A simulation-based optimization framework for integrating scheduling and model predictive control, and its application to air separation units

https://doi.org/10.1016/j.compchemeng.2018.03.009Get rights and content

Highlights

  • A new framework for the integration scheduling and MPC is proposed.

  • The framework consists of a nested decision-making structure with two control loops.

  • The outer loop is formulated as a closed-loop simulation-optimization problem.

  • The inner loop consists of an online MPC.

  • The framework is applied to an Air Separation Unit.

Abstract

The integration of dynamic process models in scheduling calculations has recently received significant attention as a mean to improve operational performance in increasingly dynamic markets. In this work, we propose a novel framework for the integration of scheduling and model predictive control (MPC), which is applicable to industrial size problems involving fast changing market conditions. The framework consists on identifying scheduling-relevant process variables, building low-order dynamic models to capture their evolution, and integrating scheduling and MPC by, (i) solving a simulation-optimization problem to define the optimal schedule and, (ii) tracking the schedule in closed-loop using the MPC controller. The efficacy of the framework is demonstrated via a case study that considers an air separation unit operating under real-time electricity pricing. The study shows that significant cost reductions can be achieved with reasonable computational times.

Introduction

Globalization and extensive information exchange supported by new technologies have given rise to an environment with fast changing market conditions, which must be taken into account in order to achieve optimal process operation. In the process systems engineering community, the search for the optimal operation has been translated to optimizing the decision-making processes across the entire enterprise. The decision-making processes can be analyzed in a hierarchical structure as presented in Fig. 1 (considering a process with product inventory). Traditionally, decisions made in upper levels of this hierarchy are communicated to lower levels, and each decision-making problem is (optimally) solved separately and independently. However, (overall) sub-optimal and infeasible solutions can be avoided by proper integration of different levels of the hierarchy (Baldea and Harjunkoski, 2014). In this work, we focus on strategies for the integration of scheduling and control problems.

The benefits that such integration can bring are intuitive, given that the dynamics of a process governed by a particular control strategy can significantly affect the behavior of scheduling relevant-variables, influencing, e.g., the time required for transitions between production setpoints and the associated operational costs. Initial efforts in the area followed the intuitive route of including the dynamic model of the process as an additional set of constraints in the scheduling problem. The result is a mixed-integer dynamic optimization problem (MIDO), and its solution provides the optimal production sequence and optimal control moves required to implement the schedule. The MIDO problem is typically discretized, resulting in a Mixed Integer Nonlinear Program (MINLP) using, for example, collocation or implicit Runge Kutta methods. This approach was proposed by Flores-Tlacuahuac and Grossmann (2006), and it was extended by Terrazas-Moreno et al. (2007) and Zhuge and Ierapetritou (2012). Alternatively, Nyström et al. (2005) proposed a decoupled modeling approach which consisted in formulating the scheduling problem (master problem) as a Mixed Integer Linear Programming (MILP) and the control problem (primal problem) as Dynamic Optimization. The problem is solved through iterations between the master and primal problems. This approach was later extended to a multiple parallel lines application by Nyström et al. (2006). These approaches, however, face considerable computational challenges associated with the use of high-fidelity representations of the process dynamics and the complexity, nonlinearities and discontinuities that this brings to the scheduling problem. In view of these difficulties, You and coworkers proposed a series of strategies to improve the computational efficiency when solving the integrated scheduling and control problem (Chu and You, 2013a, Chu and You, 2013b).

It is important to notice that, in general, issues related to the stability and safety of the dynamic process (a problem extensively studied in the control literature) have not been accounted for in integrated frameworks. This can be verified by analyzing the integrated models proposed in the literature; the majority of the respective works only consider the dynamics of the process for the transition periods. In general, a constraint establishing that the state and control variables should achieve their steady state values at the end of the transition period is imposed, and an implicit assumption that the system remains at the steady state values from that point forward is made. Such manipulation reduces the complexity of the integrated model and allows the schedule to be optimized without previous knowledge of the transition times, which should otherwise be obtained in an iterative procedure. However, it is clear from a control perspective that achieving such steady state values does not guarantee that the system remains stable, especially when considering open-loop unstable steady states. Finally, most of the existing integrated frameworks assume there is no model mismatch between the dynamic model and the process, and the control actions are usually computed offline. Such assumption is typically violated in practice, and can cause instability and safety constraint violations when implementing the integrated scheduling and control solutions.

Motivated by these issues, Zhuge and Ierapetritou (2014) proposed to integrate scheduling and model predictive control (MPC) by including explicit control laws in the scheduling problem, where control laws were derived using multi-parametric programming techniques. MPC would then address the control objective of guaranteeing stability, robustness, safety and fast tracking, while the integrated scheduling model accounted for economic objectives. Furthermore, Zhuge and Ierapetritou (2015) proposed an integrated framework that consisted of the use of two control loops for the online integration of scheduling and control. An integrated problem at the outer loop generated the production schedule and the state references for the inner loop. The inner loop tracked state references using fast model predictive control, and the exact control solution was computed online. Recently, Du et al. (2015) noticed that the use of the detailed dynamic models of the process could introduce unnecessary complexities to the scheduling problem. Rather, they chose to focus on the closed input-output process dynamics and derived low-order models of the system behavior to be included in the scheduling calculations. Such low-dimensional input-output models were referred to as time-scale bridging models (SBMs). This work was further extended by Baldea et al. (2015), where SBMs were incorporated as dynamic constraints in the expression of a model predictive control in order to shape its closed-loop behavior, while at the same time obtaining an explicit description of the closed-loop process dynamics.

