(483a) Dynamic Real-Time Scheduling and Control Under Uncertainty

In recent years, several papers have addressed the integration of scheduling and control (ISC) either in the form of a review of developments and trends^1-3, or to propose an integration framework^4-7. However, ISC under uncertainty in a real-time setting has received little attention. The present work aims to address this gap through an ISC framework that utilizes a two-stage stochastic formulation, coupled with a feedback mechanism based on measurements of plant response variables and inventory levels. We consider uncertainty in demand and process model parameters, and consider MPC as well as PI control as the plant control system.

The ISC formulation proposed in this work follows a two-level automation structure in which plant economics are considered at an upper, dynamic real-time optimization (DRTO) level, which provides a set-point trajectory to an underlying regulatory control level. A schematic representation is given in Figure 1. A key feature of our approach is the use, at the DRTO level, of the closed-loop dynamic response of the plant under the action of the regulatory control system, referred to as closed-loop DRTO (CL-DRTO)⁸. It was shown in earlier work⁸ that utilization of the open-loop plant response at the DRTO level results in suboptimal performance, particularly when the regulatory controllers are detuned, such as in the presence of dead time and right-half plane transmission zeros. Later work included scheduling decisions at the DRTO level⁶, where the DRTO problem was posed as a MILP. In this work, we extend the framework to account for uncertainty directly within the DRTO problem formulation.

We first consider the case of multiproduct plants controlled by a constrained linear MPC (Figure 1a) and operated under uncertainty in the demand level of the distinct product grades. The MPC action can be rigorously accounted for in the DRTO problem formulation via the equivalent first-order Karush-Kuhn-Tucker conditions⁸ where the complementary conditions are exactly reformulated as mixed-integer linear constraints⁶. Because the resulting problem can be computationally expensive to solve, Jamaludin and Swartz (2017) propose three different methods to approximate the closed-loop prediction, including an input-clipping approximation⁹ where the constrained MPC control action is approximated by the unconstrained counterpart for which an analytical solution exists¹⁰, coupled with a truncation mechanism for inputs that exceed physical constraint bounds. We compare the performance of both the rigorous and input-clipping approaches in the present study. The proposed ISC formulation also requires a dynamic process model. For nonlinear models, we explore linearization as well as piecewise affine approximation to maintain an MILP formulation of the DRTO problem. Demand uncertainty is accounted for using a scenario-based stochastic optimization approach in which each potential realization of the demand uncertainty constitutes a scenario¹¹. Application of the proposed ISC formulation in closed-loop simulations led to a 10% increase in the profit of a linear plant and a 60 % increase in the profit of a nonlinear plant, respectively, compared to the case where the demand uncertainty is not accounted for, and the nominal deterministic problem is solved in the DRTO level instead.

Second, we consider a linear plant that has a PI as control system (Figure 1b) and is operated under demand as well as model uncertainty. As with the MPC, the PI control action is modelled in the DRTO formulation, providing it the ability to predict the closed-loop plant response. The scenario-based stochastic approach is again used to account for uncertainty at the DRTO level. The performance of the proposed approach is evaluated through closed-loop simulations for different execution frequencies of the DRTO layer. As before, significant improvement in the expected profit of the plant is achieved by accounting for uncertainty. Increasing the execution frequency leads to considerable improvement in the performance of the nominal DRTO problem. The scenario-based formulation still shows a performance improvement over the nominal formulation, but the difference in performance is reduced, demonstrating the importance of the feedback frequency in DRTO applications under uncertainty.

The proposed ISC formulation advances the current state-of-the-art by accounting for model and demand uncertainty in the DRTO level while also predicting the closed-loop process response. It is shown to led to significant improvement in the expected profit of the plant even in the presence of uncertainties that are not accounted for in the problem formulation. We also provide valuable insights regarding the online closed-loop implementation of integrated scheduling and control frameworks. In future work, we plan to address further sources of uncertainty such as uncertainty in the cost coefficients of the DRTO objective function, and investigate decomposition approaches to reduce computation times.

References

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