(522e) Stochastic Design Optimization and Control of Modular Membrane Reactor Units

Modular systems are more beneficial than traditional stick built processes due to their flexibility, shorter lifecycle and mobility aspects [1]. The size of a modular unit is constrained by its transportation capacity, and the modular design and the implemented controller need to handle a wide range of inputs and disturbances. The optimal design and control policies for chemical systems have direct impacts on each other [2], as different designs result in systems with different reachable setpoints. Also, under the same random disturbance, an alteration in the controller changes the back-off level to prevent constraint violations. Depending on the required back-off, the system design can be further relaxed or constrained to achieve feasibility and optimality. In previous works, the design optimization and intensification of modular units have been addressed without considering the performance of the controller [3, 4]. Additionally, a back-off algorithm was proposed for the design, control, and scheduling of multiproduct systems, in which the effect of stochastic variables is assumed to be insensitive to the decision variables [5]. In this work, to solve the integrated design optimization and control of a modular unit, a two-stage stochastic programming problem is formulated considering stochastic sensitivity. In this formulation, the first stage decisions define the design specifications, and the second stage decisions specify the controller parameters.

Under the influence of stochastic disturbances, the expected values of the nonlinear functions in an optimization problem are shifted from the nominal values by some amounts corresponding to the projections of the probabilistic density functions of the random variables. Because of the differences in the expected values and the nominal values, the objective function along with some constraints of the optimization problem need to be modified to achieve a tractable deterministic form for the stochastic program. Traditional Monte Carlo approaches used for this task require a large number of samples and are computationally expensive. A modified Monte Carlo method is explored in this work based on the fact that the probability contained in a differential area of the density function must be invariant under the change of variables.

For the application of the presented work, a membrane reactor for Direct Methane Aromatization (DMA-MR) conversion of natural gas to hydrogen and benzene is optimized in terms of size and model predictive control (MPC) weighting matrices that define the controller objective function. The model of the membrane reactor used was thoroughly presented in [6]. In addition to maximizing the design specifications, the developed model is used to construct the achievable and desired output sets (AOS and DOS) as well as the feasible desired input set (DIS) for the controller. The tuning of the MPC parameters is designed to guarantee controllability and observability within the DOS and the DIS under the influence of stochastic disturbances. The dimensions of the DMA-MR are limited to ensure its modular mobility to different well locations. The stochastic first stage decision is the design of the DMA-MR considering different realizations of the random variables characterized by varying natural gas concentrations. The stochastic second stage decision is the tuning of the MPC after the feed concentration is determined.

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[4] Gazzaneo, Vitor, and Fernando V. Lima. "A multi-Layer operability framework for process design, intensification and modularization of nonlinear energy systems." Industrial & Engineering Chemistry Research (2019). DOI: 10.1021/acs.iecr.8b05482

[5] Koller, Robert W., Luis A. Ricardez‐Sandoval, and Lorenz T. Biegler. "Stochastic back‐off algorithm for simultaneous design, control, and scheduling of multiproduct systems under uncertainty." AIChE Journal 64.7 (2018): 2379-2389.

[6] Carrasco, Juan C., and Fernando V. Lima. "Novel operability‐based approach for process design and intensification: Application to a membrane reactor for direct methane aromatization." AIChE Journal 63.3 (2017): 975-983.

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