(689a) Integrated Design and NMPC-Based Control Under Uncertainty and Structural Decisions: A D-SDA and Mpcc-Based Strategy.

Studies on integration of design and control have demonstrated that the closed-loop performance of the process is directly affected by design decisions [1]. Linear model predictive control (MPC) 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. Conventionally, linear models are selected for the internal model in MPC (i.e., a linear MPC). However, for highly nonlinear processes, the use of a linear MPC may not be sufficient to maintain the process on target. In those cases, nonlinear MPC (NMPC) is preferred to control those systems; however, they may be computationally more expensive compared to linear MPC [2]. Recently, few works have tackled the integration of design and NMPC-based control for continuous optimization formulations. In a study presented by Hoffmann et al. [3], they implemented a fully discretized dynamic model in combination with an optimal economic NMPC for process design. Results showed notable changes in the process design when dynamic fluctuations (i.e., disturbances) and uncertainty are considered. Palma-Flores and Ricardez-Sandoval [4] implemented a back-off methodology for integrated design and NMPC-based control. They identified more economically attractive and improved control performance process designs when compared to those obtained from linear MPC and conventional PID controllers. Nevertheless, those studies have focused on the solution of continuous formulations. Problems involving discrete (structural) decisions is a challenging task. The introduction of an NMPC-based control for simultaneous design and control leads to a formulation with two decision makers, i.e., the design problem (upper-level problem) requires the control actions to accommodate the process design, while the NMPC problem (lower-level problem) requires the information of the process model to compute optimal control actions. This formulation, with co-dependent optimization problems, is often referred to as a bilevel problem formulation. Bilevel problems with integer decisions can be classified in several categories depending on the location of the integrality conditions [6].

In this work, we are focused on the solution of formulations with mixed-integer upper-level problems and continuous lower-level problems, i.e., a mixed-integer bilevel problem (MIBLP) of type IV [6]. The use of mathematical programs with complementarity constraints (MPCCs) allows the transformation of the original bilevel formulation into a single level optimization formulation. In the MPCC-based strategy, the lower-level problem is expressed in terms of the first-order Karush-Kuhn-Tucker (KKT) conditions of optimality. The resulting system of algebraic equations is embedded as constraints to the master optimization problem (i.e., the design problem). This leads to a single level mixed-integer nonlinear problem with complementarity constraints. In this work, we implement the discrete-steepest descent algorithm (D-SDA) introduced by Linan et al. [7] to handle the integer decisions in the formulation. D-SDA was proposed to optimally place units and streams over superstructures with ordered locations. In the D-SDA method, the mixed-integer problem is decomposed into a master problem and a primal sub-problem. In the master problem, the integer variables are treated as external variables. Given a feasible initialization for the integer variables, D-SDA executes a local search to optimize the process according to an objective function. In the local search, the master problem explores in the neighborhood of the integer decisions by fixing their values and solving the corresponding primal sub-problem, i.e., the MPCC-based formulation for design and control with fixed values on the integer variables. Therefore, the primal sub-problems are solved as conventional nonlinear programming problems (NLPs). This aims to determine the direction of search that provides the steepest decent in the objective function. The algorithm converges to a local optimum when the neighborhood search does not show improvement in the objective function. In general, MPCC-based formulations lead to highly degenerated formulations that are difficult to solve with conventional NLP solvers. Hence, the local search with the D-SDA approach may increase the CPU burden. To circumvent this issue, we explore first the implementation of a linearized MPCC-based formulation to carryout the local search in the D-SDA strategy. Once the steepest decent search direction is identified, the full nonlinear MPCC-based formulation is implemented for the optimization. To verify the validity of the linear neighbor search, the nonlinear formulation is evaluated with the linear solution. If the difference between the optimum value obtained from the linear model and the result of the evaluation using the nonlinear system is less than a user-defined tolerance, then the specific branch is solved by using the nonlinear formulation to determine a new neighbor solution to analyze. The proposed D-SDA approach offers the feature to avoid the solution of a relaxed MINLP to initialize the optimization search. Therefore, the D-SDA approach does not present zero flow issues (a common issue when conventional MINLP solvers are implemented), i.e., the presence of disjunctive constraints does not lead to nonconvexity issues when the integer decisions are relaxed. Our implementation is illustrated with a case study that aims to design a binary distillation column during an operation in closed-loop. We explore the effect on the initialization of the integer variables and the effectiveness of the methodology to converge to an optimum solution. The D-SDA strategy is compared with some of the existing deterministic solution strategies; in particular, state-of-the-art MIINLP solvers such as standard branch and bound (SBB). The results show that the initialization of the integer variables may lead to different local optimums. Moreover, the methodology decreased considerably the CPU time required to converge to an optimum when compared to those CPU times recorded by the conventional MINLP solvers. Hence, the proposed D-SDA methodology has the potential reduce the CPU cost for the solution of integration of design and NMPC-based control problems involving structural decisions. In addition, the use of a MPCC approach has the potential to guarantee optimality in the solution of the bilevel formulation thus making this approach attractive for simultaneous design and control.

References

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