Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “linear systems”

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.

At least 451 records · Page 25
All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Distributed Data-Driven Power Iteration for Strongly Connected Networks

Here, this paper presents data-driven power iteration to distributively estimate the dominant eigenvalues of an unknown linear time-invariant system. The proposed strategy only requires a single trajectory data or measurements. Furthermore, in order to perform the distributed estimation, the communication network topology can be chosen to be any strongly connected directed graphs. The proposed data-driven power iteration is demonstrated using several numerical examples and is then applied to estimate the generalized algebraic connectivity of cooperative systems and to control the epidemic spreading.

Gusrialdi, Azwirman↗
DOE OSTI2020

Restoring Distribution System Under Renewable Uncertainty Using Reinforcement Learning

Distributed energy resources (DERs) in distribution systems, including renewable generation, micro-turbine, and energy storage, can be used to restore critical loads following extreme events to increase grid resiliency. However, properly coordinating multiple DERs in the system for multi-step restoration process under renewable uncertainty and fuel availability is a complicated sequential optimal control problem. Due to its capability to handle system non-linearity and uncertainty, reinforcement learning (RL) stands out as a potentially powerful candidate in solving complex sequential control problems. Moreover, the offline training of RL provides excellent action readiness during online operation, making it suitable to problems such as load restoration, where in-time, correct and coordinated actions are needed. In this study, a distribution system prioritized load restoration based on a simplified single-bus system is studied: with imperfect renewable generation forecast, the performance of an RL controller is compared with that of a deterministic model predictive control (MPC). Our experiment results show that the RL controller is able to learn from experience, adapt to the imperfect forecast information and provide a more reliable restoration process when compared with the baseline controller.

61 RADIATION PROTECTION AND DOSIMETRY↗
DOE OSTI2020

Restoring Distribution System Under Renewable Uncertainty Using Reinforcement Learning: Preprint

Distributed energy resources (DER) in distribution systems, including renewable generation, micro-turbine, and energy storage, can be used to restore critical loads following extreme events to increase grid resiliency. However, properly coordinating multiple DERs in the system for multi-step restoration process under renewable uncertainty and fuel availability is a complicated sequential optimal control problem. Due to its capability to handle system non-linearity and uncertainty, reinforcement learning (RL) stands out as a potentially powerful candidate in solving complex sequential control problems. Moreover, the offline training of RL provides excellent action readiness during online operation, making it suitable to problems such as load restoration, where in-time, correct and coordinated actions are needed. In this study, a distribution system prioritized load restoration based on a simplified single-bus system is studied: with imperfect renewable generation forecast, the performance of an RL controller is compared with that of a deterministic model predictive control (MPC). Our experiment results show that the RL controller is able to learn from experience, adapt to the imperfect forecast information and provide a more reliable restoration process when compared with the baseline MPC controller.

61 RADIATION PROTECTION AND DOSIMETRY↗
DOE OSTI2021

Restoring Distribution System Under Renewable Uncertainty Using Reinforcement Learning

Distributed energy resources (DER) in distribution systems, including renewable generation, micro-turbine, and energy storage, can be used to restore critical loads following extreme events to increase grid resiliency. However, properly coordinating multiple DERs in the system for multi-step restoration process under renewable uncertainty and fuel availability is a complicated sequential optimal control problem. Due to its capability to handle system non-linearity and uncertainty, reinforcement learning (RL) stands out as a potentially powerful candidate in solving complex sequential control problems. Moreover, the offline training of RL provides excellent action readiness during online operation, making it suitable to problems such as load restoration, where in-time, correct and coordinated actions are needed. In this study, a distribution system prioritized load restoration based on a simplified single-bus system is studied: with imperfect renewable generation forecast, the performance of an RL controller is compared with that of a deterministic model predictive control (MPC). Our experiment results show that the RL controller is able to learn from experience, adapt to the imperfect forecast information and provide a more reliable restoration process when compared with the baseline MPC controller.

