Reduced Order Modeling In Reservoir Simulation Using the Bilinear Approximation Techniques

The reduction of the large order models of complex systems into a sufficiently low dimensional order in such a way that important dynamic characteristics of the real system are preserved, is a nontrivial task. The motivations for such a model order reduction are either to reduce computations for analysis and practical control design or to simplify the control system structure. In the present article, major techniques in model order reduction are briefly discussed. The characteristic similarities and differences are highlighted, their advantages and disadvantages are underlined, and further research requirements and trends are commented upon. Introduction and Perspective One of the central issues in the active control of complex systems such as large flexible space structures (LFSS) is the derivation of a correct mathematical model for both the controlled and the uncontrolled dynamical systems. Theoretically, there are infinitely many elastic modes or degrees of freedom (DOF) in the distributed parameter (DP) models of LFSS, usually with very low natural damping. Moreover, the flexible modes contribute to the actual deformation of the structure.' The truly infinite dimensional character of LFSS models has to be approximated by some fidelity finite (but usually very large) dimensional model. The normal approach taken by engineers to achieve this end is via modal models with a large number of modes that provide a reasonable representation of the spacecraft dynamic characteristics. However, a difficult problem still remains to be the development of a model of lowenough dimensional order that it can be utilized by the onboard controller, yet high enough dimensional order that it preserves the dynamic characteristics of the real system represented and controlled. The motivations for such a reduction are either to reduce computations for analysis and practical control design or to simplify the control system structure. Discretization procedures will not be discussed herein and it will be assumed that a large finite dimensional model has been generated somehow and for implementation and other practical considerations, the model needs to be further reduced. *This work was-supported, in part, by the Air Force Wright Aeronautical, Air Force Systems Command, Flight Dynamics Lab, Structures and Dynamics Division, while the author was employed with HR-Textron, Valencia, California. There are various approaches to model order reduction; some are or pseudo-optimal and others are ad-hoc methods based on practical considerations and engineering experience and judgment. These techniques are known by different names such as reduction, condensation, economization, aggregation, optimal projection, and other combinations thereof. It has often been pointed out that such techniques generally constitute application of Rayleigh-Ritz/Galerkin optimization, and matrix transformation methods to the eigenvalue/eigenvector formulation for structural dynamic problems. The reduction or the condensation methods are based on transformations of the coordinates in the equations of motion that essentially maintain the invariance of the quadratic forms of the potential and kinetic energies. An important feature of these methods is that the reduced order model often loses the basic characteristics of the original system. Considerable progress was made by Likins, Ohkami, and wong5 in this respect. However, they failed to develop a general enough criterion that could reduce the system model in an sense without significantly affecting the eigenvalues of the original model. The optimization or the mathematical reduction procedure, on the other hand, is based on the reduction of the eigenvalue problem to a smaller size based on some optimality criterion (usually quadratic). In these techniques, the reductions are carried out without actually resorting to approximations, i .e. , without truncatiag any coordinates or states of the original system. Balanced model reduction of linear timeinvariant dynamical systems is essentially based on the controllability and observability relations of the states of the system. Subsystem models are obtained by deleting those states that contribute the least to the controllability and observability (or the impulse response) of the original system7 and thus are only in this sense. A more recent approach to model reduction is proposed in Skelton's work,* where each state of the system model is assigned a relative to a given basis, via a quadratic criterion, and the states with the least cost are deleted in a systematic manner. The resulting reduced model is a function of the state-space basis, and thus there is no guarantee for optimality for all choices. The latest development in model order reduction techniques is the work by Hyland and erns stein.^ Herein first order necessary conditions for reduced order modeling of linear time-invariant systems are derived via a pair of modified Lyapunov equations coupled by a nonorthogonal projection. This approach reveals the possibility of multiple extrema forsome of the abovementioned methods. Copyright @ 1988 by the American Institute of Aeronautics and Astronautics. Inc. All Rights Reserved.

