(373j) Two-Stage Nonlinear Robust Optimization of Effluent Treatment Network Under Uncertainty
Robust and stochastic optimization techniques have been explored in the literature for water network optimization under uncertainty. Robust optimization solves for the worst case of uncertainty realized and presents a conservative solution to the problem that would be valid for any realization of uncertainty it was solved for. In contrast, stochastic optimization deals with uncertainty in an optimization problem by assuming that the probability distribution of the uncertainty is known and seeks to address the uncertainty through methods such as scenario-based stochastic programming, where the problem is solved for a set of discrete scenarios that represent uncertainty. The optimal synthesis of the wastewater treatment network under single contaminant load uncertainty was explored by Koppol and Bagajewicz  through a scenario-based programming approach. Karuppiah and Grossmann  addressed the optimal synthesis of a water treatment network under contaminant removal uncertainty, using Lagrangian decomposition on a linear formulation of the model using McCormick envelopes. Khor, et al.  solved the network for fixed flowrate under uncertainty using a two-stage procedure using discrete scenarios, also accounting for risk management.
In this work, we present a two-stage nonlinear robust optimization method for the optimal design and operation of the water treatment network under source flow uncertainty. In this method, the variables in the optimization model are classified into design, state and control variables. The problem utilizes a two-stage approach: the design variables are the first stage decision variables, and the state and control variables, which are dependent on uncertainty, are computed as second stage variables. This method applies decision rule approximation for the control variables as affine functions of uncertainty. In order to derive the adjustable robust counterpart to the model, the model is linearized around a specific realization of uncertainty. The final model obtained is a mixed integer nonlinear optimization problem due to the presence of bilinear terms in the component mass balance equations. The solution to this problem is a decision rule or policy for each control variable (typically, the outflow from a splitter unit). Using this decision policy, the value of the control variable may be computed instantaneously for any realization of uncertainty in the set, and consequently, the values for all state variables in the network may be calculated.
The proposed method has been applied in two case studies: a simple water treatment network illustrated in the work by Galan and Grossmann , and a modified version of the effluent treatment and steam generation network for a SAGD reservoir developed by Forshomi et al. . In both cases, the objective is to minimize the total cost of treatment, while meeting specified target concentrations, and/or demand requirements. Applying the proposed method to the case studies, we obtained water network design and operation policies for different ranges of uncertainty and thereby determining the optimal network, for each instantaneous realization of uncertainty.
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