characterize the alternative fuel cycles. The options include limited recycling in LWRs and full recycling in fast reactors and in high conversion LWRs. Fast reactor technologies studied include both oxide and metal fueled reactors. The analysis allowed optimization of the fast reactor conversion ratio with respect to desired fuel cycle performance characteristics. The following parameters were found to significantly affect the performance of recycling technologies and their penetration over time: Capacity Factors of the fuel cycle facilities, Spent Fuel Cooling Time, Thermal Reprocessing Introduction Date, and in core and Out-of-core TRU Inventory Requirements for recycling technology. An optimization scheme of the nuclear fuel cycle is proposed. Optimization criteria and metrics of interest for different stakeholders in the fuel cycle (economics, waste management, environmental impact, etc.) are utilized for two different optimization techniques (linear and stochastic). Preliminary results covering single and multi-variable and single and multi-objective optimization demonstrate the viability of the optimization scheme. (authors)

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2013

Final Scientific/Technical Report

In this work, we have built upon our results from previous DOE funding (DEFG 0204ER25655), where we developed new and more efficient methods for solving certain optimization problems with many inequality constraints. This past work resulted in efficient algorithms (and analysis of their convergence) for linear programming, convex quadratic programming, and the training of support vector machines. The algorithms are based on using constraint reduction in interior point methods: at each iteration we consider only a smaller subset of the inequality constraints, focusing on the constraints that are close enough to be relevant. Surprisingly, we have been able to show theoretically that such algorithms are globally convergent and to demonstrate experimentally that they are much more efficient than standard interior point methods.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2015

Optimal mode transformations for linear-optical cluster-state generation

In this paper, we analyze the generation of linear-optical cluster states (LOCSs) via sequential addition of one and two qubits. Existing approaches employ the stochastic linear-optical two-qubit controlled-Z (CZ) gate with success rate of 1/9 per operation. The question of optimality of the CZ gate with respect to LOCS generation has remained open. We report that there are alternative schemes to the CZ gate that are exponentially more efficient and show that sequential LOCS growth is indeed globally optimal. We find that the optimal cluster growth operation is a state transformation on a subspace of the full Hilbert space. Finally, we show that the maximal success rate of postselected entangling n photonic qubits or m Bell pairs into a cluster is (1/2) n-1 and (1/4) m-1 , respectively, with no ancilla photons, and we give an explicit optical description of the optimal mode transformations.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2022

Start-up Optimization Considering Integrated Power Electronic Systems

The startup of systems composed of power electronic systems is becoming more complex. As the systems grow, standard pre-configuration sequencing methods will struggle. This work proposes a linear programming optimization method to couple power electronic systems in a series of startup commands. This formulation is demonstrated in a real-time implementation on hardware.

Starke, Michael↗

DOE OSTI2019

An adjoint method for neoclassical stellarator optimization

Stellarators remain an enticing route to steady-state fusion power. However, to achieve the required confinement, the magnetic geometry must be highly optimized. This optimization requires navigating high-dimensional spaces, often necessitating the use of gradient-based methods. The gradient of the neoclassical fluxes is expensive to compute with classical methods, requiring $O(N)$ flux computations, where $N$ is the number of parameters. To reduce the cost of the gradient computation, we present an adjoint method for computing the derivatives of moments of the neoclassical distribution function for stellarator optimization. The linear adjoint method allows derivatives of quantities which depend on solutions of a linear system, such as moments of the distribution function, to be computed with respect to many parameters from the solution of only two linear systems. This reduces the cost of computing the gradient to the point that the finite-collisionality neoclassical fluxes can be used within an optimization loop. With the neoclassical adjoint method, we compute solutions of the drift kinetic equation and an adjoint drift kinetic equation to obtain derivatives of neoclassical quantities with respect to geometric parameters. When the number of parameters in the derivative is large ( $O(10^{2})$ ), this adjoint method provides up to a factor of 200 reduction in cost. We demonstrate adjoint-based optimization of the field strength to obtain minimal bootstrap current on a surface. With adjoint-based derivatives, we also compute the local sensitivity to magnetic perturbations on a flux surface and identify regions where tight tolerances on error fields are required for control of the bootstrap current or radial transport. Moreover, the solve for the ambipolar electric field is accelerated using a Newton method with derivatives obtained from the adjoint method.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2015

Optically resonant magneto-electric cubic nanoantennas for ultra-directional light scattering

