SEARCH · Search NASA
Results for “Asymptotic stability”
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.
Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.
Control Parameter Sensitivity Study for Inverter-Based-Resource Dominated Grids: A Small Signal Stability Approach and Framework
The growing adoption of renewable energy is driving the prevalence of inverter-based resources (IBRs) within power grids. Future power grids will integrate both grid-following IBRs (GFM-IBRs) and grid-forming IBRs (GFL-IBRs) alongside synchronous generators. Therefore, it is crucial to perform stability studies that account for all components and especially control interactions related to IBRs. Extensive research has performed to study the IBR-related stability, however, the sensitivity study of IBRs' control parameters on system stability has not been adequately studied yet, especially from a systematic way. Therefore, this paper conducts a small signal stability analysis for a generic grid with multiple types of resources and develops an analytical framework for assessing the sensitivity of control parameters affecting stability margins. To achieve that, the non-autonomous reduced-order non-linear dynamic model is developed for a generic power system with multiple synchronous generator-based resources (SGBRs), GFM-IBRs, and GFL-IBRs. Based on the analytic model, a systematic framework for parametric sensitivity on systems' asymptotic stability is developed. A parameter sensitivity analysis based on eigenvalue methods is proposed. The impact of the droop controllers of GFM-IBRs, PQ-dispatch and the PLL controller of GFL-IBR on the system asymptotic stability is discussed. This sensitivity study is aiming to provide deep insights on control parameters' impact on system stability, and gives direction for parameter tuning in case of instability.
Shock Formation and Vorticity Creation for 3d Euler
Abstract We analyze the shock formation process for the 3D nonisentropic Euler equations with the ideal gas law, in which sound waves interact with entropy waves to produce vorticity. Building on our theory for isentropic flows in [3, 4], we give a constructive proof of shock formation from smooth initial data. Specifically, we prove that there exist smooth solutions to the nonisentropic Euler equations which form a generic stable shock with explicitly computable blowup time, location, and direction. This is achieved by establishing the asymptotic stability of a generic shock profile in modulated self‐similar variables, controlling the interaction of wave families via: (i) pointwise bounds along Lagrangian trajectories, (ii) geometric vorticity structure, and (iii) high‐order energy estimates in Sobolev spaces. © 2022 Wiley Periodicals LLC.
Formation of Point Shocks for 3D Compressible Euler
We consider the 3D isentropic compressible Euler equations with the ideal gas law. We provide a constructive proof of the formation of the first point shock from smooth initial datum of finite energy, with no vacuum regions, with nontrivial vorticity present at the shock, and under no symmetry assumptions . We prove that for an open set of Sobolev‐class initial data that are a small L ∞ perturbation of a constant state, there exist smooth solutions to the Euler equations which form a generic stable shock in finite time. The blowup time and location can be explicitly computed, and solutions at the blowup time are smooth except for a single point , where they are of cusp‐type with Hölder C 1/3 regularity. Our proof is based on the use of modulated self‐similar variables that are used to enforce a number of constraints on the blowup profile, necessary to establish global existence and asymptotic stability in self‐similar variables. © 2022 Wiley Periodicals LLC.
Investigation of tearing mode stability near ideal stability boundaries via asymptotic matching techniques
A number of improvements to the TJ toroidal tearing mode code [Fitzpatrick, Phys. Plasmas 31, 102507 (2024)] are documented. The TJ code is also successfully benchmarked against the STRIDE toroidal tearing mode code [Glasser and Koleman, Phys. Plasmas 25, 082502 (2018)]. Finally, the new capabilities of the TJ code are used to investigate the stability of tearing modes in tokamak plasmas as an ideal stability boundary, associated with either an external-kink or an internal-kink mode, is approached. All elements of the tearing stability matrix are found to tend to infinity as an ideal stability boundary is approached. Furthermore, as the stability boundary is approached, the eigenfunctions of the various tearing modes in the plasma, which are decoupled by sheared plasma rotation, are all found to morph into that of the marginally stable ideal mode. However, the growth-rates and real frequencies of the various “ideal-tearing-modes” are different from one another. Moreover, the growth-rate of the ideal-tearing-mode that reconnects magnetic flux at the rational surface that lies closest to the edge of the plasma is the one that tends to a very large value as the stability boundary is approached. A relatively simple test for ideal stability that is capable of detecting stability boundaries for external-kink and internal-kink modes, even in the presence of a very close-fitting ideal wall, is described and verified.