61 RADIATION PROTECTION AND DOSIMETRY↗
DOE OSTI2021

Multirate linearly-implicit GARK schemes

Many complex applications require the solution of initial-value problems where some components change fast, while others vary slowly. Multirate schemes apply different step sizes to resolve different components of the system, according to their dynamics, in order to achieve increased computational efficiency. The stiff components of the system, fast or slow, are best discretized with implicit base methods in order to ensure numerical stability. To this end, linearly implicit methods are particularly attractive as they solve only linear systems of equations at each step. This paper develops the Multirate GARK-ROS/ROW (MR-GARK-ROS/ROW) framework for linearly-implicit multirate time integration. The order conditions theory considers both exact and approximative Jacobians. The effectiveness of implicit multirate methods depends on the coupling between the slow and fast computations; an array of efficient coupling strategies and the resulting numerical schemes are analyzed. Multirate infinitesimal step linearly-implicit methods, that allow arbitrarily small micro-steps and offer extreme computational flexibility, are constructed. The new unifying framework includes existing multirate Rosenbrock(-W) methods as particular cases, and opens the possibility to develop new classes of highly effective linearly implicit multirate integrators.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Linearizing the BPS equations with vector and tensor multiplets

We analyse the BPS equations of N = (1, 0) supergravity theory in six dimensions coupled to a vector and tensor multiplet. We show how these BPS equations can be reduced to a set of linear differential equations. This system is triangular in that each layer of equations, while linear, is quadratically sourced by the solutions of the previous layers. We examine several explicit examples and discuss the construction of new families of microstate geometries. We expect that the result presented here will open up new branches of superstrata in which the momentum is encoded in a new class of charge carriers.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Tunable Geometries in Sparse Clifford Circuits

We investigate the emergence of different effective geometries in stochastic Clifford circuits with sparse coupling. By changing the probability distribution for choosing two-site gates as a function of distance, we generate sparse interactions that either decay or grow with distance as a function of a single tunable parameter. Tuning this parameter reveals three distinct regimes of geometry for the spreading of correlations and growth of entanglement in the system. We observe linear geometry for short-range interactions, treelike geometry on a sparse coupling graph for long-range interactions, and an intermediate fast scrambling regime at the crossover point between the linear and treelike geometries. This transition in geometry is revealed in calculations of the subsystem entanglement entropy and tripartite mutual information. We also study emergent lightcones that govern these effective geometries by teleporting a single qubit of information from an input qubit to an output qubit. These tools help to analyze distinct geometries arising in dynamics and correlation spreading in quantum many-body systems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Degradation of Poly- and Perfluoroalkyl Substances (PFAS) in Water via High Power, Energy-Efficient Electron Beam Accelerator

