Low-order Model Research Articles

Experiment and modeling of stochastic ignition and combustion of fuel droplets impacting a hot surface

The ignition dynamic of liquid fuel droplets impacting hot surfaces is critical for fire-safety analysis in engineering systems, as well as for controlling wall-filming effects in IC engines. The scenario of slow fuel-leakage rates poses challenges associated with the stochastic processes of droplet splashing, break-up, turbulent mixing, and combustion. To address this, we conduct controlled experiments of liquid n-heptane droplets impacting a heated surface in the Leidenfrost regime, targeting the individual-droplet-deposition conditions. The experiment encompasses a range of surface temperatures and droplet-deposition rates. The experiments are complemented by theoretical analysis, where we developed a stochastic low-order numerical model, demonstrating good accuracy for predicting ignition probability and overall combustion dynamics. Notably, we observe a broad region of intermittent combustion behavior, with ignition probability varying based on surface temperature and droplet deposition rate. Additionally, we find that the transition to consistent ignition relies heavily on both surface temperature and deposition rate. Experimental and numerical model results shed light on the roles of the complex interplay between droplet breakup, chemical kinetics, and evaporation and mixing time scales, as well as the interaction among subsequent droplet combustion events, in governing the ignition and combustion of impacting droplet trains. The revealed dynamic of droplet/hot-surface ignition and the proposed stochastic model hold promise for advancing predictive capabilities of hot-surface-induced ignition and combustion arising from accidental leaks in flammable-liquid piping and wall-filming, particularly in the stochasticity-dominated individual-droplet-deposition regime.

Online Inertia Allocation for Grid-Connected Renewable Energy Systems Based on Generic ASF Model Under Frequency Nadir Constraint

Frequency security is one of the challenges for the power grid penetrated by renewable energy systems (RESs). Virtual inertia provision from the RESs is considered to be efficient support for frequency security enhancement. How to estimate the frequency nadir rapidly and accurately is crucial for the online inertia allocation for the RESs. This paper develops a generic average system frequency (G-ASF) model for the power grid to estimate the frequency nadir at first. The low-order generic model of the speed governing system of the SG is proposed to enhance the accuracy and flexibility of the G-ASF model. Then, based on the G-ASF model, an online inertia allocation strategy for the grid-connected RESs under frequency nadir constraint is proposed. With the pre-identified generic models of the speed governing systems, the G-ASF model of the entire power grid is periodically updated according to the grid operation state. Next to the model updating, the critical inertia of the power grid is periodically calculated under the pre-defined contingencies for frequency security verification. Then, the virtual inertia provision tasks are determined and allocated to the RESs periodically, and the grid frequency security is effectively ensured. The simulation results validate the proposed model and strategy.

Simultaneous Model Change Detection in Multivariate Linear Regression With Application to Indonesian Economic Growth Data

In this paper, we study asymptotic model change detection in multivariate linear regression by using the Kolmogorov–Smirnov function of the partial sum process of recursive residuals. We approximate the rejection region and also the power function of the test by establishing a functional central limit theorem for the sequence of the partial sum processes of the recursive residuals of the observations. When the assumed model is true, the limit process is given by the standard multivariate Brownian motion which does not depend on the regression functions. However, when the assumed model is not true (some models change), the limit process is represented by a vector of deterministic trend plus the standard multivariate Brownian motion. The finite sample size rejection region and the power of the test are investigated by means of Monte Carlo simulation. The simulation study shows evidence that the proposed test is consistent in the sense that it attains the power larger than the size of the test when the hypothesis is not true. We also demonstrate the application of the proposed test method to Indonesian economic growth data in which we test the adequacy of three‐variate low‐order polynomial model. The test result shows that the growth of the Indonesian economy is neither simultaneously constant nor linear. The test has successfully detect the appearance of a change in the model which is mainly caused by the COVID‐19 pandemic in 2020.