Semiglobal Safety-Filtered Extremum Seeking With Unknown CBFs
We introduce a safe extremum-seeking (Safe ES) algorithm which achieves the minimization of an unknown objective function while ensuring that an unknown, yet measured, control barrier function (CBF) remains above an arbitrarily small negative value for all time. In other words, “practical safety” is maintained during the entire period of convergence to the constrained extremum. Our design is based on quadratic program (QP) CBF style filters for safety, which is applied in an average and estimated sense. Using nonsmooth analysis tools, we guarantee semiglobal practical asymptotic (SPA) stability of the global constrained optimum, practical convergence to the safe set if starting in a condition violating the CBF, and practical safety for all time—semiglobally—if starting in safe set. The safety result of the paper is analogous with modern notions of SPA stability, guaranteeing that, for any small violation of safety, there exist design coefficients which guarantee that such a small violation is not exceeded. The paper outlines a set of sufficient conditions on the barrier and objective functions, and by way of a Lyapunov argument, we demonstrate that nonconvex constrained optimization problems can be solved. We present these results in the setting of a static map and a dynamical system. A simulation example illustrates the results.
Distributed State Estimation Over Time-Varying Graphs: Exploiting the Age-of-Information
Here, we study the problem of designing a distributed observer for an LTI system over a time-varying communication graph. The limited existing work on this topic imposes various restrictions either on the observation model or on the sequence of communication graphs. In contrast, we propose a single-time-scale distributed observer that works under mild assumptions. Specifically, our communication model only requires strong-connectivity to be preserved over non-overlapping, contiguous intervals that are even allowed to grow unbounded over time. We show that under suitable conditions that bound the growth of such intervals, joint observability is sufficient to track the state of any discrete-time LTI system exponentially fast, at any desired rate. We also develop a variant of our algorithm that is provably robust to worst-case adversarial attacks, provided the sequence of graphs is sufficiently connected over time. The key to our approach is the notion of a "freshness-index" that keeps track of the age-of-information being diffused across the network. Such indices enable nodes to reject stale estimates of the state, and, in turn, contribute to stability of the error dynamics.
Stability and Performance of an Undersea Kite Operating in a Turbulent Flow Field
In this paper, we examine the effects of flow disturbances resulting from turbulence on the dynamic behavior of an underwater energy-harvesting kite system that executes periodic figure-8 flight. Due to the periodic nature of the kite’s operation, we begin by assessing orbital stability using the Floquet analysis and stroboscopic intersection analysis of a Poincaré section, with the former analysis performed on a simplified “unifoil” model and the latter performed on a six-degree-of-freedom (6-DOF)/flexible tether model. With periodic stability established, a frequency-domain analysis based on a linearization about the kite’s path is used to predict the quality of flight path tracking as a function of the turbulence frequency. To validate the accuracy of these simulation-based predictions under flow disturbances, we compare the predictions of the kite’s behavior against the results of small-scale tow testing experiments performed in a controlled pool environment.
Chapter 7: Learning Stable Local Volt/Var Controllers in Distribution Grids
This chapter describes a framework to synthesize provably stable local Volt/Var controllers for distributed energy resources (DERs) in power distribution grids (DGs). The goal is to control the reactive power injections of DERs to improve the system performance as quantified by a generic optimal reactive power flow (ORPF) problem. To achieve this, we jointly design for each DER the control function, which prescribes the reactive power update rule, and the equilibrium function, which approximates the ORPF solutions from local measurements of voltages and powers. We provide conditions on the equilibrium functions and the control parameters ensuring the stability of the closed-loop system. In particular, we discuss the trade-offs between each set of conditions accounting for practical considerations, like fully exploiting the DERs' generation capabilities and reducing the optimality gap. These conditions are then translated into learning constraints on the neural networks' parameters that are enforced in the training phase. We validate our framework with numerical simulations on the IEEE 37-bus network and through a comparison with an optimized version of standard piece wise linear control rules.
