Reduced-order models

The results from the model were compared semi-quantitatively with experimental measurements to validate the model. Because of both the extremely long time constants in the system and the variations in the ambient conditions in the laboratory where the extruder was placed, it was not possible to rigorously test the model against experimental data. To conserve demands of time and materials for experiments on the extruder, numerical experiments were used to provide data for developing an optimal control system. The goal of the numerical experiments was to develop a reduced-order model suitable for optimizing the control system. 

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. 

The program SimuSolv was used to estimate the parameters of the reduced order model. The commands needed to estimate the parameters were ... 

The optimize command maximizes a statistical "likelihood function". The higher this function, the more likely is the parameter to be the correct one. In the figure below, the symbols represent points calculated by the program Topaz (the full model), and the solid lines are the values calculated from the reduced-order model using the parameters determined by the program. 

The numerical experiment started at a steady-state value of 200 C for both temperature nodes with an output of 16.89% for both heaters output number 1 was then stepped to 19.00%. If both outputs had been stepped to 19%, then both nodes would have gone to 220 C. The temperature of node 5 does not go as high, and the temperature of node 55 goes too high. In the reduced order model, the time constant x represents the effect of radial heat conduction, while the time constant X2 represents the effect of axial heat conduction. SimuSolv estimates these two parameters of the dynamic model as ... 

Figure 4. SimuSolv plot of the Topaz results (symbols) and reduced order model results (lines)...

Two PID controllers were then added to the reduced order model. Temperature at node 5 was paired with output 1, and temperature at node 55 was paired with output 2. The code required to realize the PID controllers is ... 

Figure 5. SimuSolv plot of optimized reduced order model tuning.

TAUl, TAU2 = time constants in the reduced order model. Essentially the "response time" of the nodes these are the time "lags" for thermal conduction in the extmder... 

TNI, TN5, TN51, TN55 = temperature of nodes 1, 5, 51, and 55 of the reduced order model... 

Development of a reduced-order model for metallocene-catalyzed ethylene-norbornene copolymerization reaction... 

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. 

We now derive the time-domain solutions of first and second order differential equations. It is not that we want to do the inverse transform, but comparing the time-domain solution with its Laplace transform helps our learning process. What we hope to establish is a better feel between pole positions and dynamic characteristics. We also want to see how different parameters affect the time-domain solution. The results are useful in control analysis and in measuring model parameters. At the end of the chapter, dead time, reduced order model, and the effect of zeros will be discussed. 

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. 

Let say we have a high order transfer function that has been factored into partial fractions. If there is a large enough difference in the time constants of individual terms, we may try to throw away the small time scale terms and retain the ones with dominant poles (large time constants). This is our reduced-order model approximation. From Fig. E3.3, we also need to add a time delay in this approximation. The extreme of this idea is to use a first order with dead time function. It obviously cannot do an adequate job in many circumstances. Nevertheless, this simple... 

Reduced Order Models versus Detailed CFD Models... 

Detailed CFD models of fuel cells (see Chapters 3 and 4), on the other hand, use continuum assumption to predict the 3-D distributions of the physical quantities inside the fuel cells. These models are more complex and computationally expensive compared to reduced order models especially due to the disparity between the smallest and largest length scales in a fuel cell. The thickness of the electrodes and electrolyte is usually tens of microns whereas the overall dimensions of a fuel cell or stack could be tens of centimeters. Though some authors used detailed 3-D models for cell or stack level modeling, they are mostly confined to component level modeling. In what follows, we present the governing equations for some of these models. 

Pakalapati, S.R. (2006) A new reduced order model for solid oxide fuel cell modeling, PhD Thesis, Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, WV. 

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. 

Consequently, these models often require model tuning through semiempirical correlations and data integration. Such tasks are time-consuming and problem specific as they often require information from additional experiments and pilot plant trials, with missing information leading to start-up and operational risks. The incorporation of more accurate multi-scale phenomena (at device-, meso- and even molecular scales) captured by reduced-order models (ROMs) will overcome these limitations (Lang et al., 2009, 2011). 

Composite control relies on the use of a single controller consisting of a fast component and a slow component, which are designed separately on the basis of the reduced-order models for the dynamics in the respective time scales (Figure 2.9). 

Figure 2.9 Composite control relies on separate, coordinated fast and slow controllers, designed on the basis of the respective reduced-order models, to compute a control action that is consistent with the dynamic behavior of two-time-scale systems.

In order to address these objectives, we follow the procedure outlined in Section 3.4.3 to obtain a reduced-order model of the dynamics in the slow time scale. Specifically we consider the limit of an infinitely high recycle how rate... 

Referring back to the theory introduced in Chapter 2, we can expect that the presence of terms of very different magnitudes (i.e., 0(1) and 0(e)) in the model (4.18) reflects a two-time-scale behavior in the dynamics of typical processes with recycle and purge. In what follows, we will show that this is indeed the case. Also, we will address the derivation of reduced-order models of the fast and slow dynamics, provide a physical interpretation of this dynamic behavior, and highlight its control implications. 

Let us continue with the analysis of the motivating examples introduced in Section 4.2, and present the derivation of reduced-order models for the fast and slow dynamics of the two process systems, according to the theoretical framework developed above. 

Finally, we analyzed the control implications of the presence of impurities in a process, concluding that the control of impurity levels must be addressed over an extended time horizon using the flow rate of the purge stream as a manipulated input. To close the impurity-levels loop, one should resort either to an appropriately tuned linear controller (e.g., a PI controller with long reset time) or to a (nonlinear) model-based controller that uses (an inverse of) the reduced-order model of the slow dynamics - as developed in this chapter - to compute the necessary control action. 

Each of the reduced-order models derived for the fast, intermediate, and slow dynamics (Equations (5.12), (5.21), and (5.28)) involves only one group of manipulated inputs, namely the large internal flow rates u1, the small flow rates us, and the purge flow rate up, respectively. Thus, control objectives in each of the... 

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