The goal of the 2-year workplan was to see if electron beam (EB) could be used to break down a sub-set of the larger chemical family of per and polyfluoroalkylated substances (PFAS) in an energy efficient and economical manner when compared to conventional water treatment technologies. Year one (Y1) work focused on sample EB treatment work in the Fermi National Accelerator Laboratory’s (FNALs) Accelerator Applications Demonstration and Development (A2D2) EB accelerator. While there are reportedly thousands of types of PFAS, for the point of most of the work herein, a small subset was examined, typically perfluorooctane sulfonate (PFOS) and perfluorooctanoate (PFOA). PFOA and PFOS are two of the most well studied PFAS and are studied for baseline evaluations and are considered most useful. The work from Y1 provided information about the optimal operating parameters and additives to use when treating PFOS and PFOA via EB. The data were then used to see where in a water treatment system an EB accelerator would be best suited to treat PFAS. A conventional water treatment technology, GAC, was then compared to e-beam treatment technology with respect to energy and costs for treatment. In year two (Y2), several conventional e-beam accelerator designs, and FNAL’s developmental compact SRF accelerator design, were evaluated for their suitability in PFAS treatment, from an energy efficiency and cost standpoint. Several EB parameters were evaluated and optimized for the removal of PFOA and PFOS from water at normal pressure and temperature, measured as total PFAS removal. Under the optimized test conditions both PFOA showed complete destruction to inorganic fluoride, and PFOS to inorganic fluoride and sulfate, with mass balance. The effect on PFAS removal relative to solution pH, total EB dose, EB dose rate, dissolved oxygen concentration (DO), temperature, and initial PFAS concentration were evaluated. In general, PFOA was easier to destroy than PFOS. Degradation products, typically observed under less-than-optimal EB conditions, provided insight to degradation mechanisms. Products were identified to rule out possible deleterious biproduct formation. The water radiolysis radical reaction kinetics with PFOS and PFOA were not dependent on the initial concentration over 5-orders of magnitude from 2 μg/L to 20 mg/L. This is thought to be because there was an overabundance of the reactive water radiolysis radicals relative to PFAS molecules and largely attributed to aqueous electrons. The reaction rates appeared to be diffusion limited. Testing at higher concentrations (100-200 mg/L) showed a decrease in removal efficiency, suggesting alternative kinetics, possibly second order rates, at higher concentrations. In all, we successfully defined a set of optimal EB parameters to treat PFOA and PFOS at concentrations of 20 mg/L in water with destruction efficiencies near 100%. We further tested the optimized EB parameters with other types of PFAS, including shorter and longer fluorocarbon chain homologs of PFOA and PFOS, and PFAS with alternative functional groups such as sulfonamides. Based on our results EB can be optimized as an effective destructive technology for removing PFAS from water. The conditions optimized for PFOA and PFOS were less effective with ultra-short fluorocarbon compounds like TFMS, PFES, PFPS and PFBS, and likely require re-optimization of parameters to them. In all, it was determined that from a cost and energy efficiency standpoint, EB would be best applied to waste streams with relatively high concentrations of PFOS and PFOA and is not as cost effective as GAC treatment for removing low concentrations of PFAS from water. Higher concentrations of PFAS can be found in the wastewater of conventional treatment processes such as RO and IE and therefore EB may be used to supplement such treatment technologies. Some real-world IE regeneration wash water and RO reject water containing higher concentrations of PFAS and obtained from pilot scale industrial wastewater treatment system at a fluorochemical manufacturing facility, showed that EB could remove PFAS from such types of wastewaters. The IE regenerant wash water appeared to be the most efficient of the two types of wastewaters tested. However, some further optimization of the EB parameters for the specific PFAS types present in those wastewaters may be required. Also, the effects of co-present TOC and mineral salts should be considered during such optimization efforts. From the experimental Y1 results it was seen that the aqueous electron drives degradation of the PFAS. In a hypothetical water treatment skid using EB for PFAS destruction the parameters of the system should be optimized to promote aqueous electron production. Before EB treatment, the PFAS should be preconcentrated when possible, the pH should be raised to pH 10 or higher to enhance aqueous electron production, and the water should be nitrogen purged to remove dissolved oxygen to minimize aqueous electron scavenging. An excel spreadsheet was created that calculates optimal conditions based on inlet PFAS concentration and desired outlet concentration, by optimizing the accelerator power, dose rate, water treatment rate, pH and dissolved oxygen levels to reach the desired endpoint. Given this information on accelerator operating conditions five different EB accelerator systems were compared. One EB system was a continuous-wave, linear superconducting accelerator being designed at Fermilab. Three other EB systems (IMPELA at 5% and 25% duty factor and the ILU-14) were normal conducting pulsed linear accelerators. The fifth system was an IBA Rhodotron which is a normal conducting, circular, continuous-wave accelerator. The accelerator efficiency (% of the incoming power that is used in water treatment) was the dominating factor in accelerator choice. The radio frequency (RF) power supply and the accelerator design (superconducting versus warm technology) drive the accelerator efficiency. The IBA Rhodotron was seen to be the most energy efficient commercially available technology with a wall-plug (total) power efficiency of 43% at 400 kW. The Fermilab design, with a prototype for a different application currently being fabricated, was the most energy efficient at 55% when driven by a Klystron RF power supply and as high as 77% when powered by a magnetron. As the Fermilab design was the most energy efficient by approximately 10-30%, further design work was done on the accelerator and beam delivery system specific to the destruction of PFAS in water. The Fermilab design is unique from industrial accelerators in that is superconducting. Superconducting technology allows for the acceleration of electrons without losses. The accelerator must be cooled to below the point where it is superconducting and is operated around 4 degrees Kelvin. The bulk of the design work for the accelerator is on making the accelerator as energy efficient as possible so that it does not require liquid helium and can be cooled with conduction cooling via cryocoolers. Final design work resulted in an EB accelerator that would operate at minimally 200 kW and 10 MeV. Prototype construction would cost $\$ $7.8 million dollars when driven by a Klystron power supply. A second version of the same accelerator would cost $\$ $5.5 million dollars when driven by a magnetron that is still under development. The commercially available 300 kW IBA Rhodotron cost was estimated at approximately $\$ $9 million. While it is hard to directly compare, an operational GAC system used by 3M for groundwater treatment capital cost (2022 dollars) was estimated to cost $\$ $3.3 million. While the capital expense of the EB accelerator systems was higher than GAC, the accelerator EB treatment would result in destruction of the PFAS and not just sequestration of PFAS to form a new waste stream that requires further treatment or disposal. The operating cost to destroy the PFAS via 400 kw EB system was less than $\$ $1000/kg of PFAS destroyed when treating at a 20 mg/L PFAS concentration, compared to GAC with operating costs that calculated at $\$ $27,530 per kg of PFAS sequestered when treating 100 μg/L PFOA and PFOS combined concentration.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