Integration of Vehicle Dynamic Model and System Identification Model for Extending the Navigation Service Under Sensor Failures

Accurate and reliable localization is of great importance for autonomous vehicles (AV). Mainstream localization approaches in autonomous vehicles (AV) are limited by the reliability of onboard sensors, which could be vulnerable to sensor failure, such as signal outages of the camera and signal spoofing of the global navigation satellite systems (GNSS). Different from these active or passive sensors, the vehicle dynamic model (VDM), which is the application of physical laws to a vehicle in motion, is environmentally independent and is capable of providing vehicle motion estimation continuously. However, the performance of the VDM-based motion estimation is dominated by the accuracy of the system dynamics model. To tackle this issue, this study proposes a sensor-free localization method VDM-SI by integrating system identification into the design of vehicle dynamic models (VDM). A system identification process based on low-order process models is proposed to identify the system dynamics of the AV, where the identified system responses are taken as the control input of VDM to estimate the vehicular positioning. The localization experiments in two scenarios show that the mean absolute translation error of VDM-SI can be reduced by 70% compared to conventional VDM methods. In addition, VDM-SI is experimentally proven to improve the localization performance of sensor fusion-based localization systems with high noise levels. Furthermore, in the application of re-localization after sensors fail and recover, VDM-SI shows strength in enhancing the security of AVs in extreme conditions.

A Lumped-Parameter Model of the Cardiovascular System Response for Evaluating Automated Fluid Resuscitation Systems.

Physiological closed-loop controlled (PCLC) medical devices, such as those designed for blood pressure regulation, can be tested for safety and efficacy in real-world clinical settings. However, relying solely on limited animal and clinical studies may not capture the diverse range of physiological conditions. Credible mathematical models can complement these studies by allowing the testing of the device against simulated patient scenarios. This research involves the development and validation of a low-order lumped-parameter mathematical model of the cardiovascular system's response to fluid perturbation. The model takes rates of hemorrhage and fluid infusion as inputs and provides hematocrit and blood volume, heart rate, stroke volume, cardiac output and mean arterial blood pressure as outputs. The model was calibrated using data from 27 sheep subjects, and its predictive capability was evaluated through a leave-one-out cross-validation procedure, followed by independent validation using 12 swine subjects. Our findings showed small model calibration error against the training dataset, with the normalized root-mean-square error (NRMSE) less than 10% across all variables. The mathematical model and virtual patient cohort generation tool demonstrated a high level of predictive capability and successfully generated a sufficient number of subjects that closely resembled the test dataset. The average NRMSE for the best virtual subject, across two distinct samples of virtual subjects, was below 12.7% and 11.9% for the leave-one-out cross-validation and independent validation dataset. These findings suggest that the model and virtual cohort generator are suitable for simulating patient populations under fluid perturbation, indicating their potential value in PCLC medical device evaluation.

Sparse Measurement-Based Modelling Low-Order Dynamics for Primary Frequency Regulation

Developing a low-order model for a target with dynamic analysis and control of the primary frequency regulation (PFR) is critical for power system stability. Measurement-based modelling is an important way to improve the model fit with practical systems. The traditional interpretation of PFR is based on the combined effect of each generator unit on the system frequency, so the response of each unit needs to be measured for modelling. With the integration of a large number, distribution and diversified frequency regulation of units into a power system, it will be a burden to measure the response of all generation units. Instead of revealing the PFR dynamics based on each generation unit, this paper illustrates the impact of all units on the frequency dynamics from the perspective of the system response. Only the system frequency is needed to construct a dataset for identifying the PFR model so that the measurement required is sparse. The effectiveness of the proposed method is verified using a test system containing multiple types of units in numerical simulations and real measurements of an actual power system.

Identification of Low Order Systems in a Loewner Framework

This paper considers the problem of non-parametric identification of low-order models from time-domain experimental data using a combination of Caratheodory Fejer and Loewner-based interpolation, followed by a Loewner matrix Balanced Reduction (LBR) step. As we show in the paper, the Loewner matrix is an estimator for the trace norm of a system, playing a role similar to the one played by the Hankel matrix. However, utilizing Zolotarev numbers to establish decay rate bounds for singular values reveals that the decay of singular values in the Loewner matrix is considerably faster than that in the Hankel matrix. Thus, Loewner-based methods yield lower order systems, with the same error bound, than comparable ones based on Hankel matrices. The effectiveness of our method is demonstrated through a numerical example.

