Novel Optimisation Framework for a Class of MINLP Problems in Process Design Using Branch and Bound Methods

Abstract

Process synthesis and intensification are two powerful tools for developing sustainable chemical processes that comply with stringent regulatory standards, reduce capital investment, enhance energy efficiency, and lower carbon footprints to support the transition of the chemical industry towards net-zero targets by 2050. However, most existing studies in this field rely on shortcut models, which often fail to capture the complexities of real-world processes. Ideally, rigorous unit operation models should be incorporated to ensure the applicability of the optimisation results, but doing so gives rise to a class of Mixed-Integer Nonlinear Programming (MINLP) problems that are large-scale, highly nonlinear, and nonconvex, which remain challenging to solve using the existing algorithms.

To address this, we first propose a Feasible Path-Based Branch and Bound (FPBB) algorithm, which integrates the conventional Branch and Bound (B&B) framework with a hybrid feasible path algorithm. At each node of the B&B tree, a relaxed nonlinear programming (NLP) subproblem is solved using a two-layer decomposition. The outer layer is a reduced nonlinear optimisation problem solved using Sequential Quadratic Programming or Sequential Least Squares Programming, while the inner layer is a process simulation problem handled by a hybrid steady-state and time-relaxation-based algorithm incorporating pseudo-transient continuation (PTC) models. When steady-state simulations fail to converge, PTC-assisted dynamic simulations are triggered to drive the simulation process towards the steady state solution. This significantly enlarges the basin of initial points and ensures reliable convergence from arbitrary or poor initial guesses. To further enhance computational efficiency, the FPBB algorithm employs a warm-start strategy, in which solutions from parent nodes are used to initialise NLP subproblems at their child nodes within the B&B framework. The effectiveness and robustness of the FPBB algorithm are demonstrated through five case studies involving the optimal design of dividing wall columns and conventional distillation sequence synthesis, using rigorous models. Energy consumption was reduced by 15.7% to 26.2% compared to values reported in the literature. Optimal solutions can be obtained from randomly generated starting points, eliminating the need for complex initialisation procedures and successfully overcoming convergence challenges.

However, when applying the FPBB algorithm to more complex synthesis problems, failures can occur due to numerical singularities caused by disappearing streams or units. To resolve these issues, a generalised optimisation framework is developed, consisting of: (1) an efficient superstructure representation that can be implemented within commercial process simulators, and (2) a novel mathematical modelling approach that uses conditional logic to activate or deactivate specific sets of equations, effectively handling zero-flow conditions and preventing numerical issues. The FPBB algorithm is employed as the solution algorithm. This framework is validated by three complex distillation system synthesis problems, demonstrating good convergence performance within acceptable computational time, with cost improvements of up to 45% compared to conventional configurations.

To further evaluate the capabilities of the FPBB algorithm, it is extended to complex process intensification cases, such as hybrid reactive–extractive distillation (RED) systems. Novel pseudo-transient continuation models for reactive distillation are developed, and a two-step optimisation procedure is introduced to reduce computational efforts and enhance convergence in systems involving recycle streams. Case studies show that the proposed framework yields more efficient RED designs, achieving reductions in total annualised costs ranging from 3.5% to 25.1% compared to designs reported in the literature.

In conclusion, the novel optimisation frameworks developed in this PhD thesis offer significant advantages in solving large-scale highly nonconvex and nonlinear MINLP problems involving high fidelity models. These contributions enable the reliable and efficient design of intensified and synthesised chemical processes, bridging the gap between model accuracy and computational tractability.