Document [original]

Systematic Modeling for Optimization

Erik Esche* and David Müller, Günter Wozny

Chair of Process Dynamics and Operation, Technische Universität Berlin,

Straße des 17. Juni 135, Sekr. KWT-9, Berlin, 10623, Germany

*erik.esche@tu-berlin.de

Abstract

Optimization usually requires models, which are computationally speaking less

expensive than models commonly used for simulations. At the same time,

process optimization and model predictive control etc. require dependable

accuracies in addition to the fastness. To demystify the art of preparing process

models for optimization, a workflow is presented in this contribution, which

systematically deduces models based on simplification of existing models and

experiment based deduction of computationally inexpensive correlations.

Keywords: Multiple-Scale Modeling, Optimization, Convexification,

Linearization

1. Introduction

Process systems engineers working in the field of optimization often find

themselves in one of the following situations when modeling phenomena or an

entire process: a large complex model exists, a short-cut model exists, or no

model exists at all. Whatever the status quo is, the aim in the end is to have a

model that not only describes the phenomena or process correctly, but has

adequate numerical characteristics. Among these desired characteristics are a

high degree of convexity and linearity. Often, process systems engineers will

start randomly at various points of the project to develop or modify their model,

i.e. perform experiments, establish an overly complex model not suitable for

optimization, or forget to see the whole bigger picture by focusing on the

phenomena level. A workflow systematizing the work of process engineers

working on optimization models is required to firstly speed up the model

development and model preparation process, to reduce redundancy or

repetitiveness, and to aid in the general thinking process. Thus, a step towards

systematizing the “art” of solving large-scale nonlinear programming problems

is undertaken.

In this contribution, such a systematic workflow for the development of models

for optimization purposes presented in [1] is expanded, whereby the workflow

focuses on continuously differentiable models. Then, the workflow is applied on

an absorption-desorption process. For this purpose, multi-stage model

preparation is performed starting at the phenomena level and ending on the

process level.

2. Existing Strategies for Model Reduction

Most solvers applied today for solving non-linear programming problems are

gradient based methods and converge at fulfilled KKT conditions [2]. These

conditions usually require the implementation of continuously differentiable

models. This requirement is fulfilled by most classical short-cut models such as

McCabe-Thiele method (McCabe & Thiele, 1925). Their computation is also

usually quite low. Only their accuracy leaves something to be desired. This

aspect, on the other hand, is usually not an issue for models based on neural

networks or support vector regression (Nandi et al.). In general, these types of

models show a high accuracy for the operation points they were trained with.

However, differentiability, interpolation, and especially extrapolation can be an

issue here. Lastly, reduced order models (ROM) have become fashionable in

recent years. Especially notable are Lang et al. (2009), who deduced ROMs

from CFD simulations. The only disadvantages here are the expensive set-up

procedure and the small area of applicability.

3. Model Derivation for Optimization Purposes

The model derivation for optimization is divided into two parts. Firstly, the

general workflow for process systems engineers is presented and discussed.

This is followed by a selection of possible strategies for improving the numerical

behavior of systems.

3.1. Parameter Identification and Subset Selection

The workflow consists of 16 steps, which essentially should lead to a model

suitable for optimization purposes. This whole workflow is described in [1] and

shall just shortly be revisited here. The general idea is to firstly start off with the

goal definition. The engineer must decide what type of model is required. This

leads the engineer to Step 1. Here, the following question is stated: does a

model for the system under investigation exist? If the answer is yes, this leads

the engineer to the second question concerning the accuracy of the model

(Step 2). If the accuracy is high enough, the model must be analyzed regarding

its numerical convergence behavior. If this is also acceptable, then the model is

already ready for optimization purposes. More details on all of the steps will be

given in the following case study.

Figure 1. Systematic workflow for the development of models for optimization purposes

[1].

4. Case Study: Optimization Model of a CO2 Absorption Desorption

Process

The workflow shown above will now be applied to an absorption desorption

process for the separation of CO2 from a product gas stream, which contains

larger amounts of C2H4. The process model required is to be readied for

superstructure optimization under uncertainty. In this context, a convergence

time of below 1 sec is demanded and a level of accuracy compared to

experimental data of ±5%. The absorption of carbon dioxide (CO2) using

monoethanolamine (MEA) solutions is of course one of the more extensively

investigated processes. The solubility of CO2 in aqueous solutions of MEA has

been measured for a wide range of operating conditions as published by e.g.

