(61h) Integrated Design and NMPC-Based Control Under Uncertainty and Naturally Ordered Structural Decisions: A Discrete-Steepest Descent Approach.

The significance of dynamic and controllability aspects in the initial phases of process design has been emphasized by the academia and industry. The integration of design and control has been shown to have an impact on the closed-loop performance of the process [1]. Linear model predictive control (LMPC) represents the current state-of-the-art in industry for process control since it offers the feature to explicitly introduce constraints in the controller formulation. However, for highly nonlinear processes that require a wide range of operating conditions, a LMPC may not be sufficient to maintain the process on target [2]. A promising alternative to control highly nonlinear processes is Nonlinear MPC (NMPC) [3]. Although NMPC can manage wide operational changes in highly nonlinear processes and improve the process performance, very few studies have explored the integration of design and NMPC-based control because it demands the solution of a bilevel formulation [4], i.e., the optimal process design problem (upper-level problem) is constrained by the optimization of the NMPC (lower-level problem). Attempting the solution of bilevel problems is a challenging task because they are difficult to solve using conventional NLP solvers.

In a study presented by Hoffmann et al. [5], they implemented a fully discretized dynamic model in combination with an optimal economic NMPC for process design and control. Results showed notable changes in the process design when dynamic fluctuations (i.e., disturbances) and uncertainty are considered. Palma-Flores & Ricardez-Sandoval [6] implemented a decomposition strategy that approximates the bilevel problem using a power series expansion (PSE) approach. They observed a more economically attractive process design with improved control performance compared to solutions using LMPC and conventional PID controllers. A classical KKT transformation strategy to obtain a single-level mathematical program with complementarity constraints (MPCCs) was addressed by Palma-Flores & Ricardez-Sandoval [7]. That strategy allowed to fully incorporate the NMPC’s necessary conditions for optimality into the design problem as a set of algebraic constraints leading to a single-level optimization formulation; hence, it was possible to find solutions for the MPCC-based formulation using traditional NLP solvers, i.e., they avoided the use of decomposition strategies for the solution of the bilevel problem for integrated design and NMPC-based control. Consequently, this approach guaranteed local optimality in the solution. Those studies focused on the solution of formulations involving continuous decision variables, i.e., structural decisions were not considered. In a previous work [8], a first attempt for the integration of design and NMPC-based control under uncertainty and structural decision was presented. In that work, a branch & bound (B&B) algorithm was implemented for the solution of a single-level mixed-integer MPCC (MI-MPCC). That problem can be transformed into a conventional MINLP using reformulation strategies for complementarity constraints. It was found that the B&B strategy may return sub-optimal local solutions near the initial conditions. Moreover, the complexity of the problem is likely to increase with the number of integer decisions considered within the optimization formulation. This acts as the main barrier for the implementation of conventional solutions strategies with B&B algorithms for large-scale systems, in which large CPU times, low-quality local optima, or even lack of convergence to a solution may be experienced while using conventional MINLP solvers.

In this work, we address the integration of design and NMPC-based control under uncertainty and structural decisions. This study aims to develop algorithmic framework that determines the optimal location of processing units or streams that follow a naturally ordered discrete set, e.g., a sequence of liquid-liquid separators, the number of trays in a distillation column, or the order of reaction units, among others. The binary variables associated to the superstructure can be expressed as a function of reduced variable sets called external variables. This allows the decomposition of the single-level MINLP into a master Integer Problem with Linear Constraints (IPLC) and a set of primal sub-problems. The master problem is constructed from the external variables and solved with a Discrete-Steepest Descent Algorithm (D-SDA) [9]. On the other hand, the primal sub-problems are conventional nonlinear problems (NLPs) obtained by fixing the binary variables according to the solution obtained from the master problem. Given a feasible initialization for the integer variables, the D-SDA executes a local search to optimize the process according to an objective function. In this local search, the master problem explores the neighborhoods of integer decisions by fixing their values and solving the corresponding primal sub-problems using conventional NLP solvers. This aims to determine a search direction that provides the steepest descent in the objective function. The algorithm converges to a local optimum when the neighborhood search does not show improvement in the objective function. The proposed algorithmic framework does not follow the usual definition of convexity for integer programs, but it is based on the definition of integral convexity. Consequently, a steepest-descent direction strategy can be implemented to explore the discrete search space. Thus, the proposed DSDA computes local solutions that cannot be efficiently identified using conventional MINLP solvers. Note that the decomposition strategy reduces the complexity of the formulation because all logical constraints are solved in the master problem; consequently, the primal sub-problems are simpler NLPs compared to the original MINLP. This D-SDA methodology was previously introduced by Linan et al. [9]. They implemented this method for the optimal process design of a reactive distillation column at steady-state. In this work, we illustrate the implementation of the D-SDA approach for the integration of design and NMPC-based control under uncertainty and structural decisions of a binary distillation column. This problem aims to determine the location of the feed and reflux streams, the diameter of the column, the values for the controller tuning parameters, and the operation set-points using a nonlinear process model, whereas three uncertain model parameters and process disturbances are also considered. To compare the performance and the solutions obtained with the proposed D-SDA approach, we implement an alternative benchmark methodology based on the distributed stream-tray optimization method (DSTO) presented by Lang & Biegler [10]. In the DSTO methodology, the stream locations for feed and reflux are allowed to be continuous variables in a differentiable distribution function (DDF), e.g., Gaussian distribution functions. That methodology assumes the solution of a continuous formulation that can be solved with conventional NLP solvers. The results show that the DSTO method is more sensitive to the initialization of the superstructure than the D-SDA, i.e., the DSTO method converged to local solutions that are closer to the initialization point. On the other hand, the D-SDA allowed to skip sub-optimal solution regions because the framework does not demand the relaxation of the discrete variables. This allowed the computation of a more economically attractive solution with better control performance using the D-SDA approach, i.e., a 39% more economic distillation column’s designs with fewer trays and faster control responses were obtained using the proposed D-SDA approach compared to the solution with the DSTO method.

References

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[5] Hoffmann, C., Esche, E., & Repke, J. U. (2019). Integration of Design and Control Based on Large-Scale NLP Formulations and An Optimal Economic Nmpc. In Computer Aided Chemical Engineering (Vol. 47, pp. 125-130). Elsevier.

[6] Oscar Palma-Flores and Luis A Ricardez-Sandoval. Simultaneous design and nonlinear model predictive control under uncertainty: A back-off approach. Journal of Process Control, 110:45–58, 2022.

[7] Oscar Palma-Flores and Luis A Ricardez-Sandoval. Integration of design and nmpc-based control for chemical processes under uncertainty: An mpcc-based framework. Computers & Chemical Engineering, 162:107815, 2022.

[8] Oscar Palma-Flores and Luis A Ricardez-Sandoval. Integration of design and nmpc-based control under uncertainty and structural decisions: An mpcc-based approach. IFAC-PapersOnLine, Accepted, 2023.

[9] David A Linan, David E Bernal, Luis A Ricardez-Sandoval, and Jorge M Gómez. Optimal design of superstructures for placing units and streams with multiple and ordered available locations. part i: A new mathematical framework. Computers & Chemical Engineering, 137:106794, 2020.

[10] Y-D Lang and LT Biegler. Distributed stream method for tray optimization. AIChE Journal, 48(3):582–595, 2002.