Microgrids (MGs) concepts are a systematic way to integrate the supply and demand side along with the storage devices. These mostly use renewable energy resources (RESs) viz. solar PV and wind turbines, interfaced to the low voltage distribution network via an inverter. As several components are involved, MGs has a system of systems (SoSs) architecture and its modeling leads to form a high order system. Hence, for the controllers' design purposes, these models had to be brought down to a reduced lower order system. This paper performs a comparative study of various model order reduction (MOR) techniques viz. aggregation, balanced truncation (BT), singular perturbation (SP) and Hankel norm approximation (HNA) on a two-source autonomous MG system. The standard performance evaluation methods used for a multi-input multi-output (MIMO) system i.e. the step and impulse responses, eigenvalue analysis and Bode plots were considered to determine the effectiveness of these MOR method.

In this work, a model order reduction (MOR) technique for heat-transfer system-level modeling is presented. A heat transfer solver using the finite difference method (FDM), which is appropriate for MEMS thermal analysis, is implemented. The numerical models generated by the FDM solver then were successfully reduced into low-order compact models by an Arnoldi-based model-order-reduction technique. The compact macromodels can be easily inserted into system-level or circuit simulators. We have demonstrated that the macromodel results for a thermal actuator are in good agreement with the experimental results. We also use the same technique on periodic surface heat flux problems to observe the relationship between frequency and minimum required order of the reduced model under specific error tolerance.

In steam turbines (STs), evaluation technology of thermal deformations during transient operations is increasingly important, because of a demand for improvement of operability such as shortening startup time. However, it is still difficult to predict temperatures of metal parts with sufficient accuracy due to complexity of heat transfer phenomena. The present study puts its focus on developing a method to identify metal temperatures of STs during start-up operations using particle filter (PF), a kind of sequential Bayesian filter utilizing ensemble approximation, and model order reduction (MOR) technique. In this method, a self-organizing state-space model is used to calculate time evaluations of metal temperatures, and heat transfer coefficient (HTC) parameters and steam bulk temperature parameters. And an optimal estimate of these is obtained through iterations of a short-time prediction and a correction of variables using measured temperatures. In this state-space model, state variables including HTC and bulk temperature parameters are treated as random variables, and the time evolution of each variable is modeled as follows; (1) Metal temperature is modeled using a reduced order model (ROM) in order to reduce computational time of PF. The ROM is constructed from a finite element model for unsteady heat transfer analysis using MOR technique. (2) HTC is modeled as a random walk model where its changes during one time step is randomly determined using Gaussian noise. Here, the magnitude of this noise is adjusted automatically. (3) Bulk temperature is modeled using prediction formulas with Gaussian noise added. As a validation problem, a cold start-up operation of a ST unit of a gas turbine combined cycle (GTCC) is considered. The proposed method is applied to the problem, and boundary conditions (BCs) are identified using measured temperatures at 68 measurement points. Then, the heat transfer analysis based on finite element analysis (FEA) is performed using the identified BCs. As a result of this FEA, it is confirmed that the calculated metal temperature tends to agree better with measurements compared with that of initial FEA, and that the errors of the calculated temperature at measurement points reduce by 41% on average compared with initial FEA. From this result, it is concluded that this proposed method is effective for metal temperature identification during start-up operations.

In the present work, multiple non existing model order reduction (MOR) techniques for unstable second‐order form systems (SOSs) are proposed. For unstable SOSs, continuous‐time algebraic Lyapunov equations get unsolvable that halt the reduction process. To avoid this problem, unstable SOS is first decomposed into stable and unstable portions and balanced truncation is applied to the stable part. The obtained reduced order model (ROM) for the stable portion is augmented with the unstable portion to obtain the overall reduced system. It is observed that the second‐order structure in ROM for the first technique gets lost as well as augmented unstable dynamics degrade the ROM performance. To cater to these constraints, two structure‐preserving second‐order balanced truncation techniques for unstable SOSs are proposed in second part. System is first stabilized using the Bernoulli feedback stabilization procedure and then gramians are computed for the stabilized system. Further, gramians are partitioned into position and velocity portions to achieve structure preservation in ROM and balanced truncation is applied. As second technique retains second‐order structure as well as involves stabilized system dynamics, this technique far closely approximates original system behavior. Proposed techniques are tested on multiple systems and results certify the correct development of proposed techniques that can be utilized for MOR applications of unstable SOSs. © 2021 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