Reliability assessment of off-policy deep reinforcement learning: A benchmark for aerodynamics

Abstract Deep reinforcement learning (DRL) is promising for solving control problems in fluid mechanics, but it is a new field with many open questions. Possibilities are numerous and guidelines are rare concerning the choice of algorithms or best formulations for a given problem. Besides, DRL algorithms learn a control policy by collecting samples from an environment, which may be very costly when used with Computational Fluid Dynamics (CFD) solvers. Algorithms must therefore minimize the number of samples required for learning (sample efficiency) and generate a usable policy from each training (reliability). This paper aims to (a) evaluate three existing algorithms (DDPG, TD3, and SAC) on a fluid mechanics problem with respect to reliability and sample efficiency across a range of training configurations, (b) establish a fluid mechanics benchmark of increasing data collection cost, and (c) provide practical guidelines and insights for the fluid dynamics practitioner. The benchmark consists in controlling an airfoil to reach a target. The problem is solved with either a low-cost low-order model or with a high-fidelity CFD approach. The study found that DDPG and TD3 have learning stability issues highly dependent on DRL hyperparameters and reward formulation, requiring therefore significant tuning. In contrast, SAC is shown to be both reliable and sample efficient across a wide range of parameter setups, making it well suited to solve fluid mechanics problems and set up new cases without tremendous effort. In particular, SAC is resistant to small replay buffers, which could be critical if full-flow fields were to be stored.

Periodicity and chaos of thermal convective flows in annular cylindrical domains using the method of isolation by spectral expansions

The dynamical and chaotic behaviors of natural convection flow in the semi-annular and full-annular cylindrical domains are studied using the method of isolation by spectral expansion. For each of the flow cases, spectral models with different orders are obtained and dynamical characteristics of the flow are investigated using numerical simulations. For the semi-annular case of study, results reveal that in all the generated spectral models, chaotic solution appears, but the onset of chaos between each system requires different set of control parameters. Bifurcation diagrams are provided for all the systems which show the ranges of periodic and chaotic behavior of the flow. Strange attractors are captured for all the systems with different orders. For the lowest-order model, a Lorenz-like attractor is produced, which has two distinguishable scrolls similar to Lorenz's famous attractor and for the other high-order models, the shape of strange attractor gets different. For the full-annular case of study, the physical characteristics of the flow are obtained. The results show that the produced thermal plume on the top of annulus gets higher fluctuations as the flow Rayleigh number gets higher. At a specific Rayleigh number, the flow behavior gets completely unstable, which shows a transitional regime change. At this state of flow, the domain is dominated by several vortices.

On the potential of transfer entropy in turbulent dynamical systems

Information theory (IT) provides tools to estimate causality between events, in various scientific domains. Here, we explore the potential of IT-based causality estimation in turbulent (i.e. chaotic) dynamical systems and investigate the impact of various hyperparameters on the outcomes. The influence of Markovian orders, i.e. the time lags, on the computation of the transfer entropy (TE) has been mostly overlooked in the literature. We show that the history effect remarkably affects the TE estimation, especially for turbulent signals. In a turbulent channel flow, we compare the TE with standard measures such as auto- and cross-correlation, showing that the TE has a dominant direction, i.e. from the walls towards the core of the flow. In addition, we found that, in generic low-order vector auto-regressive models (VAR), the causality time scale is determined from the order of the VAR, rather than the integral time scale. Eventually, we propose a novel application of TE as a sensitivity measure for controlling computational errors in numerical simulations with adaptive mesh refinement. The introduced indicator is fully data-driven, no solution of adjoint equations is required, with an improved convergence to the accurate function of interest. In summary, we demonstrate the potential of TE for turbulence, where other measures may only provide partial information.

