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DOE OSTI2023

Machine learning-based ethylene and carbon monoxide estimation, real-time optimization, and multivariable feedback control of an experimental electrochemical reactor

Electrochemical reduction of CO 2 gas is a novel CO 2 utilization technique that has the potential to mitigate the global climate crisis caused by anthropogenic CO 2 emissions, and enable the large-scale storage of energy generated from renewable sources in the form of carbon-based chemicals and fuels. However, due to the complexity of the electrochemical reactions, the explicit first-principles models for CO2 reduction are not available yet, and there has been a limited effort to develop process modeling, optimization and control of CO 2 electrochemical reactors. To this end, a rotating cylinder electrode (RCE) reactor has been constructed at UCLA to understand the mass transfer and reaction kinetics effects separately on the productivity. In the RCE reactor, the applied potential strongly influences the reaction energetics and the electrode rotation speed affects the hydrodynamic boundary layer and modifies the film mass transfer coefficient, which involves convective and diffusive transport. Further, the present work aims to develop a multi-input multi-output (MIMO) control scheme for the RCE reactor that integrates techniques from artificial and recurrent neural network modeling, nonlinear optimization, and process controller design. Specifically, production rates of two products from the experimental reactor, ethylene and carbon monoxide, are controlled by manipulating two inputs, applied potential and catalyst rotation speed. Process dynamics and controllability are analyzed, a feedback control strategy is designed and the controllers are tuned accordingly. The experimental electrochemical cell is employed to gather data for process modeling and implement the multivariable control system. Finally, the experimental results are presented which demonstrate excellent closed-loop performance by the control system and regulation of the outputs at three different set-points including an economically-optimal set-point.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2018

Auxetic metamaterials from disordered networks

Recent theoretical work suggests that systematic pruning of disordered networks consisting of nodes connected by springs can lead to materials that exhibit a host of unusual mechanical properties. In particular, global properties such as Poisson's ratio or local responses related to deformation can be precisely altered. Tunable mechanical responses would be useful in areas ranging from impact mitigation to robotics and, more generally, for creation of metamaterials with engineered properties. However, experimental attempts to create auxetic materials based on pruningbased theoretical ideas have not been successful. In this study, we introduce a more realistic model of the networks, which incorporates angle-bending forces and the appropriate experimental boundary conditions. A sequential pruning strategy of select bonds in this model is then devised and implemented that enables engineering of specific mechanical behaviors upon deformation, both in the linear and in the nonlinear regimes. In particular, it is shown that Poisson's ratio can be tuned to arbitrary values. The model and concepts discussed here are validated by preparing physical realizations of the networks designed in this manner, which are produced by laser cutting 2D sheets and are found to behave as predicted. Furthermore, by relying on optimization algorithms, we exploit the networks' susceptibility to tuning to design networks that possess a distribution of stiffer and more compliant bonds and whose auxetic behavior is even greater than that of homogeneous networks. Finally, taken together, the findings reported here serve to establish that pruned networks represent a promising platform for the creation of unique mechanical metamaterials.

36 MATERIALS SCIENCE↗

DOE OSTI2017

University of Rhode Island Regional Earth Systems Center

The primary objective of this program was to establish the URI Regional Earth System Center (“Center”) that would enhance overall societal wellbeing (health, financial, environmental) by utilizing the best scientific information and technology to achieve optimal policy decisions with maximum stakeholder commitment for energy development, coastal environmental management, water resources protection and human health protection, while accelerating regional economic growth. The Center was to serve to integrate existing URI institutional strengths in energy, coastal environmental management, water resources, and human wellbeing. This integrated research, educational and public/private sector outreach Center was to focus on local, state and regional resources. The centerpiece activity of the Center was in the development and implementation of integrated assessment models (IAMs) that both ‘downscaled’ global observations and interpolated/extrapolated regional observations for analyzing the complexity of interactions among humans and the natural climate system to further our understanding and, ultimately, to predict the future state of our regional earth system. The Center was to begin by first ‘downscaling’ existing global earth systems management tools for studying the causes of local, state and regional climate change and potential social and environmental consequences, with a focus on the regional resources identified above. The Center would ultimately need to address the full feedbacks inherent in the nonlinear earth systems by quantifying the “upscaled” impacts of those regional changes on the global earth system. Through an interacting suite of computer simulations that are informed by observations from the nation’s evolving climate observatories, the Center activities integrates climate science, technology, economics, and social policy into forecasts that will inform solutions to pressing issues in regional climate change science, ‘green economy’ investment and climate policy. These project objectives were designed as part of a 5-year program, which would have constituted the initial phase for the establishment of the Center. Almost immediately (i.e. before receiving even the first year of funding) we were informed that we would not be receiving any funding beyond the initial phase; one year. This seriously impacted our ability to deliver on our objectives and, with that, a re-scoping of the Center priorities was designed to fit the 1-year constraints on funding. It was decided that, given the Center’s emphasis on building IAMs, the best way to proceed was to first focus on one particularly important component of the IAM – a natural sciences model that would be useful for research and forecasting of the circualation/ecology/biogeochemistry of RI’s coastal waters. We have succeeded on that necessarily more limited objective, as we will describe below.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS1996

