Modeling reduced-order model approach

Topaz was used to calculate the time response of the model to step changes in the heater output values. One of the advantages of mathematical simulation over experimentation is the ease of starting the experiment from an initial steady state. The parameter estimation routines to follow require a value for the initial state of the system, and it is often difficult to hold the extruder conditions constant long enough to approach steady state and be assured that the temperature gradients within the barrel are known. The values from the Topaz simulation, were used as data for fitting a reduced order model of the dynamic system. 

Compared with the use of arbitrary grid interfaces in combination with reduced-order flow models, the porous medium approach allows one to deal with an even larger multitude of micro channels. Furthermore, for comparatively simple geometries with only a limited number of channels, it represents a simple way to provide qualitative estimates of the flow distribution. However, as a coarse-grained description it does not reach the level of accuracy as reduced-order models. Compared with the macromodel approach as propagated by Commenge et al, the porous medium approach has a broader scope of applicability and can also be applied when recirculation zones appear in the flow distribution chamber. However, the macromodel approach is computationally less expensive and can ideally be used for optimization studies. 

To construct the reference model, the interpretation system required routine process data collected over a period of several months. Cross-validation was applied to detect and remove outliers. Only data corresponding to normal process operations (that is, when top-grade product is made) were used in the model development. As stated earlier, the system ultimately involved two analysis approaches, both reduced-order models that capture dominant directions of variability in the data. A PLS analysis using two loadings explained about 60% of the variance in the measurements. A subsequent PCA analysis on the residuals showed that five principal components explain 90% of the residual variability. 

With higher order models, we can construct approximate reduced-order models based on the identification of dominant poles. This approach is used later in empirical controller tuning relations. 

The pole placement design predicates on the feedback of all the state variables x (Fig. 9.1). Under many circumstances, this may not be true. We have to estimate unmeasureable state variables or signals that are too noisy to be measured accurately. One approach to work around this problem is to estimate the state vector with a model. The algorithm that performs this estimation is called the state observer or the state estimator. The estimated state X is then used as the feedback signal in a control system (Fig. 9.3). A full-order state observer estimates all the states even when some of them are measured. A reduced-order observer does the smart thing and skip these measurable states. 

Typically, a non-linear system dynamic model is made up of individual lumped models of the components which at a minimum conserve mass and energy across the given component, but may also have a momentum equation if pressure drops must also be analyzed. For most dynamic problems of interest in hybrid studies, however, the momentum equation may be taken as quasi-steady (unless the solver requires the dynamic form to perform the numerical solution). Higher fidelity individual models or reduced order models (ROMs) can also be used, where the connection to the system model would be made at each subcomponent boundary. Since dynamic systems modeling is not as common as steady-state modeling, some discussion of modeling approaches will be given. There are two primary methods used to provide solutions for the pressure-flow dynamics of a system model. 

In this section we examine the primary transient phenomena that are of interest to SOFC analysis, and provide the fundamental model equations for each one. Examples for the use of these models are given in later sections. While the focus is on reduced-order models (lumped and one-dimensional), depending on the needs of the fuel cell designer, this may, or may not be justifiable. Each fuel cell model developer needs to ensure that the solution approach taken will provide the information needed for the problem at hand. For the goal of calculating overall cell performance, however, it is often that one-dimensional methods such as outlined below will be viable. 

Approaches based on parameter estimation assume that the faults lead to detectable changes of physical system parameters. Therefore, FD can be pursued by comparing the estimates of the system parameters with the nominal values obtained in healthy conditions. The operative procedure, originally established in [23], requires an accurate model of the process (including a reliable nominal estimate of the model parameters) and the determination of the relationship between model parameters and physical parameters. Then, an online estimation of the process parameters is performed on the basis of available measures. This approach, of course, might reveal ineffective when the parameter estimation technique requires solution to a nonlinear optimization problem. In such cases, reduced-order or simplified mathematical models may be used, at the expense of accuracy and robustness. Moreover, fault isolation could be difficult to achieve, since model parameters cannot always be converted back into corresponding physical parameters, and thus the influence of each physical parameters on the residuals could not be easily determined. 

A similar result was reported earlier (Georgakis 1986), when an eigenvalue analysis was used to prove that a time-scale separation is present in the transient evolution of the states 91 and 92. It is noteworthy, however, that, in contrast to the approach presented in this chapter, an eigenvalue analysis does not provide a means by which to derive physically meaningful reduced-order models for the dynamics in each time scale. 

In cases in which it is not practical to calculate the model at each iteration of an optimization algorithm, a good approach may be to use the original model as a source of computational experiments that produce data points in the same way as if we had performed a physical experiment. This Response Surface Methodology (RSM) generate a simpler model that involves explicit functions. These new models are referred as surrogate, reduced order or metamodels [1]. 

Progress has been made in both reduced-order modeling and model-based control of combustion dynamics. Advances in modeling were obtained by investigating shear-flow driven combustion instability. The authors of this chapter determined that shear-layer instability occurs when an absolutely unstable mode is present. This mode can be predicted and matched to experimental data. Also, it is shown that the temperature profile determines a transition to absolutely unstable operation. Parameters of the shear-flow modes together with a POD-based approach led to the derivation of a new reduced-order model that sheds light on the interactions between hydrodynamics, acoustics, and heat release. A RePOD... 

Experiments with two other types of mechanical stimuli were further conducted to validate the modeling and model reduction approaches. In the first experiment, the Slim-2 sample was allowed to vibrate in air freely upon an initial perturbation on the tip (Fig. 4.16(a)), and Fig. 4.16(b) shows the predicted and measured sensor outputs in response to free damp>ed oscillations. In the second experiment, a step change was applied to the tip displacement of the Big-2 sample, as recorded in Fig. 4.17(a). The corresponding predicted and measured sensor outputs are shown in Fig. 4.17(b). In Figs. 4.16 and 4.17, reduced fourth-order models were used for the prediction of sensor output, and the same parameters for the full models were adopted for the reduced ones. The match between the model predictions and the actual sensor measurements in both figures indicates that the modeling approach is effective. 

Qiao R, Alum NR (2003) Transient analysis of electro-osmotic transport by a reduced-order modelling approach. Int J Numer Methods Eng 56(7) 1023-1050... 

The calculation of the six components of the Reynolds stress tensor, that is, six second-order moments of the micro-PDF, f v,yf), is reduced to the calculation of k and the modeling of the turbulent viscosity pf As seen from (12.5.1-2), is a function of a limited number of second-order moments of the micro-PDF. Turbulent viscosity based closure models for the Reynolds-stresses can be used at relatively low computational effort. In the two-equation model approach, the turbulent viscosity is expressed in terms of the turbulent kinetic energy, k, and the turbulence dissipation rate, s, according to ... 

Reduced Order Model (ROM) The evaluation of the system response, as a function of the coordinates in the input space, is very computationally expensive, especially when brute-force approaches (e.g. Monte Carlo methods) are chosen as the sampling strategy. ROMs are used to lower this... 

Thanks to the great computer power available nowadays, CN theory models are outperformed by reduced-order physical models, e.g. DC power flow in power grids. Nonetheless, power-flow models alone lack the capability of capturing time-varying interdependencies and discrete processes, e.g. cascading failures, therefore, it is advisable to integrate them into approaches such as ABM to represent discrete event-driven processes (Schlapfer et al., 2008 Nan et al., 2013). 

Reduced-order models, i.e. CN, SD and DSCT, can enhance the feasibility of hybrid modeling and simulation approaches but the adherence to real system behavior must be validated by high-fidelity simulations. 

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