(415b) Optimal Synthesis and Heat Integration Using Generalized Disjunctive Programming with Hybrid Models

Surrogate models (SM), also known as data-driven models or metamodels, are simplified functions that replace computationally expensive simulations to estimate output data from a set of input variables. SM are especially useful to represent black-box models, whose analytical form is unknown to include in an equation-oriented optimization framework. They can be used for different purposes such as simulation, optimization and flexibility analysis (Bhosekar and Ierapetritou, 2018).

A mathematical model that includes both first principle-based and data-driven models, is referred to as a hybrid or grey-box model. This type of model has proven to provide an efficient representation of complex systems (Bajaj et al., 2018). There are particular optimization frameworks to address hybrid models since surrogate models` accuracy is uncertain over the entire feasible region and in the neighborhood of the optimal solution. On the one hand, exploration-based strategies improve SMs global performance in the entire domain to reduce the probability of not finding the global optimum. On the other hand, exploitation-based methods refine SMs in regions where optima could be potentially found.

To address process synthesis problems, Generalized Disjunctive Programming (GDP) allows a more intuitive way of formulating a mathematical problem, introducing specific process information through disjunctive constraints and logical prepositions (Chen et al., 2018). Furthermore, GDP formulations can be solved using powerful decomposition techniques such as the Logic-based Outer-Approximation (L-bOA) algorithm (Türkay and Grossmann, 1996), which proceeds by decomposing the GDP into reduced NLP subproblems and master MILP problems.

In this work, we propose a novel optimization framework for the optimal synthesis of chemical process flowsheets through first principle-based and surrogate models, i.e., hybrid models embedded within a superstructure representation that is modeled with a GDP formulation and solved with a custom implementation of the L-bOA algorithm. We build surrogate models using a combination of Simple Algebraic Regression Functions (SARFs) and Gaussian Radial Basis Functions (GRBFs). In this way, we can exploit the benefits of each type of basis. On the one hand, SARFs are low complexity, interpretable, and suitable for optimization purposes. On the other hand, GRBFs can refine the approximation models to make them exact in the interpolation points, and consequently, they can capture highly nonlinear behavior in particular domain regions. The machine learning software ALAMO (Cozad et al., 2014; Wilson and Sahinidis, 2017) is employed to generate an initial model based on SARFs.

The proposed optimization framework is shown in Fig. 1. As a first step, initial bounds for the input variables are selected for each surrogate model to generate sampling data. The Latin Hypercube sampling (LHS) technique is employed in MATLAB. Original or true models are coded in GAMS, which are evaluated for each sampling point from MATLAB. Since some simulations can be infeasible due to the problem constraints, a filtering step to discard those outcomes is implemented. Next, ALAMO is used to build an initial SM based on SARFs. Since this initial SMs accuracy might not be good enough in all sampling points, we evaluate the corresponding relative errors, and add GRBFs to represent those points whose errors are greater than a tolerance. In this way, we build the first SM based on both, SARFs and GRBFs, to carry out the first exploration step. The Error Maximization Sampling (EMS) strategy is applied in the exploration step. This method consists of maximizing the SMs relative error in the entire domain, subject to the constraints of the original model. As a result, low-accuracy domain points are determined, and an interpolation point is included through a GRBF to improve the SM performance in that region.

In the following step, we solve the GDP problem with the hybrid model in GAMS. A custom Logic-based Outer Approximation (L-bOA) algorithm is employed. We exploit the information of the L-bOA subproblems to refine the SM in the exploitation step. As some NLP subproblems or simulations might be infeasible due to the performance of SMs, we formulate mathematical problems to determine the feasible sampling point that minimizes the Euclidean distance to the NLP subproblem solution, and this point is added in the training set.

The iterative algorithm, which is shown in Fig. 1, terminates until a convergence criterion is satisfied. Otherwise, the exploration step is carried out again (the number of major iterations of the algorithm is equal to the times the GDP problem is solved).

The objective function is the maximization of the net present value (NPV), subject to general equations, disjunctions for the units and logical constraints. After validating the hybrid model through the algorithm proposed in Fig. 1, it is extended with energy integration equations to perform simultaneous optimization and heat integration (Duran and Grossmann, 1986). Therefore, a new GDP is solved to analyze the potential improvement on the plant NPV associated to energy savings.

The case study we address is the optimal synthesis of a propylene production plant via olefins metathesis. As a first step, we formulate a detailed model, which includes distillation columns represented by mass, equilibrium, summation and heat (MESH) equations, considering ideal thermodynamics, and reactor, compressor, pump, and heat exchanger models (Pedrozo et al., 2021). In a second step, we propose SMs that replace rigorous models of columns, thus obtaining a “hybrid model”, whose equation number and complexity are reduced with respect to the true model.

We analyze the effect of the number of initial sampling points over the hybrid model generation and the algorithm computational performance. The proposed iterative procedure effectively determines the optimal solution in every analyzed case, requiring higher CPU times in the cases where the initial sampling point number is lower. We perform sensitivity analysis of the NPV with respect to raw material costs. Different ethylene price scenarios were set and the same optimal flowsheet configuration was determined for each of them, using both the true and hybrid models. Numerical results show that the optimal configuration largely depends on ethylene price. Furthermore, the solutions of the hybrid GDP model exhibit higher numerical stability than the true model. This feature allows the simultaneous optimization and heat integration of the plant, which provides a valuable insight on further improvement of the plant NPV.

References

Bajaj, I., Iyer, S.S., Faruque Hasan, M.M., 2018. A trust region-based two phase algorithm for constrained black-box and grey-box optimization with infeasible initial point. Comput. Chem. Eng. 116, 306–321. https://doi.org/10.1016/j.compchemeng.2017.12.011

Bhosekar, A., Ierapetritou, M., 2018. Advances in surrogate based modeling, feasibility analysis, and optimization: A review. Comput. Chem. Eng. 250–267. https://doi.org/10.1016/j.compchemeng.2017.09.017

Chen, Q., Johnson, E.S., Siirola, J.D., Grossmann, I.E., 2018. Pyomo. GDP: Disjunctive models in python, in: Computer Aided Chemical Engineering. Elsevier, pp. 889–894.

Cozad, A., Sahinidis, N. V, Miller, D.C., 2014. Learning surrogate models for simulation‐based optimization. AIChE J. 60, 2211–2227. https://doi.org/https://doi.org/10.1002/aic.14418

Duran, M.A., Grossmann, I.E., 1986. Simultaneous optimization and heat integration of chemical processes. AIChE J. https://doi.org/10.1002/aic.690320114

Fang, H., Rais-Rohani, M., Liu, Z., Horstemeyer, M.F., 2005. A comparative study of metamodeling methods for multiobjective crashworthiness optimization. Comput. Struct. 83, 2121–2136.

Pedrozo, H.A., Reartes, S.B.R., Vecchietti, A.R., Diaz, M.S., Grossmann, I.E., 2021. Optimal Design Of Ethylene And Propylene Coproduction Plants With Generalized Disjunctive Programming And State Equipment Network Models. Comput. Chem. Eng. 107295.

Türkay, M., Grossmann, I.E., 1996. Logic-based MINLP algorithms for the optimal synthesis of process networks. Comput. Chem. Eng. 20, 959–978. https://doi.org/https://doi.org/10.1016/0098-1354(95)00219-7

Wilson, Z.T., Sahinidis, N. V, 2017. The ALAMO approach to machine learning. Comput. Chem. Eng. 106, 785–795. https://doi.org/https://doi.org/10.1016/j.compchemeng.2017.02.010