(491h) Solving a Robust Multi-Scenario Dynamic Real-Time Optimization with Closed-Loop Prediction Via Generalized Benders Decomposition

In the traditional Dynamic Real-time Optimization approach [1], the economic optimization problem is decomposed into two layers. On the upper layer, a dynamic optimization problem is solved to obtain the optimal process variables trajectories that minimize/maximize an economic criterion while satisfying all the process constraints. Then, a model predictive controller (MPC) tracks the computed process variables trajectories on the lower layer.

A variant of the traditional approach, called Closed-loop Dynamic Real-time Optimization (CL-DRTO) [2], includes the MPC behavior in the upper layer problem to consider the control performance when making economic decisions. This approach performs better than the traditional “open-loop” version, particularly in situations where the controller is detuned (for instance, due to system dead time); however, the computational cost is higher because of the CL-DRTO solution method. It consists of transforming the MPC problem into complementary constraints using its conditions of optimality, which are then moved to the constraint set of the upper layer dynamic optimization problem resulting in a single-level mathematical program with complementarity constraints (MPCC).

To further increase the computational burden of the CL-DRTO solution, including uncertainty in the economic optimization has become a key tool to ensure competitive chemical plant operation [3]. Typically, a stochastic programming framework is used, where the uncertainty is approximated by discrete realizations of the uncertain parameters. Each realization is then defined as a scenario with a given probability. Since the objective of the stochastic programming is to optimize the expected value of the objective function over all the parameter realizations, this robust CL-DRTO formulation requires the solution of an MPCC for each scenario, which then are coupled by the nonanticipativity constraints.

This work seeks to obtain the solution of the robust CL-DRTO via a decomposition strategy based on generalized Benders decomposition (BD). Here, the problem is recast such that the variables involved in the nonanticipativity constraints are designated as complicating variables. Then, the monolithic problem is reduced to a series of independent scenario subproblems. The proposed solution strategy is applied to two case studies, a linear system and a system containing several CSTRs in parallel. It is shown that the proposed strategy outperforms the solution of the monolithic problem when either the number of scenario or the size of the subproblem increases. Another interesting result is that, although convergence cannot be proved since the MPCC problem is not guaranteed to be convex, we show experimentally that we are able to obtain the same process variables trajectories via the decomposed and monolithic solution strategies.

[1] Kadam, J., Marquardt, W., Schlegel, M., Backx, T., Bosgra, O., Brouwer, P.J., Dünnebier, G., Van Hessem, D., Tiagounov, A. and De Wolf, S., 2003. Towards integrated dynamic real-time optimization and control of industrial processes. Proceedings foundations of computer-aided process operations (FOCAPO2003), pp.593-596.

[2] Jamaludin, M.Z. and Swartz, C.L., 2017. Dynamic real‐time optimization with closed‐loop prediction. AIChE Journal, 63(9), pp.3896-3911.

[3] Grossmann, I.E., Apap, R.M., Calfa, B.A., García-Herreros, P. and Zhang, Q., 2016. Recent advances in mathematical programming techniques for the optimization of process systems under uncertainty. Computers & Chemical Engineering, 91, pp.3-14.