(Shen et al., 1992) and (Jou et al., 1995). The development of the respective

kinetics started with (Clarke, 1964) and (Hikita et al., 1977). Simulation studies

have been carried out with a high degree of complexity. Yeh et al. (1999) and

Freguia et al. (2003) modeled the absorption rigorously and fitted their model to

lab-scale and field data.

4.1. Existing Mini-Plant

At Technische Universität Berlin a mini-plant exists, which features a full-scale

absorption desorption process for the removal of CO2. The absorption column

has a packing height of 5m and is operated at pressures of up to 30bar. The

desorption column has 4m of packing and is operated at temperatures of up to

130°C at a pressure of up to 2.5bar. Feed streams consisting of CO2, CH4,

C2H4, and N2 have been stripped of CO2 at varying operating conditions using a

30wt% monoethanolamine solution. The specific heat required for the removal

of CO2 and the ethylene loss in the desorption section were recorded (Stünkel,

2013).

4.2. Systematic Model Derivation

The starting point of the algorithm shown in Fig. 1 has already been discussed.

Also, given the extensive history of modeling MEA-based absorption of CO2, a

rigorous model can easily be derived (Step 1). As part of this contribution rate-

based simulations were implemented in Aspen Plus® using the Electrolyte-

NRTL package in combination with correlations for the mass and heat transfer

of the used package material (Mellapak 350 and 500) (Stünkel, 2013). The

simulations incorporating all nine ionic and non-ionic liquid as well as the five

gas components consist of roughly 2,000 state variables. To simulate a single

steady-state operation point of the mini-plant, simulations of the absorption and

desorption columns have to be prepared separately, before the recycles

between each are slowly closed. In order to switch to a different operation point,

the same procedure has to be repeated manually. Obviously, the convergence

behavior of the rigorous model as implemented in Aspen Plus® is highly

undesirable and not as required above. In addition, offsets greater than 10%

between model and experimental data from the mini-plant is observed, which

answers the question in Step 2 about the sufficiency of the accuracy. The

general behavior of the mini-plant can of course be expected to be seen in the

rigorous model (Step 5). Hence, it is initially tried to simply readjust the existing

model in Aspen Plus® (Step 10) to more closely mimic the experiments

performed in the mini-plant (Steps 11 through 14). The most important step

herein is the introduction of heat loss for each column. After this step, the

accuracy of the Aspen Plus® model is satisfactory (Step 2). However, the

convergence does not improve of course (Step 3).

4.3. Systematic Simplification of the Model (Step 4)

As described in (Esche et al., 2013) this step consists of a decomposition of an

existing model and a systematic investigation of simplification potential in each

different model part to ease the computational expense of the overall system.

The main subdivision is the classification of the governing equations into mass

balances, equilibrium equations, summations, energy and momentum balances,

and auxiliary equations. In this case, the main source for the bad convergence

behavior can be found in the bad scaling of the component balances. Several

components such as H3O+, OH-, and HCO32+ appear in tiny, but varying

amounts, close to zero, i.e. 10-9 to 10-4 mol/m3. Attempts to ease these troubles

by manually scaling the respective equations proved to be in vain as the

concentrations may vary depending on the state of the chemical equilibrium at

different positions in the plant. As these troubles cannot be resolved so that the

model does not repeatedly run into local infeasibilities, the decision was made

to find a formulation of the entire system which does without the trace

components. Given their necessity for the entire kinetic system a completely

new description thereof is required. Within Step 4 this leads to the decision to

acquire experimental data for the phenomena level.

4.4. Modeling at the Phenomena Level

Given the extensive level MEA-based CO2 absorption has already been

investigated, experimental data is available to great extents (Step 6). Therefore,

planning and performing lab-scale experiments can actually be skipped at this

point (Steps 7 and 8). Data published by Shen et al. (1992) and Kim (2007) on

the solubility of CO2 in 30wt% MEA solutions and the heat of absorption

respectively is employed to derive bivariate correlations. Fig. 2 shows the

experimental data (blue lines) and the respective correlations (colored surfaces)

for each (Step 9).

Figure 2. Solubility of CO2 (left) in a 30 wt.-\% MEA solution α depending on temperature

T and the partial pressure of CO2 pCO2 and heat of absorption of CO2 ΔhA (right)

depending on temperature T and the solubility α.