While these contributions bring significant theoretical advances for this area, they do not describe large-scale, industry-relevant applications. To prove the value of the integration of scheduling and control, closer relationships between industry and academia are essential, and efforts must be done to address real world problems. Motivated by this, Pattison et al. (2016) described the application of an integrated scheduling/control framework to the operation of an air separation unit (ASU) operating under real-time electricity pricing. Air separation units have a very high energy consumption, and the industrial gas sector utilized 19.97TWh of electricity in 2014, or about 2.55% of the amount consumed by the entire manufacturing sector in the U.S. (Pattison et al. 2016). Therefore, this industry may take advantage of the variations in electricity prices, which can change hourly (or faster) due to the change of grid-level electricity demand. Costs may be reduced by adjusting production rate according to electricity prices. The authors note that such frequent changes, however, may result in the process operating in a transient regime, possibly without ever reaching steady state conditions. This fact is contrary to the premise of most current scheduling methodologies and emphasizes the necessity to account for the dynamic behavior of the system while making decisions at the scheduling level. Therefore, the authors proposed an integrated framework where low-order dynamic models for scheduling relevant variables were derived, heuristic controls were implemented during transitions, and a schedule, which was feasible from a dynamic point of view, was obtained. This work was later extended by Pattison et al. (2017), where the authors implemented the integrated scheduling and dynamic optimization framework in a moving horizon fashion, “closing the loop” from a scheduling perspective. The authors also proposed a specially-designed observer for reconstructing the states of the dynamic models used in the scheduling calculations.

In this work, we extend the work of Pattison et al. (2016) and the proposed ASU application to consider the use of model predictive control in the operation of the plant, as described by Zhuge and Ierapetritou (2015). Specifically, we propose a nested decision-making structure, comprising two scheduling/control loops. In the outer loop, the scheduling problem is formulated as a simulation optimization problem, where the simulation involves solving a model predictive control problem for the entire scheduling horizon. The problem is solved to determine the optimal sequence of production rate setpoints. The solution is then communicated to the inner loop, which tracks the setpoints over time while ensuring feasibility, stability and fast tracking. The inner loop is solved online and handles any disturbance that may affect the control layer.

The concept of a hybrid simulation-optimization has been employed by several works addressing decision making-processes (Sahay and Ierapetritou, 2013, Wang and Zheng, 2001, Xia and Wu, 2005). However, to the best out knowledge, such concept has not been applied in the integration of scheduling and control. Hybrid simulation-optimization frameworks are usually employed when detailed mathematical models describing the system cannot be derived or cannot capture the complexities of the system under investigation. Therefore, such frameworks are suitable for the representation of complex dynamic systems operating under a model predictive control, for which explicit control laws and closed-form expressions for the closed-loop behavior may be difficult or impossible to derive. In particular, we would like to point out that, while previous contributions brought remarkable advances for the area of integrated scheduling and control, a hybrid-simulation optimization framework may be able to address some shortfalls. For example, the framework proposed by Zhuge and Ierapetritou (2014) employs multi-parametric techniques to derive explicit control laws for an MPC, which may be challenging to employ when handling high-dimensional problems. The framework also uses big-M formulations in the scheduling problem to select the critical region associated to the states of each scheduling slot, which adds complexity to the scheduling problem and may make it computationally intractable. Extending this framework to systems with a large number of input and output is difficult due to potential limitations related to the complexity of the multi-parametric solution. In a broader sense, capturing the closed-loop behavior of processes operating under MPC may also be hindered by phenomena such as saturation of controlled and manipulated variables, which effectively alter the structure of the overall closed-loop model and would likely require a hybrid system modeling approach.

This paper is organized as follows: In Section 2, the problem statement is presented, followed by a description of the ASU process. In Section 3 we describe the problem formulation, discussing the process, control strategies, and scheduling model, followed by the integrated framework description. We demonstrate the application of our framework to the ASU process in Section 4 and provide concluding remarks and future directions in Section 5.

Section snippets

Problem definition

At the supply chain level, decisions related to the assignment of clients to production facilities, demand forecasts, logistics and distribution are taken for an entire enterprise. Such decisions are transmitted to each production facility, which solves a planning problem to define production targets over long time periods (generally months). The scheduling layer then determines how to satisfy the planning targets by establishing the optimal sequence of setpoints for production and inventory

Process system representation

As noted above, the first-principles model of a chemical process is typically represented as a high-dimensional system of nonlinear differential equations (DAEs). Nevertheless, in practice, the number of process variables that are of interest to scheduling is likely much lower than the number of process states (Baldea et al., 2015). Pattison et al. (2016) postulated that, given a set of operating and quality constraints of a system, only those constraint that become active during static or

Case study

To verify the potential of the proposed framework, we present a realistic operational scenario and compare the results of the optimal schedule to the nominal operation of the plant. We assume that the storage tank has a capacity of 200 kmol of product, and the initial inventory level is 50 kmol. The energy prices for a period of 48 h are shown in Fig. 5, and the scheduling horizon starts at 00:00 h (midnight) of day 1. Knowing the energy prices profile, we expect that inventory will deplete

Conclusions

In this work, we propose a novel framework for the integration of scheduling and model predictive control. The framework includes the initial identification of a set of schedule-relevant variables, W. This step is followed by the identification of state space models which describes the dynamics of each variable belonging to W. We then proposed a model predictive control approach to track the corresponding varying set points over time, while accounting for quality and process operating

Acknowledgments

M.G.I. acknowledges financial support from NSF under grant CBET 1159244. L.D. gratefully acknowledges financial support from CNPQConselho Nacional de Desenvolvimento Científico e Tecnológico – Brazil. R.C.P. and M.B. gratefully acknowledge financial support from the National Science Foundation (NSF) through the CAREER Award 1454433 and Award CBET-1512379.

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