Design and control intended low-order model for transient investigation of an electric oven: Static mode

In general, electric ovens represent a low-efficiency class among home appliances, contributing to overall environmental changes in buildings and apartments. To improve their low efficiency, which is typically between 10% to 12%, industries are increasingly focused on developing effective technologies. One suitable strategy is to design and control the oven's center temperature, as specified by the EN 60350-1 European standard, which regulates the energy efficiency index (EEI) test. The present simulation study explains the transient thermo-fluid behavior of a domestic electric oven subjected to free convection, i.e., static mode. For this purpose, appropriate dynamic models are employed, intended for design and control. A lumped-parameter based methodology is considered to develop a low-order dynamic model, resulting in reduced computational cost and run time. A finite level of discretization is introduced into the model, enabling the capture of limited insights, such as temperature, heat flux, mass flow rate, pressure drop, etc., particularly at the cavity center, internal and external cavity wall surfaces, and at suction and exit openings. The electric oven is mainly divided into four sub-systems: door, cavity, ventilation, and cabinet. Each sub-system has a suitable level of discretization signifying a lumped mass parameter to reproduce the transient behavior within the electric oven. Heaters are considered as individual lumped masses and are integrated within the cavity sub-system. The above sub-systems employ both the first principle equations and correlations based on experimental data, giving a low-order grey-box identity to the system. In addition to conduction and convection, a non-linear form of radiation is considered to increase model accuracy. The developed model is further calibrated with available physical data. Once calibrated, the model is then compared with the remaining physical data produced at different heat loads to understand the accuracy in terms of transient temperature prediction at various temperature set points.

A lightweight vortex model for unsteady motion of airfoils

A low-order vortex model has been developed for analysing the unsteady aerodynamics of airfoils. The model employs an infinitely thin vortex sheet in place of the attached boundary layer and a sheet of point vortices for the shed shear layer. The strength and direction of the vortex sheet shed at the airfoil trailing edge are determined by an unsteady Kutta condition. The roll-up of the ambient shear layer is represented by a unique point vortex, which is consistently fed circulation by the last point vortex of the free vortex sheet. The model's dimensionality is reduced by using three tuning parameters to balance representational accuracy and computational efficiency. The performance of the model is evaluated through experiments involving impulsively started and heaving and pitching airfoils. The model accurately captures the dynamics of the development and evolution of the shed vortical structure while requiring minimal computational resources. The validity of the model is confirmed through comparison with experimental force measurements and a baseline unsteady panel method that does not transfer circulation in the free vortex sheet.

Saturation of flames to multiple inputs at one frequency

Existing experimental results show that swirling flames in annular combustors respond with a different gain to acoustic azimuthal modes rotating in either the clockwise or anti-clockwise direction. The ratio $R$ of these two gains is introduced, with $R=1$ being the conventional case of flames responding the same to the two forcing directions. To allow a difference in response to the different directions ( $R\neq 1$ ), a multiple-input single-output azimuthal flame describing function is successfully implemented in a quaternion valued low-order model of an annular combustion chamber in the current work. Theoretical studies have explored this kind of symmetry breaking between the two acoustic wave directions in the past, but it has not been backed by experimental data. One of the main features of the new model proposed in this work is the potential difference in mode shapes between the acoustic and the heat release rate modes, which has recently been observed experimentally. This results in a gain-dependent equation for the nature of the mode, which has a significant influence on the fixed points of the system. For example, one of the spinning solutions and the standing solution can disappear through a saddle node bifurcation as the parameters are varied. The presence of only a single direction for the spinning solution matches experimental observations better than the conventional models, and the proposed model is shown to qualitatively describe experimental measurements well.