High Order Strong Stability Preserving MultiDerivative Implicit and IMEX Runge--Kutta Methods with Asymptotic Preserving Properties
In this article we present a class of high order unconditionally strong stability preserving (SSP) implicit two-derivative Runge--Kutta schemes and SSP implicit-explicit (IMEX) multi-derivative Runge--Kutta schemes where the time-step restriction is independent of the stiff term. The unconditional SSP property for a method of order $p>2$ is unique among SSP methods and depends on a backward-in-time assumption on the derivative of the operator. We show that this backward derivative condition is satisfied in many relevant cases where SSP IMEX schemes are desired. We devise unconditionally SSP implicit Runge--Kutta schemes of order up to $p=4$ and IMEX Runge--Kutta schemes of order up to $p=3$. For the multiderivative IMEX schemes, we also derive and present the order conditions, which have not appeared previously. The unconditional SSP condition ensures that these methods are positivity preserving, and we present sufficient conditions under which such methods are also asymptotic preserving when applied to a range of problems, including a hyperbolic relaxation system, the Broadwell model, and the Bhatnagar--Gross--Krook kinetic equation. We present numerical results to support the theoretical results on a variety of problems.
The DESC stellarator code suite. Part 1. Quick and accurate equilibria computations
Three-dimensional equilibrium codes are vital for stellarator design and operation, and high-accuracy equilibria are also necessary for stability studies. This paper details comparisons of two three-dimensional equilibrium codes: VMEC, which uses a steepest-descent algorithm to reach a minimum-energy plasma state, and DESC, which minimizes the magnetohydrodynamic (MHD) force error in real space directly. Accuracy as measured by satisfaction of MHD force balance is presented for each code, along with the computation time. It is shown that DESC is able to achieve more accurate solutions, especially near axis. The importance of higher-accuracy equilibria is shown in DESC's better agreement of stability metrics with asymptotic formulae. DESC's global Fourier–Zernike basis also yields solutions with analytic derivatives explicitly everywhere in the plasma volume, provides improved accuracy in the radial direction versus conventional finite differences and allows for exponential convergence. Further, DESC can compute a solution with the same accuracy as VMEC in order-of-magnitude less time.
Stability Analysis of Parameter Varying Genetic Toggle Switches Using Koopman Operators
The genetic toggle switch is a well known model in synthetic biology that represents the dynamic interactions between two genes that repress each other. The mathematical models for the genetic toggle switch that currently exist have been useful in describing circuit dynamics in rapidly dividing cells, assuming fixed or time-invariant kinetic rates. There is a growing interest in being able to model and extend synthetic biological function for growth conditions such as stationary phase or during nutrient starvation. As cells transition from one growth phase to another, kinetic rates become time-varying parameters. In this paper, we propose a novel class of parameter varying nonlinear models that can be used to describe the dynamics of genetic circuits, including the toggle switch, as they transition from different phases of growth. We show that there exists unique solutions for this class of systems, as well as for a class of systems that incorporates the microbial phenomena of quorum sensing. Further, we show that the domain of these systems, which is the positive orthant, is positively invariant. We also showcase a theoretical control strategy for these systems that would grant asymptotic monostability of a desired fixed point. We then take the general form of these systems and analyze their stability properties through the framework of time-varying Koopman operator theory. A necessary condition for asymptotic stability is also provided as well as a sufficient condition for instability. A Koopman control strategy for the system is also proposed, as well as an analogous discrete time-varying Koopman framework for applications with regularly sampled measurements.
