Global Optimization via Quadratic Disjunctive Programming for Water Networks Design with Energy Recovery

TL;DR

This paper introduces quadratic and piecewise linear approximations to reformulate complex water network design problems with energy recovery into more solvable models, demonstrating improved efficiency and accuracy.

Contribution

It proposes quadratic and piecewise linear reformulations of GDP models for water networks, enabling more efficient optimization with solvers like Gurobi.

Findings

Quadratic reformulation improves solution efficiency.

Piecewise linear approximation increases accuracy.

Method applied to water treatment and energy recovery processes.

Abstract

Generalized disjunctive programming (GDP) models with bilinear and concave constraints, often seen in water network design, are challenging optimization problems. This work proposes quadratic and piecewise linear approximations for nonlinear terms to reformulate GDP models into quadratic GDP (QGDP) models that suitable solvers may solve more efficiently. We illustrate the benefits of the quadratic reformulation with a water treatment network design problem in which nonconvexities arise from bilinear terms in the mixers' mass balances and concave investment cost functions of treatment units. Given the similarities with water network design problems, we suggest quadratic approximation for the GDP model for the optimal design of a large-scale reverse electrodialysis (RED) process. This power technology can recover energy from salinity differences between by-product streams of the water sector, such as desalination brine mixed with regenerated wastewater effluents. The solver Gurobi excels in handling QGDP problems, but weighing the problem's precision and tractability balance is crucial. The piecewise linear approximation yields more accurate, yet larger QGDP models that may require longer optimization times in large-scale process synthesis problems.