A stochastic optimization framework for integrated scheduling and control under demand uncertainty

Highlights

• Performs online closed-loop scheduling while accounting for demand uncertainty.

• Control action and process dynamics are accounted for in the scheduling decisions.

• Two-stage stochastic formulation for the integrated scheduling and control problem.

• Nonlinear models are approximated via piecewise affine segments.

• The two-level control architecture is readily applicable to industrial processes.

Abstract

Increased globalization and energy market deregulation are requiring process industries to respond more rapidly to fluctuations in demand levels, and utility and raw material prices, in order to remain competitive. In this study, a two-stage stochastic approach is proposed to account for demand uncertainty in a closed-loop dynamic real-time optimization (CL-DRTO) formulation that includes scheduling decisions. The CL-DRTO problem utilizes a prediction of the closed-loop response of the plant under the action of constrained MPC. The CL-DRTO system is executed in a rolling horizon fashion to compute economically optimal operation that is communicated to the plant through set-point trajectories assigned to the plant MPC. Nonlinear plant models are approximated using linear and piecewise affine (PWA) approximations, allowing the integrated CL-DRTO problem to be formulated as a mixed-integer linear program (MILP). Case studies demonstrate significantly higher expected profit using the proposed formulation than with a deterministic formulation utilizing the expected demand.

Introduction

In the chemical process industry, scheduling is historically carried out using a steady-state process model where only an estimate of the process transition times is required. Scheduling involves defining the production sequencing, production amounts, equipment usage, among others, for a period of hours to days. The relevant scheduling information is provided as targets to an underlying decision-making layer, or applied directly to the plant.

Figure 1, adapted from Darby et al. (2011), illustrates the process automation hierarchy commonly implemented in refineries and large chemical plants. Also indicated are the time scales typically associated with the decision-making layers. Model predictive control (MPC) is the advanced control method of choice in the chemical process industry, and is typically configured to provide set-points to local plant PID-type controllers (Qin and Badgwell, 2003). Figure 1 includes real-time optimization (RTO), an economic optimization layer that traditionally utilizes a steady-state plant model, and provides set-points to the underlying MPC system. Production scheduling is the next level up. We remark that there are several variations of this process automation architecture. For example, the RTO layer may be replaced by a dynamic RTO (DRTO) system that utilizes a dynamic plant model (Kadam, Schlegel, Marquardt, Tousain, van Hessem, van den Berg, Bosgra, 2002, Tosukhowong, Lee, Lee, Lu, 2004, Jamaludin, Swartz, 2017), or the RTO layer could be entirely absent, with the scheduling decisions passed directly to the MPC system (Baldea and Harjunkoski, 2014).

Current trends toward increased globalization and deregulation of energy markets have resulted in significant variation in demand, supply, raw material costs, and energy prices, that together have created a highly dynamic environment in which plants are required to operate. Associated with the more dynamic plant operating environment is an increased frequency of scheduling decisions. Neglecting plant and control system dynamics under these conditions would generally lead to suboptimal decisions - a notion that underpins recent research activity in the integration of scheduling and control. Key conceptual paradigms toward achieving this integration are identified in Baldea and Harjunkoski (2014) and Caspari et al. (2020) as “bottom-up” and “top-down” approaches.

In the “bottom-up” approach, the scheduling task is incorporated by the control layer, allowing for direct computation of the control input values and scheduling decisions applied to the process. An example is the integration of scheduling and economic model predictive control (McAllister et al., 2019). While this approach eliminates the need for the traditional set-point tracking MPC controller, it entails significant changes to the process control architecture currently in place in industry. Conversely, the “top-down” approach consists of augmenting the optimization problem solved at the scheduling layer with information regarding the process dynamics and control. The integrated problem is solved to compute set-point targets for the process control layer that is, then, kept intact.

In the next paragraphs we present a brief literature review on different strategies to integrate scheduling and control using the more prevalent “top-down” approach. The studies differ in the formulation of the integrated scheduling and control (ISC) problem and strategies for online implementation of the optimal decisions. We refer to a formulation as “closed-loop” (CL) when it accounts for the closed-loop process behavior, that is, the process model captures the plant response under the feedback action of the plant control system. On the other hand, we use the term closed-loop implementation to mean that the ISC problem is solved online and receives feedback information from the actual process. There are some studies where the formulation is closed-loop but the implementation is open-loop (OL), and vice-versa. An open-loop implementation means that the ISC problem is solved offline, providing optimal targets that are tracked online by an appropriate controller, or optimal inputs that are applied directly to the plant. Several studies focus on the formulation and solution of the ISC problem, and hence do not provide results regarding the online implementation (Flores-Tlacuahuac, Grossmann, 2006, Pattison, Touretzky, Johansson, Harjunkoski, Baldea, 2016, Zhuge, Ierapetritou, 2014). A categorization of several studies is presented in Table 1.