Conjugate Modeling of a Closed Co-Rotating Compressor Cavity

Abstract Robust methods to predict heat transfer are vital to accurately control the blade-tip clearance in compressors and the radial growth of the disks to which these blades are attached. Fundamentally, the flow in the cavity between the co-rotating disks is a conjugate problem: the temperature gradient across this cavity drives large-scale buoyant structures in the core that rotate asynchronously to the disks, which in turn governs the heat transfer and temperature distributions in the disks. The practical engine designer requires expedient computational methods and low-order modeling. A conjugate heat transfer (CHT) methodology that can be used as a predictive tool is introduced here. Most simulations for rotating cavities only consider the fluid domain in isolation and typically require known disk temperature distributions as the boundary condition for the solution. This paper presents a novel coupling strategy for the conjugate problem, where unsteady Reynolds averaged Navier–Stokes (URANS) simulations for the fluid are combined with a series of steady simulations for the solid domain in an iterative approach. This strategy overcomes the limitations due to the difference in thermal inertia between fluid and solid; the method retains the unsteady flow features but allows a prediction of the disk temperature distributions, rather than using them as a boundary condition. This approach has been validated on the fundamental flow configuration of a closed co-rotating cavity. Metal temperatures and heat transfer correlations predicted by the simulation are compared to those measured experimentally for a range of engine-relevant conditions.

Inviscid modeling of unsteady morphing airfoils using a discrete-vortex method

Abstract A low-order physics-based model to simulate the unsteady flow response to airfoils undergoing large-amplitude variations of the camber is presented in this paper. Potential-flow theory adapted for unsteady airfoils and numerical methods using discrete-vortex elements are combined to obtain rapid predictions of flow behavior and force evolution. To elude the inherent restriction of thin-airfoil theory to small flow disturbances, a time-varying chord line is proposed in this work over which to satisfy the appropriate boundary condition, enabling large deformations of the camber line to be modeled. Computational fluid dynamics simulations are performed to assess the accuracy of the low-order model for a wide range of dynamic trailing-edge flap deflections. By allowing the chord line to rotate with trailing-edge deflections, aerodynamic loads predictions are greatly enhanced as compared to the classical approach where the chord line is fixed. This is especially evident for large-amplitude deformations. Graphical abstract

Data-driven model identification using forcing-induced limit cycles

This paper presents a data-driven model identification strategy that characterizes the behavior of a general dynamical system relative to a set of limit cycles that emerge in response to periodic forcing. Using time series data to infer the phase–amplitude dynamics associated with the underlying forced limit cycles, a low-order model can be obtained that accurately captures the dynamical behavior in response to arbitrary external inputs. The proposed strategy can be readily implemented in situations where full state measurements are unavailable and does not require any prior knowledge of the underlying model equations. This technique is applied to a model of coupled planar oscillators and to a model that considers the spike rate in a population of coupled conductance-based neurons where it outperforms two other commonly used data-driven model identification techniques.

The Hybrid Pathway to Flexible Power Turbines, Part I: Novel Autoencoder Methods for the Automated Optimization of Thermal Probes and Fast Sparse Data Reconstruction, Enabling Real-Time Thermal Analysis

Abstract The global drive toward renewable energy is imposing challenging operating requirements on power turbines. Flexible load-leveling applications must accept more frequent and demanding start-stop cycles. Full transient analyses are too computationally expensive for real-time simulation across all operating regimes so monitoring relies on sparse physical measurements. Alone, these sparse data lack the fidelity for real-time prediction of a complex thermal field. A new hybrid methodology is proposed, coupling data across a range of fidelities to bridge the limitations in the individual analyses. Combining several fidelity methods in parallel, low-order models, corrected by real-time physical measurements, is calibrated with high-fidelity simulations. The multifaceted hybrid approach enables the real-time speed of low-order analysis at high resolution. This paper series develops the critical enabling features of the hybrid method. Real-time cross-fidelity data transition is fundamental to the hybrid methodology. A novel neural network auto-encoder method is presented, facilitating complex thermal profile reconstruction. Uncovering a compressed latent space, auto-encoders leverage underlying data features for fast simulation. Coupled with a dynamic mask and top-k selection, thermal probe placement can be automatically optimized. The auto-encoder method is demonstrated on a turbine casing, reconstructing over 500 h of transient operation in real-time, whilst reducing the required number of measurements by half.