Cubic dielectric nanoparticles are promising candidates for futuristic low-loss, ultra-compact, nanophotonic applications owing to their larger optical coefficients, greater packing density, and relative ease of fabrication as compared to spherical nanoparticles; besides possessing negligible heating at nanoscale in contrast to their metallic counterparts. Here, we present the first theoretical demonstration of azimuthally symmetric, ultra-directional Kerker's-type scattering of simple dielectric nanocubes in visible and near-infrared regions via simultaneous excitation and interference of optically induced electric- and magnetic-resonances up to quadrupolar modes. Unidirectional forward-scattering by individual nanocubes is observed at the first generalized-Kerker's condition for backward-scattering suppression, having equal electric- and magnetic-dipolar responses. Both directionality and magnitude of these unidirectional-scattering patterns get enhanced where matching electric- and magnetic-quadrupolar responses spectrally overlap. While preserving azimuthal-symmetry and backscattering suppression, a nanocube homodimer provides further directionality improvement for increasing interparticle gap, but with reduced main-lobe magnitude due to emergence of side-scattering lobes from diffraction-grating effect. We thoroughly investigate the influence of interparticle gap on scattering patterns and propose optimal range of gap for minimizing side-scattering lobes. Besides suppressing undesired side-lobes, significant enhancement in scattering magnitude and directionality is attained with increasing number of nanocubes forming a linear chain. Optimal directionality, i.e., the narrowest main-scattering lobe, is found at the wavelength of interfering quadrupolar resonances; whereas the largest main-lobe magnitude is observed at the wavelength satisfying the first Kerker's condition. These unique optical properties of dielectric nanocubes thus can revolutionize their applications at visible and near-infrared regions in the fields of nanoantennas, nanolasers, photovoltaics, and even in biomedicine.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2021

Iterative Linearization for Phasor-Defined Optimal Power Dispatch

Optimal power flow (OPF) problems, which dispatch power targets to controllable generating units across a network, must generally account for non-convex constraints on power flow. Furthermore, adapting those problems so as to make them solvable with convex optimization techniques is an area of much academic and operational interest. In this paper, we present a method for solving OPF as a quadratic program by iteratively refining and re-initializing a linearized model of power flow based on the outputs of an associated nonlinear solver. The linear model on which we demonstrate this method is an adapted version of an approximation designed for use with unbalanced distribution networks. As an important benefit, the model allows for the explicit inclusion of nodal voltage phasor values in both the OPF problem's objective and its constraints, which opens the door to the idea of phasor-based control (PBC) design. We show in simulations on the IEEE 13-node test feeder that our method quickly converges to a set of phasor targets that are sufficiently precise for use in operations at the distribution level.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2018

EnergyPlus Interior Radiant Heat Exchange Runtime Performance Improvements

EnergyPlus is the flagship whole-building energy simulation program developed by the US Department of Energy. This paper describes the various approaches to improve the performance of the interior radiant heat exchange algorithm. Vectorization, optimized Basic Linear Algebra Subprograms (BLAS) library, multithreading, and Graphical Processing Unit (GPU) computation are all investigated. The approach with the best performance, while maintaining drop-in compatibility with existing EnergyPlus code, is the optimized BLAS library. However, GPU computing has the potential to reduce simulation time by orders of magnitude with major refactoring to EnergyPlus.

Adams, Mark↗

DOE OSTI2021

Parallelized POD-based suboptimal economic model predictive control of a state-constrained Boussinesq approximation

Motivated by an energy efficient building application, we want to optimize a quadratic cost functional subject to the Boussinesq approximation of the Navier-Stokes equations and to bilateral state and control constraints. Since the computation of such an optimal solution is numerically costly, we design an efficient strategy to compute a sub-optimal (but applicationally acceptable) solution with significantly reduced computational effort. We employ an economic Model Predictive Control (MPC) strategy to obtain a feedback control. The MPC sub-problems are based on a linear-quadratic optimal control problem subjected to mixed control and state constraints and a convection-diffusion equation, reduced with proper orthogonal decomposition. Finally, to solve each sub-problem, we apply a primal-dual active set strategy. The method can be fully parallelized, which enables the solution of large problems with real-world parameters.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

Multiphase Optimal and Non-Singular Power Flow by Successive Linear Approximations: Preprint

We propose an optimal power flow method for multiphase power systems. The method works for both radial and meshed networks, and is able to accommodate different types of load/source connections such as wye, delta, etc. Moreover, the method generates a sequence of non-singular points in the electrical state space and a corresponding sequence of power injections, which can be used in real-time control settings. Under certain conditions, the sequences attain the local minimum. The method has been tested using several typical networks, including the IEEE 37-bus and 123-bus test feeders.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2013

Multigrid preconditioning of linear systems for semi-smooth Newton methods applied to optimization problems constrained by smoothing operators

This article is concerned here with the question of constructing efficient multigrid preconditioners for the linear systems arising when applying semi-smooth Newton methods to large-scale linear-quadratic optimization problems constrained by smoothing operators with box-constraints on the controls. It is shown that, for certain discretizations of the optimization problem, the linear systems to be solved at each semi-smooth Newton iteration reduce to inverting principal minors of the Hessian of the associated unconstrained problem. As in the case when box-constraints on the controls are absent, the multigrid preconditioner introduced here is shown to increase in quality as the mesh-size decreases, resulting in a number of iterations that decreases with mesh-size. However, unlike the unconstrained case, the spectral distance between the preconditioners and the Hessian is shown to be of suboptimal order in general.

97 MATHEMATICS AND COMPUTING↗