(545b) On the Emergence of Symmetric Problem Structure in the Model Predictive Control of Numbered-up Modular Facilities

Recent advances in chemical engineering industries have been motivated by the desire to promote sustainability and economic feasibility. A promising approach to further enabling sustainable development and a modern circular economy is to implement distributed supply chains with modular manufacturing. Distributed manufacturing can be broadly defined as utilizing large numbers of small-scale geographically scattered facilities, which is beneficial to reducing transportation costs for geographically distributed resource supply or product demand. A pivotal facilitating technology for distributed manufacturing is that of modular production units which are small-scale, standardized units that can perform traditional or intensified chemical unit operations[1,2]. These off-site constructed modules could be transported to the aimed production site, assembled quickly with other modules with different functions to form a working facility, and can also be easily disassembled and relocated as conditions dictate. Desired throughput is achieved by "numbering-up", or installing and connecting multiple copies of the same module type.

While most existing work examines the impact of modularity on process design and supply chain management[3,4], numbering up also impacts the optimal control problem in interesting ways. In this work, we examine the emergence of symmetric structure when controlling identical modular units in parallel configuration. We show that this symmetric structure can result in two possibilities: one, where the same control action is taken on identical units (guaranteed when the problem is convex), and another where degenerate optimal solutions are generated with different control actions being applied to each unit. We also propose an approach to automatically identify symmetry by examining the values of initial conditions, and structure of the dynamic constraints. Through classifying and labeling modular units based on symmetric identification, we propose different symmetry-breaking approaches, one which removes variables corresponding to identical units from the optimization control problem and another which embeds symmetry-breaking constraints to eliminate the possibility of degenerate solutions.

To demonstrate the efficacy of one proposed approach, we present the control performance of a benchmark system with three or more modular nonisothermal CSTR's operating in parallel. Various disturbance rejections and set point tracking problems are simulated. Results suggest that local solvers most often give a result that identical modules take identical control actions. However, in some cases global optimization reveals a degenerate strategy that results in a better controller performance, as measured by a combination of integral square error and control terms. After systematically detecting symmetry and applying the appropriate symmetry breaking approach, the size of our benchmark problem and computational cost is reduced significantly by eliminating variables corresponding to the identical units in the case of no degeneracy, or by eradicating degenerate solutions via symmetry-breaking constraints. Our results also show that size reductions and speed up are even greater for larger numbered-up systems.

[1] M. Baldea, T. F. Edgar, B. L. Stanley, and A. A. Kiss, “Modular manufacturing processes: Status, challenges, and opportunities,” AIChE journal, vol. 63, no. 10, pp. 4262–4272, 2017.
[2] Y. Shao and V. M. Zavala, “Modularity measures: Concepts, computation, and applications to manufacturing systems,” AIChE Journal, vol. 66, no. 6, p. e16965, 2020.
[3] A. Allman and Q. Zhang, “Dynamic location of modular manufacturing facilities with relocation of individual modules,” European Journal of Operational Research, vol. 286, no. 2, pp. 494–507, 2020.
[4] A. Bhosekar and M. Ierapetritou, “A framework for supply chain optimization for modular manufacturing with production feasibility analysis,” Computers & Chemical Engineering, vol. 145, p. 107175, 2021.