The Hybrid Pathway to Flexible Power Turbines, Part II: Fast Data Transfer Methods Between Varying Fidelity Simulations, to Enable Efficient Conjugate Thermal Field Prediction

Abstract The global drive toward renewable energy is imposing challenging operating requirements on power turbines. Flexible load-leveling applications must accept more frequent and demanding start-stop cycles. Full transient analyses are too computationally expensive for real-time simulation across all operating regimes so monitoring relies on sparse physical measurements. Alone, these sparse data lack the fidelity for real-time prediction of a complex thermal field. A novel hybrid methodology is proposed, coupling data across a range of fidelities to bridge the limitations in the individual analyses. Combining several fidelity methods in parallel; low-order models, corrected by real-time physical measurements, are calibrated with high-fidelity simulations. A newly developed low-order thermal network code is used to predict the thermal field in real-time. High-fidelity flow characteristics are routinely transferred to the decoupled low-order solution. A critical enabling feature of this hybrid approach is the fast data interpolation between differing fidelity numerical simulations. This paper evaluates a spatial Kriging method for robust data transfer between two different fidelity mesh, tested in the case of thermal profile prediction of a power turbine. Additionally, a novel coordinate-based hash mapping process is demonstrated for the fast high-to-low fidelity data transfer. Localized hashing allows independent, parallel, nearest neighbor search at significantly reduced computational cost. The demonstrated method facilitates fast mesh pairing, necessary to support the real-time hybrid method for thermal field prediction during turbine transient operation.

Control-Oriented Modeling for Flexible Aircraft

Flexible aircraft designs with high-aspect-ratio wings and lightweight structures are increasingly adopted to achieve improved fuel efficiency. To design a controller for a flexible aircraft, a low-order model of its dynamics that captures the aeroelastic behavior is needed. One approach to establishing such a low-order model is to extend the six-degree-of-freedom rigid aircraft model with additional elastic states. However, most of the existing models along these lines are based on simplified aerodynamics models (e.g., experimentally corrected aerodynamic coefficients), which limits the applicability of such low-order models in early design stages. In this work, a semi-analytical model of flexible aircraft is established through the combination of the dynamics of elastic body and the numerical linearization of high-order structure and aerodynamics models. The proposed model is derived in the body-fixed axes formulation, represents full flight dynamics (both longitudinal and lateral), and preserves the physical meanings for the states and all the derived terms. Static residualization is also adopted to enhance the model’s accuracy with a limited number of elastic states. Using the proposed approach, low-order semi-analytical models are exemplified for different complexity aircraft representations and verified against the corresponding nonlinear high-order models.

Three-Dimensional Surface Reconstruction from Point Clouds Using Euler’s Elastica Regularization

Euler’s elastica energy regularizer, initially employed in mathematical and physical systems, has recently garnered much attention in image processing and computer vision tasks. Due to the non-convexity, non-smoothness, and high order of its derivative, however, the term has yet to be effectively applied in 3D reconstruction. To this day, the industry is still searching for 3D reconstruction systems that are robust, accurate, efficient, and easy to use. While implicit surface reconstruction methods generally demonstrate superior robustness and flexibility, the traditional methods rely on initialization and can easily become trapped in local minima. Some low-order variational models are able to overcome these issues, but they still struggle with the reconstruction of object details. Euler’s elastica term, on the other hand, has been found to share the advantages of both the TV regularization term and the curvature regularization term. In this paper, we aim to address the problems of missing details and complex computation in implicit 3D reconstruction by efficiently using Euler’s elastica term. The main contributions of this article can be outlined in three aspects. Firstly, Euler’s elastica is introduced as a regularization term in 3D point cloud reconstruction. Secondly, a new dual algorithm is devised for the proposed model, significantly improving solution efficiency compared to the commonly used TV model. Lastly, numerical experiments conducted in 2D and 3D demonstrate the remarkable performance of Euler’s elastica in enhancing features of curved surfaces during point cloud reconstruction. The reconstructed point cloud surface adheres more closely to the initial point cloud surface when compared to the classical TV model. However, it is worth noting that Euler’s elastica exhibits a lesser capability in handling local extrema compared to the TV model.