Cascade Optimization Strategy for Aircraft and Air-Breathing Propulsion System Concepts

Design optimization for subsonic and supersonic aircraft and for air-breathing propulsion engine concepts has been accomplished by soft-coupling the Flight Optimization System (FLOPS) and the NASA Engine Performance Program analyzer (NEPP), to the NASA Lewis multidisciplinary optimization tool COMETBOARDS. Aircraft and engine design problems, with their associated constraints and design variables, were cast as nonlinear optimization problems with aircraft weight and engine thrust as the respective merit functions. Because of the diversity of constraint types and the overall distortion of the design space, the most reliable single optimization algorithm available in COMETBOARDS could not produce a satisfactory feasible optimum solution. Some of COMETBOARDS' unique features, which include a cascade strategy, variable and constraint formulations, and scaling devised especially for difficult multidisciplinary applications, successfully optimized the performance of both aircraft and engines. The cascade method has two principal steps: In the first, the solution initiates from a user-specified design and optimizer, in the second, the optimum design obtained in the first step with some random perturbation is used to begin the next specified optimizer. The second step is repeated for a specified sequence of optimizers or until a successful solution of the problem is achieved. A successful solution should satisfy the specified convergence criteria and have several active constraints but no violated constraints. The cascade strategy available in the combined COMETBOARDS, FLOPS, and NEPP design tool converges to the same global optimum solution even when it starts from different design points. This reliable and robust design tool eliminates manual intervention in the design of aircraft and of air-breathing propulsion engines where it eases the cycle analysis procedures. The combined code is also much easier to use, which is an added benefit. This paper describes COMETBOARDS and its cascade strategy and illustrates the capability of the combined design tool through the optimization of a subsonic aircraft and a high-bypass-turbofan wave-rotor-topped engine.

Patnaik, Surya N.↗

NASA NTRS2015

Numerical Computation of a Continuous-thrust State Transition Matrix Incorporating Accurate Hardware and Ephemeris Models

A significant body of work exists showing that providing a nonlinear programming (NLP) solver with expressions for the problem constraint gradient substantially increases the speed of program execution and can also improve the robustness of convergence, especially for local optimizers. Calculation of these derivatives is often accomplished through the computation of spacecraft's state transition matrix (STM). If the two-body gravitational model is employed as is often done in the context of preliminary design, closed form expressions for these derivatives may be provided. If a high fidelity dynamics model, that might include perturbing forces such as the gravitational effect from multiple third bodies and solar radiation pressure is used then these STM's must be computed numerically. We present a method for the power hardward model and a full ephemeris model. An adaptive-step embedded eight order Dormand-Prince numerical integrator is discussed and a method for the computation of the time of flight derivatives in this framework is presented. The use of these numerically calculated derivatieves offer a substantial improvement over finite differencing in the context of a global optimizer. Specifically the inclusion of these STM's into the low thrust missiondesign tool chain in use at NASA Goddard Spaceflight Center allows for an increased preliminary mission design cadence.

