Comparison of dual based optimization methods for distributed trajectory optimization of coupled semi-batch processes

Abstract

The physical and virtual connectivity of systems via flows of energy, material, information, etc., steadily increases. This paper deals with systems of subsystems that are connected by networks of shared resources that have to be balanced. For the optimal operation of the overall system, the couplings between the subsystems must be taken into account, and the overall optimum will usually deviate from the local optima of the subsystems. However, for reasons, such as problem size, confidentiality, resilience to breakdowns, or generally when dealing with autonomous systems, monolithic optimization is often infeasible. In this contribution, iterative distributed optimization methods based on dual decomposition where the values of the objective functions of the different subsystems do not have to be shared are investigated. We consider connected dynamic systems that share resources. This situation arises for continuous processes in transient conditions between different steady states and in inherently discontinuous processes, such as batch production processes. This problem is challenging since small changes during the iterations towards the satisfaction of the overarching constraints can lead to significant changes in the arc structures of the optimal solutions for the subsystems. Moreover, meeting endpoint constraints at free final times complicates the problem. We propose a solution strategy for coupled semi-batch processes and compare different numerical approaches, the sub-gradient method, ADMM, and ALADIN, and show that convexification of the subsystems around feasible points increases the speed of convergence while using second-order information does not necessarily do so. Since sharing of resources has an influence on whether trajectory dependent terminal constraints can be satisfied, we propose a heuristic for the computation of free final times of the subsystems that allows the dynamic sub-processes to meet the constraints. For the example of several semi-batch reactors which are coupled via a bound on the total feed flow rate, we demonstrate that the distributed methods converge to (local) optima and The project leading to this publication has received funding from the European Union's Horizon 2020 research and innovation programme under Grant Agreement No. 723575 (CoPro, spire2030.eu/ copro) in the framework of the SPIRE PPP. Extended author information available on the last page of the article highlight the strengths and the weaknesses of the different distributed optimization methods.

A conditionally constrained compound sub‐gradient method for distributed energy coordination

“…In recent years, with the development of new technologies such as the internet of things and artificial intelligence, distributed collaborative optimization has been used in new power grids [7][8][9][10][11]. This field is developing rapidly, with huge strides being made in distributed optimization algorithms [12][13][14][15][16][17][18].…”

Bilevel optimization for joint scheduling of production and energy systems

“…The algorithm in Wenzel et al (2020) is based on extensive earlier work of the authors, e.g., in Wenzel et al (2016) an algorithm is proposed to adjust the prices for shared resources. Maxeiner and Engell (2020) compare different dual-based methods for the distributed optimization and apply the methods to a case study with coupled semi-batch reactors. The reactors are coupled by a maximum for the combined feed flow rate.…”