During lamination, residual thermo-mechanical stresses are induced in the encapsulated solar cells composing photovoltaic (PV) modules. Depending on the material and geometrical configuration of the layers of the laminate, this residual stress field can be beneficial since it may lead to a compressive stress state in Silicon and therefore crack closure effects in the presence of cracks, with a recovery of electrical conductivity in cracked solar cells. It is therefore important to investigate the distribution of thermo-mechanical stresses within the PV laminate with a view to optimizing the coupling between the electrical response and elastic deformation in the operation of PV modules. A promising approach proposed in the present thesis regards the prediction of residual stresses in composite laminates by using a shear-lag theory to model the epoxy-vinil-acetate polymeric layers, accounting for their thermo-visco-elastic response. Moreover, it will be shown that thermomechanical formulations for stress analysis of a PV laminate lead to a system of higher order ordinary differential equations or partial differential equations in which the exact solutions may be impossible to be determined in closed form and hence numerical schemes become desirable. However, the computational cost associated with the implementation of the numerical scheme may be significantly expensive. Therefore, a method to reduce the computational complexity is expected to be very important. To this aim, Model Order Reduction (MOR) techniques are applied hierarchically, first to the thermal system of a PV module in service, and then extended to coupled thermo-mechanical problems. A combination of proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM) with a modified formulation is proposed for the first-order thermal equations of photovoltaic system during service and a new coupled second-order Krylov based formulation is developed for model order reduction of the coupled thermo-mechanical model of the photovoltaic module. The results of these reduction schemes show a huge computational gain in the reduced system solutions and a high accuracy of the reduced system outputs

Sample selection based on sensitivity analysis in parameterized model order reduction

Efficient full-chip thermal simulation is among the most challenging problems facing the EDA industry today, due to the need for solution of very large systems of equations that require unreasonably long computational times. However, in most cases, temperature is not required to be computed at every point of the IC but only at certain hotspots, in order to assess the circuits compliance with thermal specifications. This makes the thermal analysis problem amenable to Model Order Reduction (MOR) techniques. System-theoretic techniques like Balanced Truncation (BT) offer very reliable bounds for the approximation error, which can be used to control the order and accuracy of the reduced models during creation, at the expense of greater computational complexity to create them. In this paper, we propose a computationally efficient low-rank BT algorithm that retains all the system-theoretic advantages in the reduction of model order for fast hotspot thermal simulation. Experimental results demonstrate a 39X order reduction and very tight accuracy bounds.

<div class="section abstract"><div class="htmlview paragraph">Battery temperature is known to have a critical influence on overall battery pack performance, from electrochemical behavior, charge acceptance, power availability, trip efficiency, safety, reliability and life-cycle costs. Temperature monitoring is critical to ensure safe and reliable battery pack operation.</div><div class="htmlview paragraph">Monitoring of cell temperatures in battery packs allows for control and estimation algorithms that can ensure homogenous pack temperature distribution, prevent excessive pack temperature rise and even infer cell core temperature, potentially allowing to both predict and mitigate onset of thermal runaway.</div><div class="htmlview paragraph">The increasing need for improved accuracy requires inclusion of more detail in the modelling stage, leading inevitably to ever larger-scale, ever more complex dynamical systems. Simulations in such large-scale settings lead in turn to unmanageably large demands on computational resources, which is the main motivation for Model Order Reduction.</div><div class="htmlview paragraph">Model Order Reduction is focused on reducing the complexity of large-scale dynamical systems, while preserving their input-output behavior. Reduced models mimic the complex behavior of large-scale dynamical systems, and can be efficiently used for design, optimization and sensitivity analysis. The resulting reduced order model can be used to replace the original system as a component in a larger simulation or it might be used to develop a simpler and faster controller suitable for real time applications.</div><div class="htmlview paragraph">Three model order reduction techniques are compared. The first method, Balanced Truncation, assumes access to the high-fidelity model structure is available whilst the other two, Nyquist and neural network approximation, assume only input/output behavioral knowledge is available. The reduced order models are compared, in simulation, to a high-fidelity model of a battery pack in terms of simulation speed-up, model accuracy and property preservation. The applicability of each technique for real-time applications, particularly in the context of xEV (grid connected electric vehicle) thermal management, together with pros and cons of each method is discussed.</div></div>