Low Thrust↗

DOE OSTI2021

Disjunctive linear separation conditions and mixed-integer formulations for aircraft conflict resolution

In this paper, we address the aircraft conflict resolution problem in air traffic control. We introduce new mixed-integer programming formulations for aircraft conflict resolution with speed, heading and altitude control which are based on disjunctive linear separation conditions. We first examine the two-dimensional aircraft conflict resolution problem with speed and heading control represented as continuous decision variables. We show that the proposed disjunctive linear separation conditions are equivalent to the classical nonlinear conditions for aircraft separation. Further, we characterise conflict-free trajectories based on aircraft velocity bounds and propose a simple pre-processing algorithm to identify aircraft pairs which are either always conflict-free, or which cannot be separated using speed and heading control only. We then incorporate altitude control and propose a lexicographic optimisation formulation that aims to minimise the number of flight level changes before resolving outstanding conflicts via two-dimensional velocity control. The proposed mixed-integer programming formulations are nonconvex, and we propose convex relaxations, decomposition methods and constraint generation algorithms to solve the two-dimensional and lexicographic optimisation formulations to guaranteed optimality. Numerical experiments on four types of conflict resolution benchmarking instances are conducted to test the performance of the proposed mixed-integer formulations. Further, the proposed method is compared against two benchmarks based on state-of-the-art approaches for the aircraft conflict resolution problem. Our numerical results show that the proposed method largely outperforms both benchmarks in terms of runtime and is able to solve significantly more instances to global optimality.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2018

Large-Time Behavior of Solutions to Vlasov-Poisson-Fokker-Planck Equations: From Evanescent Collisions to Diffusive Limit

The present contribution investigates the dynamics generated by the two-dimensional Vlasov-Poisson-Fokker-Planck equation for charged particles in a steady inhomogeneous background of opposite charges. We provide global in time estimates that are uniform with respect to initial data taken in a bounded set of a weighted L{sup 2} space, and where dependencies on the mean-free path τ and the Debye length δ are made explicit. In our analysis the mean free path covers the full range of possible values: from the regime of evanescent collisions τ→∞ to the strongly collisional regime τ→0. As a counterpart, the largeness of the Debye length, that enforces a weakly nonlinear regime, is used to close our nonlinear estimates. Accordingly we pay a special attention to relax as much as possible the τ-dependent constraint on δ ensuring exponential decay with explicit τ-dependent rates towards the stationary solution. In the strongly collisional limit τ→0, we also examine all possible asymptotic regimes selected by a choice of observation time scale. Here also, our emphasis is on strong convergence, uniformity with respect to time and to initial data in bounded sets of a L{sup 2} space. Our proofs rely on a detailed study of the nonlinear elliptic equation defining stationary solutions and a careful tracking and optimization of parameter dependencies of hypocoercive/hypoelliptic estimates.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

Efficient GPU Implementation of Automatic Differentiation for Computational Fluid Dynamics

Many scientific and engineering applications require repeated calculations of derivatives of output functions with respect to input parameters. Automatic Differentiation (AD) is a method that automates derivative calculations and can significantly speed up code development. In Computational Fluid Dynamics (CFD), derivatives of flux functions with respect to state variables (Jacobian) are needed for efficient solutions of the nonlinear governing equations. AD of flux functions on graphics processing units (GPUs) is challenging as flux computations involve many intermediate variables that create high register pressure and require significant memory traffic because of the need to store the derivatives. This paper presents a forward-mode AD method based on multivariate dual numbers that addresses these challenges and simultaneously reduces the floating-point operation count. The dimension of the multivariate dual numbers is optimized for performance. The flux computations are restructured to minimize the number of temporary variables and reduce register pressure. For effective utilization of memory bandwidth, shared memory is used to store the local flux Jacobian. This AD implementation is compared with several other Jacobian implementations on an NVIDIA V100 GPU (V100). For three-dimensional perfect-gas compressible-flow equations implemented in a practical CFD code, the AD implementation of a flux Jacobian based on multivariate dual numbers of dimension 5 outperforms all other GPU AD implementations on V100. Its performance is comparable with the optimized hand-differentiated version. Finally, the implementation achieves 75% of the peak floating-point throughput and 61 % of the peak global device memory bandwidth usage.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2020

