Two-layered dynamic control for simultaneous set-point tracking and improved economic performance

Highlights

• Dynamic multi-objective control solved using two-layer hierarchical architecture.

• Upper layer handles priority-based multi-objectives using lexicographic approach.

• Pareto-based trade-off of tracking cost proposed to improve economic performance.

• Optimal trajectory implemented by lower layer MPC.

• Two case studies demonstrate flexibility and performance of proposed approach.

Abstract

This work introduces a multi-objective optimization strategy to handle conflicting set-point tracking and economic objectives in a two-layer hierarchical control framework. A dynamic multi-objective real-time optimizer (DMO), incorporated in the upper layer, handles multiple control objectives with set-point tracking being the higher priority objective and computes optimal plant trajectories. This plant-wide trajectory information is communicated to the lower-layer model predictive control (MPC) operating at a faster sampling rate. The conventional weight-based and lexicographical method for the DMO are discussed. A new algorithm is conceptualized based on the lexicographical method to handle prioritized objectives. The proposed algorithm modifies the higher priority tracking objective and establishes improved economic performance compared to the conventional techniques, with minimal effect on the conflicting tracking objective, through a systematic choice of the preferred Pareto solution. The proposed algorithm’s efficacy, within the hierarchical framework, is analyzed using two case studies: A polymerization reactor and a multi-unit reactor–separator system.

Introduction

Staying competitive and profitable in the current market-driven scenario requires a control framework for optimal management of multiple conflicting objectives. Conventional control structures employ a single plant-wide objective based on optimizing economic targets for maximizing profit. However, performance targets such as product quality, production goals, and environmental norms cannot be ignored in the pursuit of profitable plant operation [1]. In fact, some plant-wide objectives such as product quality set-points may be higher priority objective(s), since the off-spec product is not sellable [2]. Hence, there is a need to develop multi-objective control strategies handling objectives, such as set-point tracking, in addition to the traditional economic targets of maximizing profits. However, conflicting behavior of objectives and associated computational demands of a multi-objective control problem are issues that need to be addressed while designing the controller.

Traditionally, a plant-wide economic objective is solved at a steady-state RTO (real-time optimization) layer based on the process-related decisions received from the planning and scheduling layer. The RTO uses a nonlinear model to compute steady-state set-points for the underlying MPC (model predictive control) layer [3]. Although steady-state RTO is prevalent in the process industry, it has several drawbacks: Long-term transients limits how often optimization can be performed [4], it is unable to handle higher frequency disturbances or dynamic cost parameters [3], and model inconsistency with the MPC layer can potentially lead to infeasible solutions [5]. One type of approach that addresses these issues are single-layered approaches, where the RTO layer is integrated with the lower-layer MPC [6] or the economic cost function itself is solved in the MPC layer [7]. Idris and Engell [8] demonstrated the computational issues of implementing a single-layer economic-MPC on a pilot-scale plant. Findeisen and Allgöwer [9] discussed loss in performance and instability due to delay in computing the optimal solution. Although significant developments in computational power make integrated single-layer control attractive [10], concomitant developments in process systems and the increasing importance of process intensification in plants comprising of multiple units present significant challenges to the single-layer approach. Consequently, computation of the optimal trajectory within the stipulated control interval is rendered computationally intractable [5]. Since the focus of the current work is on multiple conflicting control objectives, it is imperative to design a control strategy to handle the computational demands of optimization-based control algorithms. Hence, inspired from earlier works on dynamic hierarchical control [4], [5], [11], we employ a two-layer control framework, with the multi-objective optimization (MOO) solved in the upper dynamic RTO layer, operating at a slower time-scale to handle the computational issues associated with the MOO problem.

Several applications of MO control have been discussed in the literature. For example, polymerization processes require managing multiple process targets, such as maximizing conversion, minimizing undesired products, and achieving the desired quality parameters of the final polymer [12], [13], [14]. Often, the multiple objectives are prioritized, with safety or product quality targets being at a higher priority than economic objective. Anilkumar et al. [15] used a lexicographic approach to handle multiple set-point tracking objectives based on priority. He et al. [16] used lexicographic optimization to handle a similar case of prioritized set-point tracking and economic objectives. A few other applications of multi-objective optimization include fed-batch bioreactor [17], solar refrigeration plant [18], path tracking control with predefined speed profile [19], and autonomous vehicle control to reach the destination with minimum energy and minimum time [20]. Haßkerl et al. [21] formulated a weight-based control scheme for a reactive distillation system comprising of an economic cost function and a regularization term that penalizes violation on product purity. The respective weight parameters were tuned meticulously to obtain the desired trade-off between the two objectives. Tian et al. [22] and Li et al. [23] implemented a single-layer multi-objective control, where the desired trade-off between the economic and the tracking objective is manipulated through user-defined tuning parameter. Thus, different aspects of multi-objective control comprising of tracking and economic objective were studied using single-layered control frameworks, which requires the optimal solution be available within a sampling instance. However, this becomes challenging as the complexity of the system increases, particularly for multi-objective control. Secondly, most of the earlier works either involved offline optimization or required user intervention to select the required optimal solution. The current work addresses these issues through a multi-layered control framework equipped to handle the computational rigor of multi-objective problems. Different optimization methodologies are discussed, which can be applied depending on the nature of the control problem.

The scope of this work includes a two-layer hierarchical control framework for managing priority-based multiple objectives, aimed at tracking a process parameter to its desired set-point and simultaneously maximizing the profit. An upper-layer dynamic multi-objective optimizer (DMO) computes the optimal plant trajectory for a lower-layer MPC. MPC is preferred in the design of the two-layer framework as it can deal with multiple variables and process constraints [3]. We evaluate various multi-objective optimization techniques by quantifying the performance requirement of the control objectives. A popular technique uses weight to scalarize the multiple objectives into a single objective function. However, tuning these weighing parameters is a demanding task because the objectives belong to different domains can often lead to ill-conditioned or highly nonlinear tuning behavior[15], [24]. In contrast, the lexicographic approach can handle priority-based optimization [17], [25]. However, the optimal tracking solution may pose a too stringent constraint to the subsequent economic optimization. Hence, a second strategy is to minimize the tracking error only at the terminal point of the horizon [15]. In principle, this is analogous to communicating the desired set-point, allowing for a compromise in the performance during the transients. To exploit the advantages of the two techniques, we propose a new algorithm, which computes multiple Pareto solutions through modifications in the tracking function. The preferred solution is chosen online from the Pareto-optimal set. This solution is determined based on the solution closest to the ideal solution of the objective. It is easy to understand that it is impossible to achieve the ideal solution of the multiple objectives simultaneously due to their conflicting nature, and the preferred Pareto solution offers the best trade-off. The optimal trajectories are computed from the preferred solution and are communicated to a lower-layer MPC.

The performance of various approaches is evaluated using two case studies. The first investigation is a free radical polymerization reactor [26], which involves tracking of a quality parameter and maximizing the net profit as prioritized and conflicting objectives. The second case study is a “plant-wide” reactor–separator system, with a similar set of objectives.