Two-Stage Robust Optimization of Water Treatment Network Design and Operations under Uncertainty

Abstract

This paper presents a study of two-stage adaptive nonlinear robust optimization in dealing with uncertainty in the optimal design and operation of water treatment systems. This work aims to obtain (i) a robust process design and (ii) robust operational policies for the water treatment network, which are easily applicable for any realization of uncertainty that the problem has been modeled to handle. The approach uses a two-stage nonlinear robust optimization technique, which is based on the linearization around the nominal value of the uncertainty. The intractable adaptive robust optimization model is transformed into its tractable robust counterpart by applying the linear decision rule over the control variables. To handle the issue of concentration target violation caused by an inaccurate linear approximation model, a modified optimization model with additional constraints on the extreme points of the uncertainty set is further presented. The performance and advantages of the approach are then analyzed and contrasted on two case studies, a small water treatment model and a larger SAGD water treatment network.

A generalized cutting‐set approach for nonlinear robust optimization in process systems engineering

“…To simplify the two‐stage robust counterpart, we employ DR, which have been ubiquitously used in the area of ARO to convert second‐stage variables into a function of the uncertain parameters q and new first‐stage decision variables d (the parameterization of the DR themselves). ARO with affine DR, that is, an affine relationship between uncertain parameters and second‐stage variables, was first proposed by Ben‐Tal et al 11 In process systems engineering applications, affine DR have also been used to solve two‐stage RO problems in the contexts of water treatment networks 28 and steel‐making processes 33 . ARO with generalized affine DR for mixed‐integer linear optimization models has also been recently demonstrated by Avraamidou and Pistikopoulos 34 via a multi‐parametric programming approach.…”

“…To address this, Bertsimas et al 24 proposed a local search algorithm for identifying robust feasible solutions to uncertain optimization problems with non‐convex inequality constraints. Additionally, there have been recent advances in the development of novel methods and applications of RO methods to nonlinear process systems engineering models, including general nonlinear programming robust counterpart formulations, 25 robust counterparts with local linearization of nonlinear uncertain constraints and a novel sampling algorithm, 26 application to the pooling problem utilizing a cutting‐plane solution algorithm, 27 application to water treatment network operation, 28 robust counterpart derivation for the synthesis of fuel refineries under cost uncertainty, 29 and design and operation of process systems with resilience to disruptive events, 30 to name but a few.…”

Economic and Flexible Design under Uncertainty for Steam Power Systems Based on Interval Two-Stage Stochastic Programming

“…ΔH boi,n denotes steam enthalpy drop between inlet and outlet steam enthalpy of boiler n (kJ/ kg). (8) The pollutant emission constraints…”

“…The existing methods applied to deal with uncertainty in SPS mainly include robust optimization (RO) − and two-stage stochastic programming (TSP). − RO is considered as a successful approach to deal with uncertainty in the input data: for a given set U containing all relevant scenarios, i.e., all sufficiently likely realizations of the uncertain parameters, a solution is sought that is feasible for every scenario in U and that is worst-case optimal under this constraint. , In general, the RO model can usually be expressed as where x denotes the decision variable vector, ξ denotes the uncertainty parameter, U denotes the bounded closed convex set, f ( x ,ξ) denotes the objective, and g j ( x ,ξ) denotes the constraint functions.…”

Application of an Interval Two-Stage Robust (ITSR) Optimization Model for Optimization of Water Resource Distribution in the Yinma River Basin, Jilin Province, China

“…Tan [23] developed a distribution network dispatching method based on interval-two-stage-robust stochastic planning, which reduced the risk of distribution system network loss. In view of the uncertainty in the optimal design and operation of the water treatment system, Kammammettu [24] studied the two-stage adaptive nonlinear robust optimization problem and developed a robust operation strategy for the water treatment network by using the robust method in order to avoid the risk of excessive pollutant emissions in the water treatment process.…”