HYDRODYNAMIC ANALYSIS OF MACROALGAE LOCAL MODEL USING COMPUTATIONAL FLUID DYNAMICS

In this article, a local scale, fully nonlinear coupled fluid-structural interaction (FSI) sugar kelp model has been developed using a computational fluid dynamics (CFD) method. In this model, to be consistent with available experimental data, the sugar kelp is approximated as elongated rectangles with smoothed isosceles triangles at the ends and a single kelp model with one end fixed in a channel with constant current model is developed. Several different current speeds are simulated, and the resulting drag forces and calculated drag coefficients are validated by comparison with experimental data from the literature. In a previous study, a global scale model was developed using a computational structural dynamics (CSD) method to simulate macroalgae farming system and guide the system configuration design. In the global scale model, the hydrodynamic forces are calculated using Morison’s equation and the kinematics and dynamics of the sugar kelp are simplified and the group of kelps attached to the long line is modeled as a slender structure with the same length and an effective diameter such that the volumes are consistent with the real physical system. This simplified model matches the weight and buoyancy but adjusting the hydrodynamic properties when the general hydrodynamic coefficients are employed. Therefore, optimal hydrodynamic coefficients used in global scale model were determined to obtain the hydrodynamic force more accurately. The validated local scale model is then be applied to determine the hydrodynamic coefficients of the simplified sugar kelp model for global dynamic analysis.

hydrodynamic modeling, macroalgae, CFD model↗

DOE OSTI2022

Dryness controls temperature-optimized gross primary productivity across vegetation types

Temperature response of gross primary productivity (GPP) is a well-known property of ecosystem, but GPP at the optimum temperature (GPP_T opt ) has not been fully discussed. Our understanding of how GPP_T opt responds to warming and water availability is highly limited. Here, in this study, we analyzed data at 326 globally distributed eddy covariance sites (79°N-37°S), to identify controlling factors of GPP_T opt . Although GPP_T opt was significantly influenced by soil moisture, global solar radiation, mean annual temperature, and vapor pressure deficit in a non-linear pattern (R 2 = 0.47), the direction and magnitude of these climate variables’ effects on GPP_T opt depend on the dryness index (DI), a ratio of potential evapotranspiration to precipitation. The spatial pattern showed that soil moisture did not affect GPP_T opt across energy-limited sites with DI < 1 while dominated GPP_T opt across water-limited sites with DI >1. The temporal pattern showed that GPP_T opt was lowered by warming or low precipitation in water-limited sites while energy-limited sites tended to maintain a stable GPP_T opt regardless of changes in air temperature. Vegetation types in humid climates tended to have higher GPP_T opt and were more likely to benefit from a warmer climate since it was not restricted by water conditions. This study highlights that the response of GPP_T opt to global warming depends on the dryness conditions, which explains the nonlinear control of water and temperature over GPP_T opt . Our finding is essential to realistic prediction of terrestrial carbon uptake under future climate and vegetation conditions.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2025

Exploring the transition from continuous turbulence fluctuations to bursting ELMs in high SOL density regimes

BOUT++ turbulence simulations of the DIII-D reveal that the density profile between the separatrix and pedestal plays a crucial role in the dynamics of edge localized modes (ELMs) and edge plasma turbulent transport. Nonlinear simulations demonstrate that small ELMs in the DIII-D hybrid scenario under high SOL density conditions are predominantly driven by local ballooning modes near the separatrix, stabilizing global instabilities while enhancing localized pressure fluctuations. A key control parameters for ELM dynamics is the separatrix-to-pedestal density ratio, n e,sep /n e,ped . A high ratio indicates a shallow gradient, favoring small ELMs, while a lower ratio signals a steep gradient, which increases the likelihood of large ELMs. Comprehensive parameter scans, including n e,sep /n e,ped , density gradient profiles near the separatrix, and resistivity, reveal the critical role of these parameters in shaping transitions between turbulence-driven transport and ELM bursting. The scans demonstrate that in high SOL density regimes, small ELMs can result from either global resistive MHD instabilities or local ballooning modes near the separatrix, depending on the steepness of the separatrix density gradient. These findings also highlight the transition from continuous turbulence to small ELMs. The post-crash peak in pressure fluctuations, δP rms serves as a critical metric for identifying transition from continuous turbulence fluctuations to ELM bursting. Larger δP rms values correlate with ELM bursts driven by local or global instabilities, whereas smaller values indicate turbulence-dominated transport. Drift-Alfvén and resistive ballooning turbulence enhance the entrainment of fluctuations from the pedestal to the SOL, contributing to the complex interplay of dynamics in this regime. These findings emphasize the importance of separatrix density shaping and pedestal gradient control for optimizing ELM behavior in ITER and future fusion devices.