<p style='text-indent:20px;'>We consider model order reduction of a nonlinear cable-mass system modeled by a 1D wave equation with interior damping and dynamic boundary conditions. The system is driven by a time dependent forcing input to a linear mass-spring system at one boundary. The goal of the model reduction is to produce a low order model that produces an accurate approximation to the displacement and velocity of the mass in the nonlinear mass-spring system at the opposite boundary. We first prove that the linearized and nonlinear unforced systems are well-posed and exponentially stable under certain conditions on the damping parameters, and then consider a balanced truncation method to generate the reduced order model (ROM) of the nonlinear input-output system. Little is known about model reduction of nonlinear input-output systems, and so we present detailed numerical experiments concerning the performance of the nonlinear ROM. We find that the ROM is accurate for many different combinations of model parameters.

Balanced truncation (BT) model order reduction (MOR) is known for its superior accuracy and computable error bounds. Balanced stochastic truncation (BST) is a particular BT procedure that provides a general, structure-independent MOR framework to preserve both passivity and stability of original models. Its application toward large scale systems, however, has been limited by the complexity of solving large size continuous time algebraic Riccati equations (CAREs). This paper introduces a novel quadratic extension of the alternating direction implicit (ADI) iteration, called QADI, that efficiently solves a CARE. A Cholesky factor variant of QADI, called CFQADI, further exploits low rank matrices and and produces solution in factor form that greatly accelerates BST. Remarkable efficiency of the proposed BST/(CF)QADI integration is demonstrated with numerical examples.

This paper presents three types of linear model order reduction (MOR) technique, namely singular value decomposition (SVD)-based, Krylov-based, and modal truncationbased type applied to large-scale wind farm models. The first type includes a Balanced Truncation (BT) and Alternating Direction Implicit (ADI)-based BT method, while the second type encompasses a Rational Krylov (RK), and Iterative Rational Krylov Algorithm (IRKA) method. In the third type, a Subspace Accelerated MIMO Dominant Pole Algorithm (SAMDP) method is used. The effectiveness of these methods are tested on practicalsized wind farms with 90, 120 and 210 doubly-fed induction generators (DFIGs). Merits and demerits of each method are discussed in detail. The reduced order model (ROM) of wind farm is validated against the full order model (FOM) in term of frequency domain indices and waveform agreement at the point of common coupling (PCC).

Reduced-order deep learning for flow dynamics. The interplay between deep learning and model reduction

Numerical simulations for flow and transport in subsurface porous media often prove computationally prohibitive due to property data availability at multiple spatial scales that can vary by orders of magnitude. A number of model order reduction approaches are available in the existing literature that alleviate this issue by approximating the solution at a coarse scale preserving fine scale features. We attempt to present a comparison between two such model order reduction techniques, namely (1) adaptive numerical homogenization and (2) generalized multiscale basis functions. We rely upon a non-linear, multi-phase, black-oil model formulation, commonly encountered in the oil and gas industry, as the basis for comparing the aforementioned two approaches. An expanded mixed finite element formulation is used to separate the spatial scales between non-linear, flow, and transport problems. A numerical benchmark is setup using fine scale property information from the 10th SPE comparative project dataset for the purpose of comparing accuracies of these two schemes. An adaptive criterion is employed by both the schemes for local enrichment that allows us to preserve solution accuracy compared to the fine scale benchmark problem. The numerical results indicate that both schemes are able to adequately capture the fine scale features of the model problem at hand.

This paper presents some new findings in model order reduction of multi input-multi output (MIMO) systems of modern power grid with wind generation. A widely-used model reduction technique called Balanced Truncation (BT) is compared with a relatively new Interpolatory approach of Iterative Rational Krylov Algorithm (IRKA). It is demonstrated that both BT and IRKA can produce an acceptable accuracy during model reduction for power grids with synchronous generators. Whereas, when modern grid with wind generation is considered, BT fails to produce a stable reduced order model (ROM) while IRKA is still able to derive accurate stable ROM retaining the poorly-damped inter-area modes of the system.