The underlying functions of the correlations are determined with respect to the

general curvature of the data. For this purpose the bivariate dependency is

especially useful. Based on the solubility and heat of absorption correlations no

description of the ionic composition is required. The system is reduced to three

liquid components (CO2, MEA, and H2O) and four gas components (CO2, H2O,

CH4, and C2H4). At the phenomena level these correlations are joined with a

basic description of the vapor pressure of water using the Antoine equation, a

correlation estimating the solubility of C2H4 in MEA solutions derived from data

published by Carroll et al. (1998). At this point, no additional information will be

given on the actual modeling seeing as it has already been published in (Esche

et al., 2013).

4.5. Fitting to Mini-plant Data

To model the whole mini-plant for the absorption desorption process, the

absorption column is divided into 40 equilibrium stages, for which component

mole balances are formulated as well as energy balances. One equilibrium

stage is used to model both the flash and the desorption column. Tray

efficiencies are introduced into the solubility equations for each equilibrium

stage. In addition, U

∗

A-based heat loss correlations and the heat of evaporation

of water are added. The liquid recycle from the desorption column back to the

absorption is modeled as a forwarding of the CO2 load, whilst MEA and H2O are

reinitiated to ease the computational complexity. This extended model is then

fitted to actual experimental mini-plant data reported by Stünkel (2013). The

experiments were carried out in the mini-plant varying the absorption pressure

between 5 and 32 bar, the gas load factor between 0.25 and 0.42 Pa0.5, the

feed concentrations of carbon dioxide between 0.14 and 0.26 mol/mol, the CO2

removal rate between 70 and 100%, as well as the flow of the scrubbing liquid

from 10 to 60 kg/h. It was found, that all parameters above can be set to

constant values, while the efficiency for the desorption column can be described

depending on five characteristic variables: the absorption pressure, the gas load

factor, the feed concentration of CO2, the removal rate of CO2, and the

scrubbing liquid flow. For the subsequent parameter estimation the solubility

correlation parameters, the stage efficiencies, and U

∗

A are used as decision

variables. This concludes Steps 10 to 14 in the systematic in Fig. 1.

4.6. Numerical Stabilization Strategies

The resulting, fitted model can represent all experimental points with a margin

of error of ± 3% with respect to the energy required per captured kilogram of

CO2. So the reevaluation in Step 2 gives a positive reply. However, when given

to Step 3, the model at this point still is not adequate. For this reason, it is

handed back to Step 4, in which at this point no simplifications but rather

reformulations are carried out. Among these are:

(1)

(2)

(3)

(4)

Eq. (1) and (2) are applied as reformulations for various fractions appearing in

the system. Whenever b can be expected to take values close to 0, the

reformulation from Eq. (2) is employed. This substitution is more

computationally expensive but stabilizes the fraction more reliably by also fixing

the non-differentiability. Eq. (1) instead does not rectify this, but seems to be

sufficient for most cases, in which b only infrequently turns zero during an

optimization or simulation iteration step. The substitution in Eq. (3) similarly is

only applied to cases, in which a is frequently smaller than zero. Lastly, Eq. (4)

is employed for reformulating equations which can be written explicitly in terms

of the logarithmic function. For the model derived here these four reformulations

were chosen. It would also have been possible to substitute the problematic

terms in the equations and add additional equations for those. However, seeing

as the system in hand is already quite sparse, the additional equations would

only increase the computational complexity as there would be no additional

benefit from a further increase in sparsity. Optimization studies carried out with

this model showed a fast and reliable convergence from different starting points

and usually converged within a second on a 64bit AMD Athlon X2 Dual Core

Processor 3800+. Hence, with respect both to accuracy and computational time

(Steps 2 and 3), this model is suitable for optimization.

5. Conclusions and Outlook

So far the systematic introduced in this article has been applied to two

fundamentally different systems and applied successfully to derive

computationally fast and accurate models for optimization. For future work, the

workflow will be reapplied to more complex systems and extended if necessary.

Additionally, the numerical stabilization strategies will be extended and

formalized further. The workflow as a whole will be implemented into the online

modeling platform MOSAIC.

Acknowledgments

The authors acknowledge the support from the Cluster of Excellence “Unifying

Concepts in Catalysis” and the Collaborative Research Center SFB/TR 63

InPROMPT “Integrated Chemical Processes in Liquid Multiphase Systems” both

coordinated by the Technische Universität Berlin and funded by the German

Research Foundation (Deutsche Forschungsgemeinschaft “DFG”).

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