Li, Nami [Lawrence Livermore National Laboratory (↗

DOE OSTI2021

Hybridizing Machine Learning and Physically-based Earth System Models to Improve Prediction of Multivariate Extreme Events (AI Exploration of Wildland Fire Prediction)

Focal Areas: This project responds to two focal areas identified in the DOE Call for AI4ESP White Papers: 1) Predictive modeling through the use of artificial intelligence (AI) techniques, and 2) insights gleaned from complex data using explainable AI and big data analytics. Science Challenge: Large wildland fires (hereafter wildfires) appearing as high-impact compound climate extreme events are closely related to hydroclimate and water cycle extremes that modulate surface fuel supply and combustibility. These compound events have multivariate climatic features (e.g., temperature, precipitation, relative humidity, wind, lightning) and societal drivers (e.g., forest management, land use change, human caused ignitions). Meanwhile, they induce strong feedbacks to the coupled atmosphere, biosphere, and hydrosphere by perturbing regional and global radiation budget as well as ecological, biogeochemical, and water cycles across multiple spatiotemporal scales. The nonlinear interactions between these natural and anthropogenic components of the Earth system are too complex to be completely and adequately represented in today’s Earth system models (ESMs). The inherent stochastic nature of fire activity at all scales further increases the difficulty of its prediction using ESMs that are usually developed from deterministic equations and parameterizations. Besides, concurrence of long-term (decadal to interdecadal) global climate change and fire regime shifts overlapping with short-term (intraseasonal to interannual) variations of regional fire weather and burning activity confound predictability of these compound extreme events. We propose to address the above scientific challenges by using machine learning (ML)-based data-driven modeling techniques to integrate observations and physically-based ESMs’ simulations in a computationally efficient hybrid prediction system. This prediction system is supposed to characterize the wildfire’s sensitivity to climate and exogenous drivers at high resolution (~ 0.25°) on subseasonal to seasonal (S2S) timescales providing improved predictability and explainability. We will use the system to help identify: (1) What are the computational elements of a hybrid system needed to predict compound climate extreme events such as global wildfires? (2) What are the key drivers (either natural or anthropogenic) that modulate short-term variations of multivariate fire weather and burning activity over different regions? How can one take advantage of those driver-response relationships to improve the predictability of large wildfires on S2S time scales? (3) What are the underlying physical mechanisms and sources of improved predictability? Which ML techniques are optimal in revealing and adapting these mechanisms?

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS2001

Discrete-State Simulated Annealing For Traveling-Wave Tube Slow-Wave Circuit Optimization

Algorithms based on the global optimization technique of simulated annealing (SA) have proven useful in designing traveling-wave tube (TWT) slow-wave circuits for high RF power efficiency. The characteristic of SA that enables it to determine a globally optimized solution is its ability to accept non-improving moves in a controlled manner. In the initial stages of the optimization, the algorithm moves freely through configuration space, accepting most of the proposed designs. This freedom of movement allows non-intuitive designs to be explored rather than restricting the optimization to local improvement upon the initial configuration. As the optimization proceeds, the rate of acceptance of non-improving moves is gradually reduced until the algorithm converges to the optimized solution. The rate at which the freedom of movement is decreased is known as the annealing or cooling schedule of the SA algorithm. The main disadvantage of SA is that there is not a rigorous theoretical foundation for determining the parameters of the cooling schedule. The choice of these parameters is highly problem dependent and the designer needs to experiment in order to determine values that will provide a good optimization in a reasonable amount of computational time. This experimentation can absorb a large amount of time especially when the algorithm is being applied to a new type of design. In order to eliminate this disadvantage, a variation of SA known as discrete-state simulated annealing (DSSA), was recently developed. DSSA provides the theoretical foundation for a generic cooling schedule which is problem independent, Results of similar quality to SA can be obtained, but without the extra computational time required to tune the cooling parameters. Two algorithm variations based on DSSA were developed and programmed into a Microsoft Excel spreadsheet graphical user interface (GUI) to the two-dimensional nonlinear multisignal helix traveling-wave amplifier analysis program TWA3. The algorithms were used to optimize the computed RF efficiency of a TWT by determining the phase velocity profile of the slow-wave circuit. The mathematical theory and computational details of the DSSA algorithms will be presented and results will be compared to those obtained with a SA algorithm.

