(575a) Sampling Domain Reduction for Surrogate Model Generation – Applied to Hydrogen Production with Carbon Capture

The definition of the sampling
domain impacts both the performance of a surrogate model and the number of
sampling points required to obtain a satisfactory fit. The most commonly
applied approach to limit the sampling domain is by use of simple box
constraints for each of the independent variables. This leads to a choice
between two problems; 1. The chosen box constraints are tight, which limits the
applicability of the surrogate model, and 2. the chosen box constraints are
large, causing weak bounds on the relevant sampling domain and thereby sampling
in operating regions never encountered in the application of the surrogate
model. The latter is particularly a problem in chemical engineering, in which the
component flow rates are normally depending on each other. Here, the
application of box constraints and an adaptive sampling algorithm may result in
extensive sampling in regions far outside the nominal operating conditions.
This, in turn, may cause sampling in regions that exhibit highly nonlinear characteristics
that are not relevant or prevailing in the regions of nominal operations
conditions [1].
Structured sampling domain reduction through incorporation of constraints from
known physical relations between the chosen independent variables may
significantly improve the numerical efficiency of adaptive surrogate model
generation.

If an inlet stream to the
surrogate model is the overall feed to the system, it is straightforward to
implement proportional or inverse proportional dependencies in-between the
component flow rates [2].
This already limits the size of the sampling domain to relevant regions, as compositions
far away from the nominal operation conditions are rarely encountered. However,
this approach fails if the feed stream to the surrogate model is the product of
a chemical reaction or, to a lesser extent, the product of a separation. As a
motivating example, consider the product stream of a steam methane reformer (SMR)
which is fed to the water-gas shift reactors (WGS). Two contradictory
conclusions may be drawn for the dependency between methane and hydrogen:

1.       the more methane is in the feed to the water-gas shift reactors,
the more hydrogen is in the feed due to a larger inlet flow rate of methane to the
steam methane reformer while maintaining a similar extent of reaction
(proportional dependency);

2.       the more methane is in the feed to the water-gas shift reactors,
the less hydrogen is in the feed due to a reduced extent of reaction in the
steam methane reformer (inverse proportional dependency).

However, it is not possible to draw
a conclusion on the exact nature of the dependency. Hence, we propose to use a
data-driven approach to solve this problem and identify a constrained sampling
domain of relevance that is, regions which can be achieved in practice based on
the outlet conditions of the previous unit operations. In the first step of the
approach, it is therefore necessary to sample points for the feed composition
of the surrogate model with  components. Here, the previous
unit operations are used for creating outlet points. Based on the already sampled
points (denoted by superscript cal),
it is possible to calculate in total  inequality constraints
given by

The first set of inequality constraints corresponds to box
constraints and define upper and lower bounds of each component flow rate. The
second set of constraints limits the ratios of two component flow rates,
whereas the last set limits the sum of two component flow rates. In the context
of the outlined issues above, the second set of constraints provides bounds on
proportional dependencies in-between the component molar flow rates, whereas
the third set provides bounds on inverse proportional dependencies. Note that
these inequalities always define a convex set and thus polytopic constraints in
.

The sampling of
the points requires evaluations of the previous sections in the detailed model,
e.g. the SMR in the case of sampling for a WGS. This can act as
showstopper for the reduction of the sampling domain. However, if a surrogate
model has been fitted to the previous section, then this surrogate model can be
used for the calculation of the inequality constraints at limited computational
expenses. This is for example the case in the procedure outlined in [3]
and as well a part of the philosophy of the ALAMO approach [4].

While implementation of inequality
constraints in the sampling is in general relatively straight forward, the
complexity depends on the chosen sampling approach. Adaptive sampling
algorithms frequently utilize a black-box solver for finding regions for optimal
sampling of the simulator model due to a lack of access to the code of the
simulator. This approach enables addition of inequality constraints for sampling
domain reduction, provided that the black-box solver admits general constraints.
Static (predefined) sampling approaches require, however, that points are
placed within the bounds directly. One possibility is to use only the box
constraints for defining a set of sampling points, discard all points which are
infeasible, and then select an optimal subset of the feasible points. This
approach is utilized in the ARGONAUT algorithm [5],
but may result in a large fraction of discarded points. We implement an
iterative surrogate-model generation approach, using a LASSO-based approach [6]
with polynomial basis functions for surrogate fit and complexity reduction of
the surrogate model, together with an adaptive sampling technique with linear
constraints for reducing the sampling domain as described above.

The sampling domain reduction is applied
to a model of a SMR in Aspen HYSYS. A WGS reactor is located after the SMR and
shall be modelled as a new surrogate model. Five chemical components in the outlet
of the SMR are identified to have dependencies, CH₄, H₂O,
CO, CO₂, and H₂. If proportional dependencies are
incorporated in the feed to the SMR, it is possible to reduce the sampling
domain to 1 % of the size of the box constraints, whereas if we do not
incorporate proportional dependencies in the feed to the SMR, we can reduce the
sampling domain to 14 % of the total sampling domain. Figure 1 is
illustrating the dependencies between steam, methane, and hydrogen based on the
sampled points and the inequality constraints when proportional dependencies
are incorporated in the feed to the SMR. These dependencies are especially
pronounced between hydrogen and steam showing the necessity to incorporate
structured sampling domain reduction to improve the sampling for surrogate
model generation.

Figure 1:
Illustration of the dependencies in-between a)
methane and steam, b) methane and hydrogen, and c) steam and hydrogen including
the bounds illustrated in Eqs. (1)-(3).

References:

[1]         J. Straus and S. Skogestad, “Surrogate
model generation using self-optimizing variables,” Comput. Chem. Eng.,
vol. 119, pp. 143–151, Nov. 2018.

[2]         J. Straus and S. Skogestad, “Use of
Latent Variables to Reduce the Dimension of Surrogate Models,” in Computer
Aided Chemical Engineering, 2017, vol. 40, pp. 445–450.

[3]         J. Straus and S. Skogestad, “Minimizing
the complexity of surrogate models for optimization,” in Computer Aided
Chemical Engineering, 2016, vol. 38, pp. 289–294.

[4]         A. Cozad, N. V. Sahinidis, and D. C.
Miller, “Learning surrogate models for simulation-based optimization,” AIChE
J., vol. 60, no. 6, pp. 2211–2227, Jun. 2014.

[5]         F. Boukouvala and C. A. Floudas,
“ARGONAUT: AlgoRithms for Global Optimization of coNstrAined grey-box
compUTational problems,” Optim. Lett., vol. 11, no. 5, pp. 895–913, Jun.
2017.

[6]         H. Zou, “The Adaptive Lasso and Its
Oracle Properties,” J. Am. Stat. Assoc., vol. 101, no. 476, pp. 1418–1429, Dec. 2006.