Flores-Tlacuahuac and Grossmann (2006) compute the optimal cyclic production sequencing and control input values for multiproduct CSTRs. The manipulated inputs are computed directly, thus the strategy is characterized here as an open-loop formulation. Process transitions times are calculated interactively as part of the proposed solution algorithm that involves the solution of mixed-integer nonlinear programming (MINLP) problems for fixed transition times. Zhuge and Ierapetritou (2012) present an ISC formulation for online closed-loop operation of a nonlinear process. The closed-loop implementation consists of solving the ISC problem and applying the optimal input values to the process. The ISC problem is re-solved and new input values computed when the difference between the process states and the ISC prediction exceeds a given threshold. In Zhuge and Ierapetritou (2015), the nonlinear process model is approximated via piecewise affine (PWA) segments, allowing formulation of the ISC problem as a mixed-integer linear programming (MILP) problem. The optimal state profiles serve as reference values for the plant MPC, based on the PWA model, to track. Large deviations from the reference states trigger execution of the ISC calculation.

A known challenge in incorporating dynamics into the scheduling problem is the resulting computational complexity (Flores-Tlacuahuac and Grossmann, 2006). The situation becomes worse when accounting for the control action to model the closed-loop process dynamics. To deal with this issue, a number of studies replace the complex high fidelity process model with a reduced-order model. Du et al. (2015) present an ISC scheme for optimal production in a cyclic manufacturing process. Closed-loop dynamics, based on the action of an input-output feedback linearizing controller, are represented through a so-called time scale-bridging model (SBM) that is incorporated into the scheduling formulation. An optimal set-point trajectory determined from the solution of the ISC problem is applied in an open-loop fashion, and tracked online by an input-output linearizing controller. Pattison et al. (2016) utilize scheduling-relevant low-order dynamic models to represent the closed-loop transition dynamics in an ISC formulation. Continuous-time, nonlinear Hammerstein-Wiener models are used in an air separation application study. Dias et al. (2018), in an ISC formulation applied to an air separation plant, account for the closed-loop plant dynamics through a simulation-based optimization approach. A linear state-space plant model is identified using open-loop response data generated from a high-fidelity plant model. The ISC optimization problem is solved in an iterative manner, in which flow set-point iterates are generated by an optimization module, then applied to a simulation module in which the predicted plant response under the action of MPC is determined. Once converged, the optimal set-points are applied to the plant MPC system. The scheduling layer is implemented in open-loop, although the authors remark that it can be applied in a moving horizon fashion. Pattison et al. (2017) utilize low-order, scheduling-relevant Hammerstein-Wiener SBMs in an air separation production scheduling application, and incorporate feedback to the integrated scheduling system in a moving horizon framework.

Zhuge and Ierapetritou (2014) utilize a multi-parametric MPC (mp-MPC) formulation that expresses the MPC solution as a piecewise affine function of the plant state (Bemporad et al., 2002), and embed this representation within a scheduling optimization formulation, together with a piecewise affine approximation of a nonlinear plant model. The authors solve the problem as a MINLP, and also consider a linear approximation of the objective function to yield a MILP. Burnak et al. (2018) propose an ISC framework that utilizes an mp-MPC formulation based on a linear state-space model identified from a high-fidelity plant model. The mp-MPC is embedded in the high-fidelity model, which is used to generate a linear state-space model that approximates the closed-loop dynamics of the system. This model is used in a mp-MILP formulation of the scheduling problem, which allows for fast computation.

Simkoff and Baldea (2019) and Remigio and Swartz (2020) propose integrated scheduling and control formulations in which embedded MPC subproblems, used to generate the closed-loop plant response along the prediction horizon, are replaced by their equivalent first-order Karush-Kuhn-Tucker (KKT) conditions. This draws on the closed-loop DRTO (CL-DRTO) formulation proposed in Jamaludin and Swartz (2017b), where the performance benefit of a closed-loop over an open-loop predicted response is demonstrated. Replacing the MPC subproblems by their KKT conditions yields a single-level mathematical program with complementarity constraints (MPCC), for which efficient solution techniques exist. In Simkoff and Baldea (2019), the binary decisions introduced with the scheduling constraints are also reformulated using complementary constraints, maintaining an overall MPCC structure. Remigio and Swartz (2020), on the other hand, reformulate the complementarity constraints introduced via the MPC KKT conditions using binary variables. Since a linear dynamic process model and linear objective function are used, the resulting problem is a MILP.

In this paper, we present an ISC formulation for explicit handling of demand uncertainty. We build on the work of Remigio and Swartz (2020), but introduce a number of significant extensions, including the use of a piecewise affine formulation (Kvasnica et al., 2011) for approximating a nonlinear dynamic process model to maintain the ISC problem an MILP. The ISC problem is formulated as a two-stage stochastic programming problem, whose solution (including the production sequence) is communicated to the plant via set-point trajectories provided to the plant MPC. The ISC scheme is applied in a moving horizon manner, incorporating feedback from the plant and storage units. To the authors’ knowledge, there is only one recent study that tackles uncertainty in the ISC formulation. Simkoff and Baldea (2020) propose an open-loop two-stage stochastic formulation for scheduling and control of a chlor-alkali plant where the random variables are the electricity cost and demand. Different from Simkoff and Baldea (2020), the formulation presented in this study accounts for the closed-loop process response, and the ISC problem is solved online. We compare the performance of the deterministic and stochastic formulation for closed-loop operation of a linear and nonlinear process.