Wilson, Jeffrey D.↗

NASA NTRS2007

Iterative-Transform Phase Retrieval Using Adaptive Diversity

A phase-diverse iterative-transform phase-retrieval algorithm enables high spatial-frequency, high-dynamic-range, image-based wavefront sensing. [The terms phase-diverse, phase retrieval, image-based, and wavefront sensing are defined in the first of the two immediately preceding articles, Broadband Phase Retrieval for Image-Based Wavefront Sensing (GSC-14899-1).] As described below, no prior phase-retrieval algorithm has offered both high dynamic range and the capability to recover high spatial-frequency components. Each of the previously developed image-based phase-retrieval techniques can be classified into one of two categories: iterative transform or parametric. Among the modifications of the original iterative-transform approach has been the introduction of a defocus diversity function (also defined in the cited companion article). Modifications of the original parametric approach have included minimizing alternative objective functions as well as implementing a variety of nonlinear optimization methods. The iterative-transform approach offers the advantage of ability to recover low, middle, and high spatial frequencies, but has disadvantage of having a limited dynamic range to one wavelength or less. In contrast, parametric phase retrieval offers the advantage of high dynamic range, but is poorly suited for recovering higher spatial frequency aberrations. The present phase-diverse iterative transform phase-retrieval algorithm offers both the high-spatial-frequency capability of the iterative-transform approach and the high dynamic range of parametric phase-recovery techniques. In implementation, this is a focus-diverse iterative-transform phaseretrieval algorithm that incorporates an adaptive diversity function, which makes it possible to avoid phase unwrapping while preserving high-spatial-frequency recovery. The algorithm includes an inner and an outer loop (see figure). An initial estimate of phase is used to start the algorithm on the inner loop, wherein multiple intensity images are processed, each using a different defocus value. The processing is done by an iterative-transform method, yielding individual phase estimates corresponding to each image of the defocus-diversity data set. These individual phase estimates are combined in a weighted average to form a new phase estimate, which serves as the initial phase estimate for either the next iteration of the iterative-transform method or, if the maximum number of iterations has been reached, for the next several steps, which constitute the outerloop portion of the algorithm. The details of the next several steps must be omitted here for the sake of brevity. The overall effect of these steps is to adaptively update the diversity defocus values according to recovery of global defocus in the phase estimate. Aberration recovery varies with differing amounts as the amount of diversity defocus is updated in each image; thus, feedback is incorporated into the recovery process. This process is iterated until the global defocus error is driven to zero during the recovery process. The amplitude of aberration may far exceed one wavelength after completion of the inner-loop portion of the algorithm, and the classical iterative transform method does not, by itself, enable recovery of multi-wavelength aberrations. Hence, in the absence of a means of off-loading the multi-wavelength portion of the aberration, the algorithm would produce a wrapped phase map. However, a special aberration-fitting procedure can be applied to the wrapped phase data to transfer at least some portion of the multi-wavelength aberration to the diversity function, wherein the data are treated as known phase values. In this way, a multiwavelength aberration can be recovered incrementally by successively applying the aberration-fitting procedure to intermediate wrapped phase maps. During recovery, as more of the aberration is transferred to the diversity function following successive iterations around the ter loop, the estimated phase ceases to wrap in places where the aberration values become incorporated as part of the diversity function. As a result, as the aberration content is transferred to the diversity function, the phase estimate resembles that of a reference flat.

Dean, Bruce H.↗

DOE OSTI2020

Inexact convex relaxations for AC optimal power flow: Towards AC feasibility

Convex relaxations of AC optimal power flow (AC-OPF) problems have attracted significant interest as in several instances they provably yield the global optimum to the original non-convex problem. If, however, the relaxation is inexact, the obtained solution is not AC-feasible. The quality of the obtained solution is essential for several practical applications of AC-OPF, but detailed analyses are lacking in existing literature. Here, this paper aims to cover this gap. We provide an in-depth investigation of the solution characteristics when convex relaxations are inexact, we assess the most promising AC feasibility recovery methods for large-scale systems, and we propose two new metrics that lead to a better understanding of the quality of the identified solutions. We perform a comprehensive assessment on 96 different test cases, ranging from 14 to 3120 buses, and we show the following: (i) Despite an optimality gap of less than 1%, several test cases still exhibit substantial distances to both AC feasibility and local optimality and the newly proposed metrics characterize these deviations. (ii) Penalization methods fail to recover an AC-feasible solution in 15 out of 45 test cases. (iii) The computational benefits of warm-starting non-convex solvers have significant variation, but a computational speedup exists in over 75% of cases.

24 POWER TRANSMISSION AND DISTRIBUTION↗