OPTIMAL DESIGN AND HEAT INTEGRATION WITH HYBRID MODELS USING GENERALIZED DISJUNCTIVE PROGRAMMING

PEDROZO, HÉCTOR ALEJANDRO; DIAZ, MARÍA SOLEDAD; RODRIGUEZ REARTES, SABRINA BELÉN; GROSSMANN, IGNACIO E.; BERNAL, D.E.

Congreso; AICHE ANNUAL MEETING 202; 2021

Surrogate models (SM), also so-called data-driven models or metamodels, are simplified functions that can replace computationally demanding simulations to estimate output data from a set of input variables. SM are especially useful to represent black-box models, which 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, it 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 special optimization frameworks to address hybrid models since the accuracy of surrogate models is uncertain beforehand in the entire feasible region and in the neighborhood of the optimal solution. On the one hand, exploration-based strategies improve the global performance of the SM in the entire domain to reduce the probability of skipping the global optimum. On the other hand, exploitation-based methods refine SM in regions where optima could be potentially found. To address process design 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). Besides, 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. The objective of this work is to develop a novel optimization framework for chemical process optimal design through first principle-based and surrogate models, which are embedded within a superstructure representation that in turn is modeled as a GDP model, and solved with a custom implementation of the L-bOA algorithm. We build surrogate model using a combination of Simple Algebraic Regression Functions (SARF) and Gaussian Radial Basis Functions (GRBF). In this way, we can profit from the benefits of each type of basis. On one side, SARF are low complexity, interpretable, and suitable for optimization purposes. On the other hand, GRBF can refine the approximated models to make them exact in the interpolation points, and consequently, it can capture highly nonlinear behavior in special domain regions. The machine learning software ALAMO (Cozad et al., 2014; Wilson and Sahinidis, 2017) is employed to generate an initial model based on SARF. We also use MATLAB as platform for data handling and solving linear systems to find GRBF coefficients. The proposed optimization framework is shown in Fig. 1. As a first step, initial bounds for 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 could be infeasible due to problem constraints, a filtering step to discard those outcomes is carried out. Then, ALAMO is used to build an initial SM based on SARF. Since the accuracy of this initial SM could not be good enough in all sampling points, we evaluate the corresponding relative errors, and we add GRBF to represent those points whose errors are greater than a tolerance. In this way, we build the first SM based on both, SARF and GRBF, 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 relative error of the SM in the whole domain, subject to the constraints of the original model. As a result, low accurate domain points are determined, and an interpolation point is included through GRBF to improve the SM performance in that region. In the following step, we solve the hybrid GDP problem in GAMS. A custom Logic-based Outer Approximation (L-bOA) algorithm is employed. We profit from the information of L-bOA subproblems to refine the SM in the exploitation step. Each optimal solution from NLP subproblems is a candidate for the optimal solution, so we evaluate the SM at these points to assess their accuracy, and eventually to include these points in the training data. As some NLP subproblems or simulations could be infeasible due to the performance of SM, we formulate mathematical problems to find the feasible sampling point that minimizes the Euclidean distance to the solution from the NLP subproblem, and this point is added in the training set. The algorithm iterates as shown in Fig. 1 until a convergence criterion is satisfied. If not satisfied, the exploration step is executed again (the number of major iterations of the algorithm is equal to the times the GDP problem is solved).The objective function is the net present value (NPV) maximization, subject to general equations, disjunctions for units, as well as logical equations. After validation of 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.As case study, we consider a propylene production via olefins metathesis plant. 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, as well as reactor, compressor, pump, and heat exchanger models (Pedrozo et al., 2021). In a second step, we propose a SM that replaces rigorous models of columns, 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 performance of the algorithm. As a result, the proposed iterative procedure is effective in determining the optimal solution in every case, but with higher CPU times for the cases where the initial sampling points number is lower. We perform sensitivity analysis of the NPV with respect to the main raw materials. Different ethylene price scenarios were set and the optimal configuration was determined for each of them, using both the true model and the hybrid model. Numerical results show that the plant optimal configuration highly depends on ethylene price. Furthermore, the hybrid model GDP solution shows 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.