Linear complexity

We present factorization and solution phases for a new linear complexity direct solver designed for concurrent batch operations on fine-grained parallel architectures, for matrices amenable to hierarchical representation. We focus on the strong-admissibility-based $\mathscr{H}^{2}$ format, where strong recursive skeletonization factorization compresses remote interactions. We build upon previous implementations of $\mathscr{H}^{2}$ matrix construction for efficient factorization and solution algorithm design, which are illustrated graphically in stepwise detail. The algorithms are ‘blackbox’ in the sense that the only inputs are the matrix and right-hand side, without analytical or geometrical information about the origin of the system. We demonstrate linear complexity scaling in both time and memory on four representative families of dense matrices up to one million in size. Parallel scaling up to 16 threads is enabled by a multi-level matrix graph coloring and avoidance of dynamic memory allocations thanks to prefix-sum memory management. An experimental backward error analysis is included. We break down the timings of different phases, identify phases that are memory-bandwidth limited, and discuss alternatives for phases that may be sensitive to the trend to employ lower precisions for performance.

Boukaram, Wajih↗
DOE OSTI2021

On the possibility of footprint compression with one lens in nonlinear accelerator lattice

Electromagnetic interaction of colliding beams along with other nonlinear fields often limits the beams' lifetimes and luminosities. Nonlinearities result in the spread of betatron frequencies (footprint) and, thus, may enhance dynamic diffusion of particles due to high order resonances. One of the possible ways to eliminate nonlinearities and overcome the corresponding difficulties is compensation of nonlinear forces, but, in practice, it is hardly possible to obtain exact linearity of the system. The compensation with a single nonlinear lens cannot cope with distributed nonlinearities, nonlinearities due to parasitic crossings, etc. Here, we present a method to compute parameters of nonlinear element (lens) that eliminates both the footprint and resonance strength without achieving full compensation.

43 PARTICLE ACCELERATORS↗
DOE OSTI2022

Sensitivity of time-dependent density functional theory to initial conditions

Time-dependent density-functional theory is mathematically formulated through nonlinear coupled time-dependent three-dimensional partial differential equations, and it is natural to expect a strong sensitivity of its solutions to variations of the initial conditions, akin to the butterfly effect ubiquitous in classical dynamics. Since the Schrödinger equation for an interacting many-body system is, however, linear and mathematically the exact equations of the density-functional theory reproduce the corresponding one-body properties, it would follow that the Lyapunov exponents are also vanishing within a density-functional theory framework. Whether for realistic implementations of the time-dependent density-functional theory the question of the absence of the butterfly effect and whether the dynamics provided is indeed a predictable theory was never discussed. At the same time, since the time-dependent density-functional theory is a unique tool allowing us to study the nonequilibrium dynamics of strongly interacting many-fermion systems, the question of predictability of this theoretical framework is of paramount importance. Here our analysis, for a number of quantum superfluid many-body systems (unitary Fermi gas, nuclear fission, and heavy-ion collisions) with a classical equivalent number of degrees of freedom O(10 10 ) and larger, suggests that its maximum Lyapunov exponents are negligible for all practical purposes.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2020