Performance measurement of emerging 3- and 4-terminal tandem solar cells
Tandem solar cells are not limited to the conventional two-terminal (2-T) configuration. Multi-terminal designs like three-terminal (3-T) and four-terminal (4-T) devices have gained increasing attention in the PV community due to their relaxed current-matching requirement between subcells and their potential for enhanced energy yield. However, reliable and standardized methods for evaluating the performance of multi-terminal tandems remain underdeveloped. This work addresses this gap by providing comprehensive measurement guidelines tailored to these advanced configurations. We examine key coupling mechanisms between subcells, including the shared electrical load in 3-T devices and optical luminescent coupling in both 3-T and 4-T devices, to enable accurate and consistent performance evaluation. Furthermore, we propose two stabilized measurement methods for emerging 3-T tandem cells incorporating perovskite subcells: (1) a two-dimensional maximum-power-point tracking (MPPT) approach that continuously tracks both subcells' maximum power points (P MAX ) until convergence to stabilized outputs, and (2) a hybrid approach that combines MPPT for one subcell with stabilized current recording under fixed voltage biases near the P MAX of the other, allowing robust extraction of the overall stabilized P MAX (termed “MPPT + asymptotic P MA X scan” method). These methods directly address the dynamic current responses inherent to perovskite-containing tandems, providing a foundation for meaningful and reproducible performance comparisons.
Constraints on OPF Surrogates for Learning Stable Local Volt/Var Controllers
We consider the problem of learning local Volt/Var controllers in distribution grids (DGs). Our approach starts from learning separable surrogates that take both local voltages and reactive powers as arguments and predict the reactive power setpoints that approximate optimal power flow (OPF) solutions. We propose an incremental control algorithm and identify two different sets of slope conditions on the local surrogates such that the network is collectively steered toward desired configurations asymptotically. Our results reveal the trade-offs between each set of conditions, with coupled voltage-power slope constraints allowing an arbitrary shape of surrogate functions but risking limitations on exploiting generation capabilities, and reactive power slope constraints taking full advantage of generation capabilities but constraining the shape of surrogate functions. AC power flow simulations on the IEEE 37-bus feeder illustrate their guaranteed stability properties and respective advantages in two DG scenarios.
Unsupervised Learning for Equitable DER Control: Preprint
In the context of managing distributed energy resources (DERs) within distribution networks (DNs), this work focuses on the task of developing local controllers. We propose an unsupervised learning framework to train functions that can closely approximate optimal power flow (OPF) solutions. The primary aim is to establish specific conditions under which these learned functions can collectively guide the network towards desired configurations asymptotically, leveraging an incremental control approach. The flexibility of the proposed methodology allows to integrate fairness-driven components into the cost function associated with the OPF problem. This addition seeks to mitigate power curtailment disparities among DERs, thereby promoting equitable power injections across the network. To demonstrate the effectiveness of the proposed approach, power flow simulations are conducted using the IEEE 37-bus feeder. The findings not only showcase the guaranteed system stability but also underscore its improved overall performance.
A stable corridor for toroidal plasma compression
In this study, a toroidal plasma compressed by a collapsing flux conserver is analyzed to reveal stable scenarios of operation to high compression ratios. The resistive and ideal MHD stability is calculated in full toroidal geometry, using the asymptotic matching method in realistic conditions, and comparing with nonlinear simulations. The near edge current profile, controlled by toroidal field ramping during compression, is shown to be critical to stability due to coupling between poloidal components of the least stable mode. The extension of a length of shaft on axis is also found to be critical at high compression, as the resulting good curvature region in magnetic field stabilizes pressure driven modes that would otherwise be unstable. This work extends from previous studies, which initially showed the existence of a stable scenario, to include findings of more extensive stable zones, detailed effects of geometry, and nonlinear simulations of the instabilities. The nonlinear simulations of the compression are consistent with the linear analyses, confirming both the conservation and stability properties.
Structural stability and artificial buckling modes in topology optimization
Abstract This paper demonstrates how a strain energy transition approach can be used to remove artificial buckling modes that often occur in stability constrained topology optimization problems. To simulate the structural response, a nonlinear large deformation hyperelastic simulation is performed, wherein the fundamental load path is traversed using Newton’s method and the critical buckling load levels are estimated by an eigenvalue analysis. The goal of the optimization is to minimize displacement, subject to constraints on the lowest critical buckling loads and maximum volume. The topology optimization problem is regularized via the Helmholtz PDE-filter and the method of moving asymptotes is used to update the design. The stability and sensitivity analyses are outlined in detail. The effectiveness of the energy transition scheme is demonstrated in numerical examples.