Simulating coupled surface–subsurface flows with ParFlow v3.5.0: capabilities, applications, and ongoing development of an open-source, massively parallel, integrated hydrologic model

Surface flow and subsurface flow constitute a naturally linked hydrologic continuum that has not traditionally been simulated in an integrated fashion. Recognizing the interactions between these systems has encouraged the development of integrated hydrologic models (IHMs) capable of treating surface and subsurface systems as a single integrated resource. IHMs are dynamically evolving with improvements in technology, and the extent of their current capabilities are often only known to the developers and not general users. This article provides an overview of the core functionality, capability, applications, and ongoing development of one open-source IHM, ParFlow. ParFlow is a parallel, integrated, hydrologic model that simulates surface and subsurface flows. ParFlow solves the Richards equation for three-dimensional variably saturated groundwater flow and the two-dimensional kinematic wave approximation of the shallow water equations for overland flow. The model employs a conservative centered finite-difference scheme and a conservative finite-volume method for subsurface flow and transport, respectively. ParFlow uses multigrid-preconditioned Krylov and Newton–Krylov methods to solve the linear and nonlinear systems within each time step of the flow simulations. The code has demonstrated very efficient parallel solution capabilities. ParFlow has been coupled to geochemical reaction, land surface (e.g., the Common Land Model), and atmospheric models to study the interactions among the subsurface, land surface, and atmosphere systems across different spatial scales. This overview focuses on the current capabilities of the code, the core simulation engine, and the primary couplings of the subsurface model to other codes, taking a high-level perspective.

58 GEOSCIENCES↗
DOE OSTI2023

Multilevel-in-width training for deep neural network regression

A common challenge in regression is that for many problems, the degrees of freedom required for a high-quality solution also allows for overfitting. Regularization is a class of strategies that seek to restrict the range of possible solutions so as to discourage overfitting while still enabling good solutions, and different regularization strategies impose different types of restrictions. In this paper, we present a multilevel regularization strategy that constructs and trains a hierarchy of neural networks, each of which has layers that are wider versions of the previous network's layers. Here, we draw intuition and techniques from the field of Algebraic Multigrid (AMG), traditionally used for solving linear and nonlinear systems of equations, and specifically adapt the Full Approximation Scheme (FAS) for nonlinear systems of equations to the problem of deep learning. Training through V-cycles then encourage the neural networks to build a hierarchical understanding of the problem. We refer to this approach as multilevel-in-width to distinguish from prior multilevel works which hierarchically alter the depth of neural networks. The resulting approach is a highly flexible framework that can be applied to a variety of layer types, which we demonstrate with both fully connected and convolutional layers. We experimentally show with PDE regression problems that our multilevel training approach is an effective regularizer, improving the generalize performance of the neural networks studied.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

A general-purpose method for Pareto optimal placement of flow rate and concentration sensors in networked systems – With application to wastewater treatment plants

The advent of affordable computing, low-cost sensor hardware, and high-speed and reliable communications have spurred installation of ubiquitous sensors in complex engineered systems. However, ensuring reliable data quality remains a challenge. Exploitation of redundancy among sensor signals can help improving the precision of measured variables, detecting the presence of gross errors, and identifying faulty sensors. The cost of sensor ownership, maintenance efforts in particular, can still be cost-prohibitive however. Maximizing the ability to assess and control data quality while minimizing the cost of ownership thus requires a careful sensor placement. To solve this challenge, in this work we develop a generally applicable method to solve the multi-objective sensor placement problem in systems governed by linear and bilinear balance equations. Importantly, the method computes all Pareto-optimal sensor layouts with conventional computational resources and requires no information about the expected sensor quality.

42 ENGINEERING↗
DOE OSTI2025

Thermophysical Properties of NaCl–UCl 3 –PuCl 3 Molten Salts: A Combined Computational and Experimental Study

Actinide-bearing molten salts for use as fuels are an essential part of next generation molten salt reactors. Yet, numerous multicomponent salt mixtures are underdeveloped or have not been investigated. Here, this study, based on a combination of experimental and modeling techniques, is dedicated to determining and understanding a variety of properties of the ternary system of NaCl–UCl 3 –PuCl 3 , which represents a scenario for burnup of NaCl–UCl 3 fuel, at two compositions (∼10 and 5 mol % PuCl 3 in eutectic NaCl–UCl 3 pseudobinary) and a range of temperatures. Evaluation of the heat flow and mass loss data showed the 0.61NaCl–0.30UCl 3 –0.09PuCl 3 salt had a melting temperature of 551 ± 5 °C. Two additional thermal effects were observed occurring at approximately 410 and 494 °C. The transition occurring at 410 °C may be due to the presence of oxide in the salt. Extrapolation of thermodynamic data indicates the transition occurring at 494 °C is due to the formation of a liquid phase. Experimental testing determined the density of this system is a linear function of temperature and can be represented by the equation ρ = 4.014–0.0010T(°C), R 2 = 0.992. Additionally, by using atomistic modeling, we found that increasing the PuCl 3 content from 5 to 10 mol % led to the formation of larger Pu 3+ clusters and slower transport of ions.

11 - NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2021

Using extended MHD to explore lasers as a trigger for x-pinches

X-pinches have been shown to be a source of extremely intense x-ray emissions useful for diagnosing plasma dynamics and imaging biological objects. The most striking feature of an x-pinch is the hotspot, the point source from where all the x rays come from. Unfortunately, the exact timing and location of the hotspot are still unpredictable. Since an x-pinch hotspot forms from instabilities (like an m = 0 mode), we will computationally explore whether we can use a high-power laser to control the timing and location of these instabilities. Our goal is to reduce the temporal and spatial jitter associated with the x-ray burst. Using an extended MHD (XMHD) code, we explore the non-relativistic instability generation using a current profile of a 250 kA linear transformer driver system and laser characteristics of SLAC's Matter in Extreme Conditions laboratory. Our results include both laser-penetration results using a boundary-defined electromagnetic (EM) wave and instability results from a power-deposition method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Flowing plasma rearrangement in the presence of static perturbing fields

Charged particles interacting with electromagnetic waves have a portion of their energy tied up in wave-driven oscillations. When these waves are localized to the exhaust of linear magnetic confinement systems, this ponderomotive effect can be utilized to enhance particle confinement. The same effect can be derived for particles moving via an E×B drift into a region of a static perturbation to the electromagnetic fields which has a large wave vector component in the direction of the motion. In this work, we use a simplified slab model to self-consistently solve for the electromagnetic fields within the fluid flowing plasma of a static flute-like (k∥=0) perturbation and evaluate the resulting ponderomotive potential. We find that two types of perturbations can exist within the flowing plasma, which are an O wave and an X wave in the frame moving with the fluid. In the case of tenuous plasma, these perturbations are magnetostatic or electrostatic multipole-analog perpendicular to the guiding magnetic field in the lab frame, respectfully. For denser plasmas, the O wave-like perturbation is screened at the electron skin depth scale, and the X wave-like perturbation is a combination of a similar perpendicular electric perturbation and parallel magnetic perturbation. The ponderomotive potential generated in the X wave-like case is gyrofrequency-dependent and can be used as either potential barriers or potential wells, depending on the direction of the flow velocity.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Fluid temperature predictions of geothermal borefields using load estimations via state observers

Fluid temperature predictions of geothermal borefields usually involves temporal superposition of its characteristic g-function, using load aggregation schemes to reduce computational times. Assuming that the ground has linear properties, it can be modeled as a linear state space system where the states are the aggregated loads. However, the application and accuracy of these models is compromised when the borefield is already operating and its load history is not registered or there are gaps in the data. This paper assesses the performance of state observers to estimate the borefield load history and obtain accurate fluid predictions. Results show that both Time-Varying Kalman Filter (TVKF) and Moving Horizon Estimator (MHE) provide predictions with average and maximum errors below 0.1 $^\circ C$ and 1 $^\circ C$ respectively. MHE outperforms TVKF in terms of n-step ahead output predictions and load history profile at the expenses of about 5 times more computational time.

Figueroa, Iago C.↗